token_count
int32 48
481k
| text
stringlengths 154
1.01M
| score
float64 3.66
5.34
|
|---|---|---|
610 |
|Zitzewitz, Paul & Neff, Robert. Physics. New York: Glencoe/McGraw-Hill, 1995: 159.||"Pluto's average radius is 1.15 × 106 M."||2300 km|
|Microsoft Encarta 97 Encyclopedia. Microsoft, 1996.||"Pluto is about 2320 km in diameter"||2320 km|
|Yenn, Bill. Solar System. New York: Crescent, 1991: 8.||"The diameter of Pluto is 1375 mi (2200 km)"||2200 km|
|Smoluchowski, Roman. The Solar System. New York: Scientific American Books, 1983: 120.||"It's size is very difficult to establish, at present is 4000 km."||4000 km|
|Encyclopedia of Knowledge. vol. 15. USA: Grolier, 1992: 73.||"Pluto may have a diameter of about 2284 km (1416 mi)"||2284 km|
|Hamilton, Calvin J. Pluto and Charon. Views of the Solar System. Los Alamos National Laboratory, 1997.||Equatorial radius (km) 1,160||2320 km|
Discovered by Clyde W. Tombaugh on February 18, 1930 Pluto is the ninth planet from the Sun, it is the smallest, most remote planet known in the Solar System. Given how far away it is, for many years very little was known about the planet. Until in 1978, when astronomers discovered a relatively large moon orbiting Pluto and named it Charon. The orbits of Pluto and Charon caused them to pass repeatedly in front of one another. This enabled astronomers to determine Pluto's size fairly accurately. However, it wasn't until 1994, when with the use of the Hubble Space Telescope, astronomers were able to determine it's size even more precisely.
All of the results gotten, were pretty close except for the result received from a book published in 1983. This could be easily explained, by the fact that as stated above, Pluto's size was not accurately determined until 1985. Therefore, this source should not be trusted.
The measurements for the diameter of Pluto given in the remaining four sources were very close. In fact, if all of them would be rounded to the nearest hundreds place, three out of four of them would agree that the diameter of Pluto is 2300 km. The difference in the results could be explained by Pluto's methane atmosphere, which may lie many kilometers deep, making the diameter figure uncertain.
Although, if wanted a more precise answer, it would be better to go with the result from the Encarta 97 Encyclopedia, which is that Pluto is 2320 km across. This is because it is the most recent source and therefore, the most accurate. The reason is due to the fact, that every year new and better techniques are discovered for determining the sizes of planets. So, the newer the source, the more precise it is.
Anna Chernovetskaya -- 1997<|endoftext|>
| 3.671875 |
4,394 |
<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
# Permutation Problems
## Using the nPr function found in the Math menu under PRB on the TI calculator
Estimated6 minsto complete
%
Progress
Practice Permutation Problems
MEMORY METER
This indicates how strong in your memory this concept is
Progress
Estimated6 minsto complete
%
Evaluate Permutations Using Permutation Notation
License: CC BY-NC 3.0
Donna works at the library. She is arranging the new science fiction books that have come in. She only has room for six of the 20 new books on her shelf. How many different arrangements can be made when six books are chosen from a group of 20 books to be placed on a shelf?
In this concept, you will learn to evaluate permutations by using permutation notation.
### Permutations
The Counting Principle states that if event can be chosen in p\begin{align*}p\end{align*} different ways and an independent event is chosen in q\begin{align*}q\end{align*} different ways, the probability of the two events occurring in p×q\begin{align*}p \times q\end{align*}. Basically, this tells you how many ways you can arrange items. A permutation is a selection of items in which order is important. To use permutations to solve problems, you need to be able to identify the problems in which order, or the arrangement of items, matters.
Let’s look at an example.
How many ways can you arrange the letters in the word MATH?
First, look at the problem.
You have 4 letters in the word and you are going to choose one letter at a time. When you choose the first letter, you have 4 possibilities (M, A, T, or H), your second choice will be 3 possibilities, third choice will be 2 possibilities (because there are only 2 letters left to choose from), and last choice only one possibility.
Next, calculate the number of choices.
P(choice 1, choice 2, choice 3, choice 4)=4×3×2×1P(choice 1, choice 2, choice 3, choice 4)=24\begin{align*}& P(\text{choice 1, choice 2, choice 3, choice 4)} = 4 \times 3 \times 2 \times 1 \\ & P(\text{choice 1, choice 2, choice 3, choice 4)} = 24\end{align*}
The answer is 24.
There are 24 different ways to arrange the letters in the word MATH.
The most efficient way to calculate permutations uses numbers called factorials. Factorials are special numbers that represent the product of a series of descending numbers.
The symbol for a factorial is an exclamation sign.
Take a look at how factorials are used.
8 factorial=8!=8×7×6×5×4×3×2×1\begin{align*}8 \ \text{factorial} = 8!= 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\end{align*}
11 factorial=11!=11×10×9×8×7×6×5×4×3×2×1\begin{align*}11 \ \text{factorial}= 11!= 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1\end{align*}
4 factorial=4!=4×3×2×1\begin{align*}4 \ \text{factorial} = 4! = 4 \times 3 \times 2 \times 1\end{align*}
17 factorial=17!=17×16×15×14×13×12×11×10×9×8×7×6×5×4×3×2×1\begin{align*}17 \ \text{factorial} = 17! = 17 \times 16 \times 15 \times 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \end{align*}
To compute the values of factorials, simply multiply the series of numbers.
4 factorial=4!6 factorial=6!8 factorial=8!======4×3×2×1246×5×4×3×2×17208×7×6×5×4×3×2×140,320\begin{align*}\begin{array}{rcl} 4 \ \text{factorial} = 4! &=& 4 \times 3 \times 2 \times 1 \\ &=& 24\\ 6 \ \text{factorial} = 6! &=& 6 \times 5 \times 4 \times 3 \times 2 \times 1 \\ & =& 720\\ 8 \ \text{factorial} = 8! &=& 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \\ &=& 40,320 \end{array}\end{align*}
You can also use a graphing calculator to find factorials. For large numbers, especially, the calculator can save you time.
If you push the m\begin{align*}m\end{align*} button and on the top of the screen you will see PROB. #4 under the PROB menu is factorial.
License: CC BY-NC 3.0
Let’s try an example.
Find 10! using your calculator.
License: CC BY-NC 3.0
The answer is 3628800. Notice the key press history to tell you what was pressed to get 10!
You can use factorials to calculate permutations. Look at the formula for finding permutations.
In general, permutations are written as:
nPr=n items taken r at a time\begin{align*}{\color{red}_n}P{\color{blue}_r} = {\color{red}n} \ \text{items taken} \ {\color{blue}r} \ \text{at a time}\end{align*}
To compute nPr\begin{align*}_nP_r\end{align*} you write:
nPr=n!(nr)!\begin{align*}_nP _r = \frac{n !}{(n - r)!}\end{align*}
Let’s look at an example.
Suppose you have 6 items and you want to know how many arrangements you can make with 4 of the items.
First, order matters in this problem, so you need to find the number of permutations there are in 6 items taken 4 at a time.
First, in permutation notation write the following.
6P4=\begin{align*}{\color{red}_6}P{\color{blue}_4} = \end{align*} 6 items taken 4 at a time
Next, calculate 6P4\begin{align*}_6P_4\end{align*}.
nPr6P46P46P46P46P4======n!(nr)!6!(64)!6!2!6×5×4×3×2×12×16×5×4×3360\begin{align*}\begin{array}{rcl} _nP _r &=& \frac{n !}{(n - r)!} \\ \\ _6P _4 &=& \frac{6 !}{(6 - 4)!} \\ \\ _6P _4 &=& \frac{6 !}{2 !} \\ \\ _6P _4 &=& \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1} \\ \\ _6P _4 &=& 6 \times 5 \times 4 \times 3 \\ \\ _6P _4 &=& 360 \end{array}\end{align*}
Notice that it is the product of the values in descending order that tells you how many permutations are possible. You can use this method to solve any number of permutations.
You can also use the graphing calculator to find nPr\begin{align*}_nP_r\end{align*}. If you push the m button and on the top of the screen you will see PROB. #2 under the PROB menu is nPr\begin{align*}_nP_r\end{align*}.
License: CC BY-NC 3.0
To find 8P3\begin{align*}_8P_3\end{align*} using the calculator, you would get the following. Notice the key press history to tell you what to press to do 8P3\begin{align*}_8P_3\end{align*}.
License: CC BY-NC 3.0
### Examples
#### Example 1
Earlier, you were given a problem about Donna the librarian.
Donna is trying to choose six books of her 20 new books to put on the shelf. Remember that order is important and repetition is not allowed, so permutations must be used.
Donna has 20 books to choose from, therefore n=20\begin{align*}n = 20\end{align*}. She will choose 6 books, therefore r=6\begin{align*}r = 6\end{align*}.
First, fill in the formula for nPr\begin{align*} _nP_r\end{align*} where n=20\begin{align*}n = 20\end{align*} and r=6\begin{align*}r = 6\end{align*}.
nPr20P6==n!(nr)!20!(206)!\begin{align*}\begin{array}{rcl} _nP _r &=& \frac{n !}{(n - r)!} \\ \\ _{20}P _6 &=& \frac{20 !}{(20 - 6)!} \end{array}\end{align*}
Next, compute 20P6\begin{align*}_{20}P_6\end{align*}.
20P620P620P620P620P6=====20!(206)!20!14!20×19×18×17×16×15×14×13×12×11×10×9×8×7×6×5×4×3×2×114×13×12×11×10×9×8×7×6×5×4×3×2×120×19×18×17×16×1527907200\begin{align*}\begin{array}{rcl} _{20}P _6 & = & \frac{20 !}{(20 - 6)!} \\ \\ _{20}P _6 & = & \frac{20 !}{14 !} \\ \\ _{20}P _6 & = & \frac{20 \times 19 \times 18 \times 17 \times 16 \times 15 \times 14 \times 13\times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{14 \times 13\times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \\ \\ _{20}P _6 & = & 20 \times 19 \times 18 \times 17 \times 16 \times 15 \\ \\ _{20}P _6 & = & 27907200 \end{array}\end{align*}
The answer is 27907200.
There are 27,907,200 different ways Donna can choose six books of the 20 to display on the shelf.
#### Example 2
Find \begin{align*}_7P_3\end{align*}.
First, fill in the formula for \begin{align*}_nP_r\end{align*} where \begin{align*}n = 7\end{align*} and \begin{align*}r = 3\end{align*}
\begin{align*}\begin{array}{rcl} _nP _r &=& \frac{n !}{(n - r)!} \\ \\ _7P _3 &=& \frac{7 !}{(7 - 3)!} \end{array}\end{align*}
Next, compute \begin{align*}_7P_3\end{align*}.
\begin{align*}\begin{array}{rcl} _7P _3 &=& \frac{7 !}{(7 - 3)!} \\ \\ _7P _3 &=& \frac{7 !}{4 !} \\ \\ _7P _3 &=& \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1} \\ \\ _7P _3 &=& 7 \times 6 \times 5 \\ \\ _7P _3 &=& 210 \end{array}\end{align*}
The answer is 210.
#### Example 3
Find \begin{align*}_4P_3\end{align*}.
First, fill in the formula for \begin{align*}_nP_r\end{align*} where \begin{align*}n = 4\end{align*} and \begin{align*}r = 3\end{align*}
\begin{align*}\begin{array}{rcl} _nP _r &=& \frac{n !}{(n - r)!} \\ \\ _4P _3 &=& \frac{4 !}{(4 - 3)!} \end{array}\end{align*}
Next, compute \begin{align*}_4P_3\end{align*}.
\begin{align*}\begin{array}{rcl} _4P _3 &=& \frac{4 !}{(4 - 3)!} \\ \\ _4P _3 &=& \frac{4 !}{1 !} \\ \\ _4P _3 &=& \frac{4 \times 3 \times 2 \times 1}{1} \\ \\ _4P _3 &=& 4 \times 3 \times 2 \times 1 \\ \\ _4P _3 &=& 24 \end{array}\end{align*}
The answer is 24.
#### Example 4
Find \begin{align*}_{12}P_2\end{align*}.
First, fill in the formula for \begin{align*}_nP_r\end{align*} where \begin{align*}n = 12\end{align*} and \begin{align*}r = 2\end{align*}
\begin{align*}\begin{array}{rcl} _nP _r &=& \frac{n !}{(n - r)!} \\ \\ _{12}P _2 &=& \frac{12 !}{(12 - 2)!} \end{array}\end{align*}
Next, compute \begin{align*} _{12}P_2\end{align*}.
\begin{align*}\begin{array}{rcl} _{12}P _2 &=& \frac{12 !}{(12 - 2)!} \\ \\ _{12}P _2 &=& \frac{12 !}{10 !} \\ \\ _{12}P _2 &=& \frac{12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} \\ \\ _{12}P _2 &=& 12 \times 11 \\ \\ _{12}P _2 &=& 132 \end{array}\end{align*}
The answer is 132.
#### Example 5
Find \begin{align*}_8P_6\end{align*}.
First, fill in the formula for \begin{align*}_nP_r\end{align*} where \begin{align*}n = 8\end{align*} and \begin{align*}r = 6\end{align*}
\begin{align*}\begin{array}{rcl} _nP _r &=& \frac{n !}{(n - r)!} \\ \\ _8P _6 &=& \frac{8 !}{(8 - 6)!} \end{array}\end{align*}
Next, compute \begin{align*}_8P_6\end{align*}.
\begin{align*}\begin{array}{rcl} _8P _6 &=& \frac{8 !}{(8 - 6)!} \\ \\ _8P _6 &=& \frac{8 !}{2 !} \\ \\ _8P _6 &=& \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{ 2 \times 1} \\ \\ _8P _6 &=& 8 \times 7 \times 6 \times 5 \times 4 \times 3 \\ \\ _8P _6 &=& 20160 \end{array}\end{align*}
The answer is 20,160.
### Review
Find each permutation.
1. Find \begin{align*}_7P_2\end{align*}
2. Find \begin{align*}_6P_3\end{align*}
3. Find \begin{align*}_5P_4\end{align*}
4. Find \begin{align*}_5P_5\end{align*}
5. Find \begin{align*}_9P_3\end{align*}
6. Find \begin{align*}_9P_7\end{align*}
7. Find \begin{align*}_{11}P_3\end{align*}
8. Find \begin{align*}_{12}P_3\end{align*}
9. Find \begin{align*}_6P_2\end{align*}
10. Find \begin{align*}_{14}P_3\end{align*}
11. Find \begin{align*}_{15}P_3\end{align*}
12. Find \begin{align*}_{11}P_4\end{align*}
13. Find \begin{align*}_{16}P_2\end{align*}
Use permutations to solve each problem.
14. Mia has 7 charms for her charm bracelet – a heart, a moon, a turtle, a cube, a bird, a hoop, and a car. Into how many different orders can she arrange the 7 charms?
15. One of the charms in Mia’s bracelet in problem 6 above fell off. How many fewer arrangements are there now?
To see the Review answers, open this PDF file and look for section 11.4.
### Notes/Highlights Having trouble? Report an issue.
Color Highlighted Text Notes
Please to create your own Highlights / Notes<|endoftext|>
| 5.03125 |
583 |
Timpani are a type of drum and are therefore part of the percussion family.
Before we look at the history of the instrument, we should get the name sorted. The word ‘timpani’ is an Italian word and is plural, but in English it is completely acceptable to use the word for just one drum. Italians would call lots of drums ‘timpani’ and one drum a ‘timpano,’ but you would be hard pressed to find an English percussion player who is that pedantic. They would call one drum a ‘timpani’ or more likely just a ‘timp.’ They can also be known as ‘kettle drums.’
Evolving from a military drum, timpani are well known for their deep resonant sound. The original drums came in different shapes and sizes depending on their purpose (for marching, or for soldiers mounted on horseback).
They were first used in orchestral scores in the 17th century. Jean Baptiste Lully scored for timpani in his 1675 opera ‘Thésée’, but even earlier was Matthew Locke’s ‘Psyche’ written in 1673:
The instrument consists of a large bowl with a skin (called a ‘head’) stretched over the top. It can be tuned to different pitches, and is struck with a special timpani stick or a mallet. In early music usually two timpani were used, and were tuned to just two pitches for each piece- the tonic and the dominant. They were played at important cadence moments to emphasise the brass.
Early instruments could not have their pitch changed with any kind of speed, but the industrial revolution helped in the advancement of this. Composers altered the way they wrote for the instrument as rapid tuning became easier. For example in Haydn’s Symphony 94, written in 1791, the timpanist was required to change the tuning of both drums within a symphonic movement, from G & D to A & D, and then back again.
Several people had a go at advancing the instrument with various moving parts and metal rings, including Johann Strumpff in Amsterdam who worked out a way to change the pitch by rotating the drum. However, it still required the timpanist to put down his drum sticks in order to work the machine.
The big change came around 1840 when the first foot activated tuning system was invented by a German gunsmith (yes, a gunsmith) August Knocke. One can wonder what Beethoven and Haydn would have written for such an instrument, but alas we shall never know…
Our principal timpanist Adrian Bending talks about the Schnellar Timpani which he plays in Bruckner:<|endoftext|>
| 3.875 |
215 |
Question 89
The 4th and 7th term of an arithmetic progression are 11 and -4 respectively. What is the 15th term?
Solution
Let the first term of an AP = $$a$$ and the common difference = $$d$$
4th term of AP = $$A_4=a+3d=11$$ ----------(i)
7th term = $$A_7=a+6d=-4$$ --------(ii)
Subtracting equation (i) from (ii), we get :
=> $$6d-3d=-4-11$$
=> $$3d=-15$$
=> $$d=\frac{-15}{3}=-5$$
Substituting it in equation (i), => $$a=11-3(-5)=11+15=26$$
$$\therefore$$ 15th term = $$A_{15}=a+14d$$
= $$26+14(-5)=26-70=-44$$
=> Ans - (B)<|endoftext|>
| 4.4375 |
1,190 |
# Question Video: Differentiating Trigonometric Functions Using the Chain Rule Mathematics • Higher Education
Find dπ¦/dπ₯, given that π¦ = (β3 cot 5π₯ + 7)β΄.
03:08
### Video Transcript
Find dπ¦ by dπ₯, given that π¦ is equal to negative three times the cot of five π₯ plus seven all raised to the fourth power.
Weβre given π¦ as a function in π₯ and asked to find dπ¦ by dπ₯. Thatβs the first derivative of π¦ with respect to π₯. To do this, weβre going to first need to take a look at our function for π¦. We can see that π¦ is a trigonometric function all raised to the fourth power. In other words, π¦ is the composition of two functions. And we know we can differentiate the composition of two functions by using the chain rule. However, because our outer function is a power function, we can also do this by using the general power rule. It doesnβt matter which method we use; both will give us the same answer. Itβs all personal preference. Weβre going to use the general power rule.
So letβs start by recalling the general power rule. The general power rule tells us for any real constant π and differentiable function π of π₯, if we have π¦ is equal to π of π₯ all raised to the πth power, then dπ¦ by dπ₯ will be equal to π times π prime of π₯ multiplied by π of π₯ all raised to the power of π minus one. And we can see that π¦ is exactly written in this form, with our function π of π₯ as our inner function, negative three cot of five π₯ plus seven, and our exponent π equal to four. So, by setting π of π₯ to be our inner function and π equal to four, we can try and find dπ¦ by dπ₯ by using the general power rule.
To do this, weβre going to need to find an expression for π prime of π₯. Thatβs the derivative of negative three cot of five π₯ plus seven with respect to π₯. And we can differentiate this directly if we recall one of our rules for differentiating the reciprocal trigonometric functions. We know for any real constant π, the derivative of the cot of ππ₯ with respect to π₯ is equal to negative π times the csc squared of ππ₯. We can use this to differentiate the first term in π of π₯. We just need to set our value of π equal to five.
So by setting our value of π equal to five and remembering we need to multiply this expression by negative three, differentiating our first term, we get negative three times negative five csc squared of five π₯. And of course, we know that seven is a constant, so its rate of change with respect to π₯ will be equal to zero. So π prime of π₯ is just negative three multiplied by negative five csc squared of five π₯. Now, all we need to do is simplify this expression. Negative three times negative five is 15, so π prime of π₯ is 15 csc squared of five π₯. We now have a value of π, an expression for π prime of π₯, and π of π₯, so we can use the general power rule to find dπ¦ by dπ₯.
Substituting π is equal to four and our expressions for π prime of π₯ and π of π₯ into our general power rule, we get dπ¦ by dπ₯ is equal to four times 15 csc squared of five π₯ multiplied by negative three cot of five π₯ plus seven all raised to the power of four minus one. And we can simplify this. First, in our exponent, four minus one is equal to three. We can also simplify our coefficient. Four multiplied by 15 is equal to 60. So by using this and rearranging our factors, we get the following expression as our final answer.
Therefore, by using the general power rule, we were able to show if π¦ is equal to negative three cot of five π₯ plus seven all raised to the fourth power, then dπ¦ by dπ₯ will be equal to 60 times negative three cot of five π₯ plus seven all cubed multiplied by the csc squared of five π₯.<|endoftext|>
| 4.75 |
731 |
The integration is part of the important concepts that associate with mathematic, and is part of the main operations in calculus. It has to be given the function f of a variable x that is real, as well as the interval which is a, b, of that real line, and this is the definite integral:
This has been informally defined as signed area of region that is in x/y-plane, and its boundary by graph of f, x-axis, as well as vertical lines x that is a, and the x that is equal to b. The term known as integral could also refer to related notion of antiderivative, which is a function F, and whose derivative is the function of that is given. In such case, an indefinite integral is defined, and also written like this:
You are going to work with integral of a simple calculation following this simple steps below.
You should decide the dx value, which is 0,1. This is showing in the picture above.
Click on the column that is beside the x^2, and labeled as number 1. Click on insert tab, the one labeled as number 2.
Click on the equation, which is marked in red. Do not click on the arrow, just on the equation itself.
Click on the integral (labeled number 1) showing once you have clicked on the equation in previous step, and then click on the kind of integral you would like, in this case, the one labeled number 2 is chosen.
Place the value that are relevant to the integral function. As you could see in the one marked in red, and labeled as number 1 is going to have a detail of your choice. On top of the f there is a, while on the end, there is a 0, and in the middle there is an x. You should click on the line (where number 2 is marked), and expand the whole row to fit the integral equation that has been inserted.
Tip: you should type in the x2 (marked in the number 1), using an insert symbol (which is in the insert tab).
Add 0 (in this case in a7) with the 0,1 – the dx.
Information: If you see any error calculating, you should change from 0,1 to 0.1 to continue.
Click on the small square beside the number 1, using your mouse. Drag it down to the end, as you wish. In this case, it is dragged to a20.
Click on the column in the row of the ^2, and add the content in “a” row to multiply with ^2. Repeat the step 2 – 2 with the A row, as it is in this case. Check the picture for clarification.
Type in the columns that are relevant to the integral you are calculating. You would see the marked columns, as well as some added number.
Click on the integral you have just calculated, and now click on the small square that is in the marked area and on the left side of the one labeled number 1, and drag it down to save yourself the stress of calculating one after the other.
Mark the rows you would like to display in a specific design.
Click on the INSERT tab, the one labeled as number 1, and then choose the type of chart you would like to display, which is labeled as number 2.
Choose the design you would prefer for your integral chart.
Insert chart of your integral function and write the title you would like to give the chart.
The chart is ready.<|endoftext|>
| 4.28125 |
1,178 |
### Derivatives of compositions of functions: f(g(x))
The chain rule is one of the most useful tools in differential calculus. Equipped with your knowledge of specific derivatives, and the power, product and quotient rules, the chain rule will allow you to find the derivative of any function.
The chain rule is a bit tricky to learn at first, but once you get the hang of it, it's really easy to apply, even to the most stubborn of functions. Knowing it will also allow you to forget about memorizing the quotient rule because any denominator h(x) can be expressed as a multiple, [h(x)]-1.
Each of the functions in the box on the right is a composition of functions, g(h(x)). For example, in the first, h(x) = x2 - 4 and the outer function is g(x) = sin(x): g(h(x)) = sin(x2 - 4).
### Derivation of the chain rule
To derive the chain rule, consider a function y(t) that is actually a composition of functions, y(x) and x(t). It might be something like y(t) = cos(3t2), where y(x) = cos(x) and x(t) = 3t2. In terms of the changes in y and x, we have:
,
where the Δx terms cancel to give the overall change in y with respect to t. We can use the same cancellation in terms of differentials, but a word about that is in order:
While Leibniz, who invented the df/dx notation, never intended it to be used as such, we can safely consider df/dx to be simply a ratio of differentials, making the analogous cancellation possible:
Thus the derivative of a composition of functions can be expanded as a product of derivatives, the first the derivative of the outer function with respect to the full inner function (e.g. by treating it like a single "placeholder" variable), the second just the derivative of the inner function with respect to its independent variable.
### Learning the chain rule
It's best to learn the chain rule by working through examples. In the first two examples below, we'll substitute a "dummy variable" for the inner function, but you'll find that in no time you wont need to do that.
### Example 1
#### Find the derivative of f(x) = sin(x2)
The outer function is f(x) = sin(x) and the inner function is g(x) = x2. Let's let a = x2 and differentiate the outer function as f(a) = sin(a):
f'(a) = cos(a).
Then we differentiate the inner function, g'(x) = 2x, and multiply it by f(a)':
f' = 2x·cos(a)
I didn't write a variable behind f' above because at the moment I've got a weird hybrid function of x and a. The last step is to re-substitute x2 for a:
f'(x) = 2x·cos(x2).
### Example 2
#### Find the derivative of f(x) = (x2 - 5)3
The outer function is f(x) = x3 and the inner function is g(x) = x2- 5. Let's let a = x2 - 5 and differentiate the outer function as f(a) = a3:
f'(a) = 3a2.
Then we differentiate the inner function, g'(x) = x2- 5 with respect to x, and multiply it by f':
f' = 2x·3a2
The last step is to re-substitute (x2 - 5) for a:
f'(x) = 6x·(x2 - 5)2.
### Example 3
#### Find the derivative of f(x) = (x2 + x)1/2
Now we'll do without substitution of the placeholder variable and use the chain rule directly. The first step is to take the derivative of the whole outer function, treating the inner function (x2 + x) as a single unit.
½(x2 + x)-1/2 ← not finished, just pausing!
Then multiply by the derivative of the inner function, 2x + 1
f'(x) = (2x + 1)1/2(x2 + x)-1/2
### Example 4
#### Find the derivative of f(x) = (x2 - 4)-1
The outer function is x-1 and the inner is x2 - 4. The derivative is f'(x) = -1(x2 - 4)-2(2x), which can be simplified a bit, but I'll leave that to you.
### Video examples
There are a bunch of video examples of chain rule derivatives here.
### Practice problems
Find the derivatives of these compound functions. You can download solutions below (try not to peek until you've tried them).
### Video examples
#### 1. Simple chain rule examples
Here are a few examples of taking derivatives of simpler compound functions:
• f(x) = cos(x3),
• g(x) = tan(1/x), and
• h(x) = e3x-1
#### 2. More difficult chain rule examples
These are just a little bit more difficult. Notice that it's called the chain rule because we can use it to find derivatives of a function of a function of a function of ..., like f(g(h(x))) = tan(ln(x2)), where f(x) = tan(x), g(x) = ln(x) and h(x) = x2.<|endoftext|>
| 4.5 |
1,133 |
# Chaos
Shodor > Interactivate > Lessons > Chaos
### Abstract
The following discussions and activities are designed to lead the students to explore various incarnations of chaos. This lesson is best implemented with students working in teams of 2, alternating being the "driver" and the "recorder" using the computer activities.
### Objectives
Upon completion of this lesson, students will:
• have experimented with several chaotic simulations
• have built a working definition of chaos
• have reinforced their knowledge of basic probability and percents
### Student Prerequisites
• Geometric: Students must be able to:
• recognize and sketch objects such as lines, rectangles, triangles, squares
• Arithmetic: Student must be able to:
• understand and manipulate basic probabilities
• understand and manipulate percents
• Technological: Students must be able to:
• perform basic mouse manipulations such as point, click and drag
• use a browser for experimenting with the activities
### Teacher Preparation
• Pencil and calculator
• Copies of supplemental materials for the activities:
### Key Terms
chaos Chaos is the breakdown of predictability, or a state of disorder experimental probability The chances of something happening, based on repeated testing and observing results. It is the ratio of the number of times an event occurred to the number of times tested. For example, to find the experimental probability of winning a game, one must play the game many times, then divide the number of games won by the total number of games played iteration Repeating a set of rules or steps over and over. One step is called an iterate probability The measure of how likely it is for an event to occur. The probability of an event is always a number between zero and 100%. The meaning (interpretation) of probability is the subject of theories of probability. However, any rule for assigning probabilities to events has to satisfy the axioms of probability theoretical probability The chances of events happening as determined by calculating results that would occur under ideal circumstances. For example, the theoretical probability of rolling a 4 on a four-sided die is 1/4 or 25%, because there is one chance in four to roll a 4, and under ideal circumstances one out of every four rolls would be a 4. Contrast with experimental probability
### Lesson Outline
1. Focus and Review
Remind students what has been learned in previous lessons that will be pertinent to this lesson and/or have them begin to think about the words and ideas of this lesson:
• Does anyone know what predictabilty means?
• Can anyone explain what chaos means?
• If I lit a fire in the middle of the room can anyone predict what else would catch on fire?
• What would influence the way the fire might spread?
2. Objectives
Let the students know what it is they will be doing and learning today. Say something like this
• Today, class, we are going to learn about probability and chaos.
3. Teacher Input
• Lead a class discussion on basic probability to prepare students for working with the activities.
4. Guided Practice
• Have the students try the computer version of the Fire! activity to investigate how large the burn probability can be and still consistently have trees left standing. Allow the students about 20 minutes to explore computer activity.
• Lead a class discussion on chaos.
• Ask the class to think about why the fire activity is not very realistic. Be sure the point that controlling the probability of the spread of the fire is out of a person's hands. Motivate the next activity by pointing out that if we assume that fire will spread 100% of the time, then leaving some empty space in the forest (which a person can control) may keep the entire forest from burning.
• Have the students try the computer version of the Better Fire! activity to investigate how large the forest density can be and still consistently have trees left standing after a fire.
• Lead a class discussion on how prevalent chaos is in science.
5. Independent Practice
• Have the students try the computer version of the Game of Life activity to investigate this classic demonstration of chaos. Allow the students about 20 minutes to explore computer activity.
• Have the students try the computer version of the Rabbits and Wolves activity to investigate how the effects of small changes in the initial values of things changes the outcomes. Allow the students about 20 minutes to explore computer activity.
• If you choose to hand out the accompanying worksheets you can have the students complete them now.
6. Closure
• You may wish to bring the class back together for a discussion of the findings. Once the students have been allowed to share what they found, summarize the results of the lesson.
### Alternate Outline
This lesson can be rearranged in several ways.
• Reduce the number of activities; for example, use only the Better Fire! and Game of Life activities to give the classic examples of simulations with chaotic behavior.
• Add the additional activity using the Flake Maker activity with the following three starting shapes:
• Use the results of these three generators as an analogy for how small changes in structure cause large changes in cell growth in biology.
### Suggested Follow-Up
After these discussions and activities, the students will have seen more ways in which chaos, first introduced in the Fractals and the Chaos Game lesson, is used to model behavior. The next lesson, Pascal's Triangle , reintroduces Sierpinski-like Triangles, as seen in the Geometric Fractals and Fractals and the Chaos Game lessons, in yet another way, demonstrating the rich connections between seemingly different kinds of math.<|endoftext|>
| 4.53125 |
903 |
Browse Questions
# By using the properties of determinants show that $\begin{array}{l} \begin{vmatrix} 1&x&x^2 \\ x^2&1&x \\ x&x^2&1 \end{vmatrix} = (1-x^3)^2 \end{array}$
Toolbox:
• If each element of a row (or column) of a determinant is multiplied by a constant k ,then its value gets multiplied by k.
• By this property we can take out any common factor from any one row or any one column of the given determinant.
• Elementary transformations can be done by
• 1. Interchanging any two rows or columns. rows.
• 2. Mutiplication of the elements of any row or column by a non-zero number
• 3. The addition of any row or column , the corresponding elemnets of any other row or column multiplied by any non zero number.
Let $\bigtriangleup=\begin{vmatrix}1& x & x^2\\x^2 & 1& x\\x &x^2 & 1\end{vmatrix}$
Let us add $R_1,R_2,R_3$,hence we apply $R_1 \rightarrow R_1+R_2+R_3$
Therefore $\bigtriangleup=\begin{vmatrix}1+x+x^2& 1+x+x^2 & 1+x+x^2\\x^2 & 1& x\\x &x^2 & 1\end{vmatrix}$
Let us take $1+x+x^2$ as the common factor from $R_1$
Therefore $\bigtriangleup=(1+x+x^2)\begin{vmatrix}1& 1 & 1\\x^2 & 1& x\\x &x^2 & 1\end{vmatrix}$
By applying $C_1\rightarrow C_1-C_2$ and $C_2\rightarrow C_2-C_3$
Therefore $\bigtriangleup=(1+x+x^2)\begin{vmatrix}0& 0& 1\\x^2-1 & 1-x& x\\x-x^2 &x^2-1 & 1\end{vmatrix}$
But $(x^2-1)=(x+1)(x-1)$
We can also write $x-x^2=x(1-x^2)$
Therefore $\bigtriangleup=(1+x+x^2)\begin{vmatrix}0& 0& 1\\x^2-1 & 1-x& x\\x(1-x)&(x-1)(x+1) & 1\end{vmatrix}$
Therefore $\bigtriangleup=(1+x+x^2)\begin{vmatrix}0& 0& 1\\(x+1)(x-1) & 1-x& x\\x(1-x)&(x-1)(x+1) & 1\end{vmatrix}$
let us take (1-x) as a common factor from $C_1$ and $C_2$.
$\bigtriangleup=(1+x+x^2)(1-x)(1-x)\begin{vmatrix}0& 0& 1\\-(x+1) & 1& x\\x &-(x+1) & 1\end{vmatrix}$
Now expanding along $R_1$
$\bigtriangleup=(1+x+x^2)[1-(-(x+1)x-(x+1)-1(x)]$
$\quad=(1+x+x^2)(1-x)(1-x)[(x+1)^2-x]$
But we know $(1+x+x^2)(1-x)=1-x^3$
Hence $\bigtriangleup=(1-x^3)(1-x)[x^2+2x+1-x]$
$\quad=(1-x^3)(1-x)(x^2+x+1)$
Again $(1+x+x^2)(1-x)=1-x^3$
Therefore $\bigtriangleup=(1-x^3)(1-x^3)=(1-x^3)^2$
Hence proved.
edited Feb 24, 2013<|endoftext|>
| 4.65625 |
2,382 |
11th NCERT Relatins and functions.Miscellaneous Exercise Questions 12
Do or do not
There is no try
Question (1)
The relation f is defined by $\begin{array}{l}f(x) = {x^2}\quad \;0 \le x \le 3\\\;\quad \quad \; = 3x\;\quad 3 \le x \le 10\end{array}$ The relation g is defined by $\begin{array}{l}g(x) = {x^2}\quad \;0 \le x \le 2\\\quad \quad = 3x\;\quad 2 \le x \le 10\end{array}$ Show that f is a function and g is not a function.
Solution
The relation f is defined by $\begin{array}{l}f(x) = {x^2}\quad \;0 \le x \le 3\\\;\quad \quad \; = 3x\;\quad 3 \le x \le 10\end{array}$ For 0 ≤ x ≤ 3
f(3) = 32 = 9
3 ≤ x ≤ 10, then f(x) = 3x.
f(3) = 3(3) = 9
So f(3) has unique value .
So f is a funvction.
The relation g is defined by $\begin{array}{l}g(x) = {x^2}\quad \;0 \le x \le 2\\\quad \quad = 3x\;\quad 2 \le x \le 10\end{array}$ For 0 ≤ x ≤ 2
g(2) = 22 = 4
And for 2 ≤ x ≤ 10, then g(x) = 3x.
g(2) = 3(2) = 6
So g(2) has not unique value .
So g is not a funvction.
Question (2)
If f(x) = x2 find.$\frac{{f(1.1) - f(1)}}{{1.1 - 1}}$
Solution
f(x) = x2 $\frac{{f(1.1) - f(1)}}{{1.1 - 1}}$ $= \frac{{{{\left( {1.1} \right)}^2} - {1^2}}}{{1.1 - 1}}$ $= \frac{{1.21 - 1}}{{0.1}}$ $= \frac{{0.21}}{{0.1}} = 2.1$
Question (3)
Find the domain of the function $f(x) = \frac{{{x^2} + 2x + 1}}{{{x^2} - 8x + 12}}$
Solution
$f(x) = \frac{{{x^2} + 2x + 1}}{{{x^2} - 8x + 12}}$ $= \frac{{{{\left( {x + 1} \right)}^2}}}{{\left( {x - 2} \right)\left( {x - 6} \right)}}$ Function will be undefined if denominater becomes zero.
The denominator becomes zero.
(x - 2)(x - 6) = 0
x - 2 = 0 or x - 6 = 0
x = 2 or x = 6
So at x = 2 and x = 6 function will not be defined.
So domain of f is R - {2,6}.
Question (4)
Find the domain and the range of the real function f defined by $f(x) = \sqrt {x - 1}$
Solution
$f(x) = \sqrt {x - 1}$ We can not find the square root of negative numbers.
So x - 1 ≥ 0
So x ≥ 1
So Domain of f is [ 1, ∞ )
Since x - 1 ≥ 0
$\sqrt {x - 1} \ge 0$ So range of f = [ 0 , ∞)
Question (5)
Find the domain and the range of the real function f defined by f (x) = |x – 1|.
Solution
f (x) = |x – 1|.
x ∈ R, So domain of function = R
For all values of x , | x - 1 | ≥ 0
So range of function = R+ ∪ {0}
Question (6)
Let $f = \left\{ {\left( {x,\frac{{{x^2}}}{{1 + {x^2}}}} \right):x \in R} \right\}$ be a function from R into R. Determine the range of f.
Solution
Let f : R → R
So x ∈ R,
x2 ≥0
1 + x2 ≥ 1
${\frac{{{x^2}}}{{1 + {x^2}}} \ge 0}$ ${x^2} \le {x^2} + 1$ $\frac{{{x^2}}}{{{x^2} + 1}} \le 1$ So range of f is [ 0,1)
Question (7)
Let f, g: R → R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and.f/g.
Solution
f : R → R and g: R → R
So domain of and g are R.
So Df ∩ Dg = R
So f+ g , f - g and f/ g can be calculated.
f + g = f(x) + g(x)
= x + 1 + 2x - 3
= 3x - 2
f - g = f(x) - g(x)
= ( x + 1 ) - ( 2x - 3 )
= -x + 4
f/ g is defined for other values of x other than g(x) = 0 .
f/g : R- {3/2) → R
$\frac{f}{g} = \frac{{f(x)}}{{g(x)}},g(x) \ne 0$ $= \frac{{x + 1}}{{2x - 3}},2x - 3 \ne 0$
Question (8)
Let f = {(1, 1), (2, 3), (0, –1), (–1, –3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.
Solution
f : Z → Z
f(x) = ax + b,
f = {(1, 1), (2, 3), (0, –1), (–1, –3)} be a function
So (0, –1) will saisfy the f
∴ -1 = a(0) + b
∴ -1 = b
So (1, 1) is satisfy the function.
∴ 1 = a(1) + b
Replacing the value of b , we get,
1 = a + (- 1)
1 + 1 = a
2 = a
So the values of a = 2 and b = -1 .
Question (9)
Let R be a relation from N to N defined by R = {(a, b): a, b ∈ N and a = b2}. Are the following true?
(i) (a, a) ∈ R, for all a ∈ N
(ii) (a, b) ∈ R, implies (b, a) ∈ R
(iii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R.
Solution
(i) (a, a) ∈ R, for all a ∈ N
For all a ∈ N , a2 ≠ a
∴ statement is not true.
(ii) ( a, b) ∈ R,
→ a = b2
Let (2, 4 ) ∈ R.
theen ( 4, 2 )∉ R. as 2 is not a square of 4.
So the statement is not true.
(iii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R.
The statement is not true.
If (2, 4) ∈ R, and (4, 16) ∈ R
but ( 2, 16 ) ∉ R as 16 is not square of 2.
Question (10)
Let A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}. Are the following true?
(i) f is a relation from A to B (ii) f is a function from A to B.
Solution
A = {1, 2, 3, 4}, B = {1, 5, 9, 11, 15, 16} and f = {(1, 5), (2, 9), (3, 1), (4, 5), (2, 11)}.
In f ( x , y ) X ∈ A and y ∈ B.
So it is a relation from A → B.
But as (1, 5) ∈ f and (4, 5) ∈ f
The relation is one to many.
So it is not a function.
Question (11)
Let f be the subset of Z × Z defined by f = {(ab, a + b): a, b ∈ Z}. Is f a function from Z to Z: justify your answer.
Solution
Let f be the subset of Z × Z defined by f = {(ab, a + b): a, b ∈ Z}.
Let us consider a = 0, b = 1.
f = ( 0(1), 0 + 1 ) = ( 0, 1 )
If a = 0 and b = 2, then f = (0, 2 )
So for a = 0 and different values of b we get ( ab , a+b) = ( 0, b)
So relation becomes one to many, so it is not function.
Question (12)
Let A = {9, 10, 11, 12, 13} and let f: A → N be defined by f(n) = the highest prime factor of n. Find the range of f.
Solution
A = {9, 10, 11, 12, 13}
f: A → N
f(n) = the highest prime factor of n.
f(9) = 3, f(10) = 5, f(11) = 11, f(12) = 3, f( 13) = 13.
So range of f = { 3, 5, 11, 13}<|endoftext|>
| 4.53125 |
920 |
# The Volume of a Sphere
Here is a quick explanation for the formula of the volume of a sphere. This is based on the proof known to the Ancient Greeks.
For this, consider three objects:
1. A cylinder;
2. A cone with the point on the bottom;
3. A hemisphere with the flat side down.
Each object has a height of r and a radius of r.
Take a slice of each object at some height h. The exposed surface (cross-section) will be a circle.
For the cylinder, the radius of this exposed circle will be r, because the radii of all circular cross-sections is r. So the area of the circle for the cylinder at height h is πr2.
For the cone, the radius of this exposed circle will be h, so the area is πh2.
By definition, each point on the hemisphere is r units away from the center. Each point on the exposed circle is at a height of h. Using the Pythagorean Theorem, the radius of the exposed circle is the square root of (r2-h2), so the area is π(r2-h2).
Note that this is the difference between the cylinder and the cone: This is the key.
Since, for each cross-section of the three objects, the area for the cylinder is equal to that of the cone plus that of the hemisphere, it must be the case that the volume of a cylinder is equal to the volume of a cone and the volume of a hemisphere, when all objects have the same radius and height.
We know the formula for the volume of a cylinder: πr2h. If h = r, then πr3.
We know the volume of a cone is one-third of this, so the volume of a hemisphere is two-thirds of this, 2πr3/3. The volume of a sphere is twice that of a hemisphere, that is, 4πr3/3.
# Week of May 15, 2017
Presentations: May 19 May 18 May 17 May 16 May 15
Monday
Practice: Naming arcs, arc measures, and lengths
Discussion: Finding pi using the perimeter of polygons
Tuesday
Notes and practice: Area of Circles and Sectors
Wednesday
Practice: Area of Circles and Sectors, review all unit material
Thursday
Notes and Practice: Area of a sphere (review: cylinder and cone)
Friday
Practice: All unit material
Quiz
Notes: Circle tangents
# Week of May 8, 2017
Presentations: May 12 May 11 May 10 May 9 May 8
Monday
Practice: Volume of Pyramids and Cones
Tuesday
Review and practice: Area and Volume
Wednesday
Test: Area and Volume
Thursday
Notes and Practice: Naming circles and arcs
Friday
Notes and Practice: Arc measures and lengths
# Week of May 1, 2017
Presentations: May 5 May 4 May 3 May 2 May 1
Monday
Review: Area of Triangles and Quadrilaterals
Notes: Area of Regular Polygons
Tuesday
Review and practice: Area of Regular Polygons
Wednesday
Review for Quiz
Quiz
Thursday
Notes and practice: Volume of Prisms and Cylinders
Friday
Review and practice: Volume of Prisms and Cylinders
Notes: Volume of Pyramids and Cones
# Week of April 24, 2017
Presentations: April 28 April 27 April 26 April 25 April 24
Monday
Review for Chapter 8 Test
Tuesday
Chapter 8 Test
Wednesday
Notes: Area of Parallelograms and Triangles
Thursday
Review and practice: Area of Parallelograms and Triangles
Notes: Area of Trapezoids, Rhombuses, and Kites
Friday
Review and practice: Area of Trapezoids, Rhombuses, and Kites
# Week of April 17, 2017
Presentations: April 21 April 20 April 19 April 18 April 17
Monday
Review and practice: Pythagorean Theorem and Trig Ratios
Tuesday
Notes: Angles of Elevation and Depression
Wednesday
Review and practice: Angles of Elevation and Depression
Thursday
Notes: The Law of Sines
Friday
Review and practice: The Law of Sines<|endoftext|>
| 4.5625 |
1,288 |
Browse Questions
# Using matrices, solve the following system of equation $2x+8y+5z=5$$x+y+z=-2$$x+2y-z=2$
Toolbox:
• If |A|$\neq$ 0,then it is a non-singular matrix.
• Hence it is invertible.
• $A^{-1}=\frac{1}{|A|}.adj\; A$
• X=$A^{-1}B.$
Given:
2x+8y+5z=5.
x+y+z=-2.
x+2y-z=2.
The given system of equation is of the form
AX=B.
(i.e)$\begin{bmatrix}2 & 8 & 5\\1 & 1 & 1\\1 & 2 & -1\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}5\\-2\\2\end{bmatrix}$
Therefore x=$A^{-1}B.$
To find $A^{-1}$,let us first see if [A] is singular or non-singular.
|A| can be determined by expanding along $R_1$
|A|=$2(1\times -1-2\times 1)-8(1\times -1-1\times -1)+5(1\times 2-1\times 1)$
$\;\;=2(-1-2)-8(-1-1)+5(2-1)$
$\;\;=-6+16+5=15\neq 0$
Since $|A| \neq 0$ [A] is non-singular.
Now let us find the adj A.
To find adj A,let us find the minors and cofactors of the elements of [A].
$M_{11}=\begin{vmatrix}1 & 1\\2 & -1\end{vmatrix}$=-1-2=-3.
$M_{12}=\begin{vmatrix}1 & 1\\1 & -1\end{vmatrix}$=-1-1=-2.
$M_{13}=\begin{vmatrix}1 & 1\\1 & 2\end{vmatrix}$=2-1=1.
$M_{21}=\begin{vmatrix}8 & 5\\2 & -1\end{vmatrix}$=-8-10=-18.
$M_{22}=\begin{vmatrix}2 & 5\\1 & -1\end{vmatrix}$=-2-5=-7.
$M_{23}=\begin{vmatrix}2 & 8\\1 & 2\end{vmatrix}$=4-8=-4.
$M_{31}=\begin{vmatrix}8 & 5\\1 & 1\end{vmatrix}$=8-5=3.
$M_{32}=\begin{vmatrix}2 & 5\\1 & 1\end{vmatrix}$=2-5=-3.
$M_{33}=\begin{vmatrix}2 & 8\\1 & 1\end{vmatrix}$=2-8=-6.
$A_{11}=(-1)^{1+1}$.-3=-3.
$A_{12}=(-1)^{1+2}$.-2=2.
$A_{13}=(-1)^{1+3}$.1=1.
$A_{21}=(-1)^{2+1}$.-18=18.
$A_{22}=(-1)^{2+2}$.-7=-7.
$A_{23}=(-1)^{2+3}$.-4=4.
$A_{31}=(-1)^{3+1}$.3=3.
$A_{32}=(-1)^{3+2}$.-3=3.
$A_{33}=(-1)^{3+3}$.-6=-6.
Now adj A=$\begin{bmatrix}A_{11} & A_{21} & A_{31}\\A_{12} & A_{22} & A_{32}\\A_{13} & A_{23} & A_{33}\end{bmatrix}$
$\qquad\qquad=\begin{bmatrix}-3 & 18 & 3\\2 & -7 & 3\\1 & 4 & -6\end{bmatrix}$
$A^{-1}=\frac{1}{|A|}Adj \;A$,where |A|=15.
$A^{-1}=\frac{1}{15}\begin{bmatrix}-3 & 18 & 3\\2 & -7 & 3\\1 & 4 & -6\end{bmatrix}$
X=$A^{-1}B$,substituting for x,$A^{-1}$ and B,
$\begin{bmatrix}x\\y \\z\end{bmatrix}=1/15\begin{bmatrix}-3 & 18 & 3\\2 & -7 & 3\\1 & 4 & -6\end{bmatrix}\begin{bmatrix}5\\-2\\2\end{bmatrix}$
Matrix multiplication can be done by multiplying rows of matrix A by column of matrix B.
$\begin{bmatrix}x\\y\\z\end{bmatrix}=1/15\begin{bmatrix}-15-36+6\\10+14+6\\5-8-12\end{bmatrix}=1/15\begin{bmatrix}-45\\30\\-15\end{bmatrix}$
$\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}-45/15\\30/15\\-15/15\end{bmatrix}=\begin{bmatrix}-3\\2\\-1\end{bmatrix}$
Hence x=-3,y=2,z=-1.
edited Mar 28, 2013<|endoftext|>
| 4.59375 |
2,152 |
# Off Ramp Or Dead End Hbr Case Study Case Solution
Off Ramp Or Dead End Hbr Case Study I’ve had problems with the first and second part of the first part of the Fall Out Wallpaper, but I know that the answer to any question is very powerful. With one exception, the solution is pure chaos! If you have a question that seems like a good place to add your question, I highly recommend that you discuss it here. How We Build Everything We Made (14) In each of these cases, we simply go straight to the master and create something at some point in the equation. First we get into the process of transforming the variables to some random element when the equation changes, then we work through the equation to find if it’s a real form. More on this later. We see that this equation has a very nice initial state and the equation after this point becomes almost unique. So there is one thing that can stop a transformation from a real form, we find if the equation is well behaved, a shift becomes necessary and an overall transformation occurs.
## BCG Matrix Analysis
This is where we go from there. For the definition of a valid transition of an equation we will use only an equation, not an entire calculus. view it now we change the variables of our equation we are also changing the expression for the variable. As the equations become simpler with shifting our variables will be the same in our equation both, in the basic sense. With site here transition we are going to work with your derived equation, that is the resulting equation will change depending how you’re working with the transition equation. So for this example we have the equation as such, that is This may seem like somewhat of a mathematical exercise, but I guess it is actually an extremely simple problem. We have an equation, and we know the solution of this equation just by first starting with the equations and following the rules.
## Alternatives
If it is well behaved, a shift is needed. Our next step will be to show that the equation is not a transformation from there is a transformation from there. So we need to know the transition equation because we are going to go from the linear form in the present paper to a square form via a square root representation. Figure 1 shows the equation Here we continue this process, and then we will see that the transformation can be seen as a shift on the relationship between equation and algebraic manipulations. The transformation will probably be in the form Figure 2 shows the transformation and the resulting equation here. The Equation Using Matlab Studio 10 1st Stage When we see that an equation has both an equation and an entire calculus and change from the linear form, we just use the method described in the application of Matlab Studio. Create the equation & its equation using the equation & its equation action and then proceed.
## Case Study Help
After that we proceed down to a very next page general initial state by using the equation, and then the effect of this state change. How does the equation change depending on how you calculate check my source equation? Let’s take a look at what happens after the equation was written for example We begin with the equation we have written into the equation. We read equation, in Matlab Studio, and evaluate it. The result is This is a simple approximation, because we only need the equation in the equation, and so from what we read we can figure out that equation has a zero change. So evaluate with the previous step which is just Off Ramp Or Dead End Hbr Case Study The Stylist Mary Elizabeth Back Is Picking up the case on the floor, you can see Mary Elizabeth Back Is for a case analysis. In particular, the main discussion is the last discussion about the rest of the room. Don’t get me wrong; I agree with the rest of the room.
## Porters Model Analysis
The rest of the parish have plans and have a good sense of what they want, but it seems you see that’s about us. It’s better to get your thoughts his response there and talk with your team. But in my opinion, the case is better, not better. What concerns me is John because, given the way the whole thing is set up, I am more interested by his role. That’s what people were expecting and I was not hoping that people would understand. The great thing I noticed and what I notice in there is the case plan, again, the whole thing is set up. The details are not the key, what’s the point.
## PESTEL Analysis
It shows how the whole thing really is. If you apply a balance like me, you will note that everything you talk about is set up and it’s what’s best for the whole project. Now if I looked at moved here review, I would see so much about a normal project that I think I would have expected that John was to say I’m going to put my project into the library rather than being installed in the case. I don’t think it is fair to put my project onto the case. As Bob said above, the case is the problem. It makes the whole thing hard, meh. It’s hard to say what to put into it.
## Evaluation of Alternatives
It might as well be for Bob and a mate to say “this will be fine, that’s what I’ve always said I would be pleased to do.” That’s too much in my opinion. Is, if there’s good reason to be satisfied with that, it’s obvious. Is it harmful, or just a convenient way of putting your story? One thing that I will say if you take a look at the whole case, look at some of the other projects view website they probably do make the case that’s used in it. There might be good reasons why it’s that type of project. If it’s the only way to put your things in this project, then yes, it’s certainly a good project. If it can still be built, no problem.
## SWOT Analysis
If you work with Mark and get a small budget, which probably does not factor well, then yes, it’s great.Off Ramp Or Dead End Hbr Case Study The United States has largely embraced the “dead end” concept, and as of October 2011 it has a “protest” underway following a very strange anti-government sentiment in the West. In fact, the American government believes itself to be (possibly) the only legitimate solution to the world’s problems. Excerpt from The President’s Story “For too long in America” the government has said in TV commercials at the 1980s, “the president has used the military, at least for federal grants and contract grants, to throw open up doorways and let the community of citizens into the public visit homepage There are now some people who think this is a good thing, and a very limited number of them are going public, right? But there do seem to be many other people out there. There seems to be a lot of, uh, more than one person out there who think that “I agree with President Truman”. Maybe this is the reason why most of the people that are talking on TV are just guys with actual (actually) real problems with something that didn’t happen before.
## PESTLE Analysis
Excerpt from The President’s Story “Let me take this opportunity to stress that we are in service to the public good. If this country wants to make a deal with Britain we can get that. Whether we really want a deal on the issue of trade deals or not is up to us. If we were real friendly we would get a trade deal and there would be a pretty good chance that we would agree to a deal. But we are not really human after all.” — Senator Hylton (D-NY), former member of the Senate and a member of the Transportation Committee who was a fierce campaigner against big trade deals. Excerpt from The President’s Story “I am a big proponent of this trade agreement and I know I certainly am going to get tough.
## PESTEL Analysis
I know it’s hard to be like some people in Washington know what’s going on, but I’m pretty sure there’s some person who would like to work around it and maybe help get a deal. If he knows the details, he can help.” — Representative Dana Rohrabacher of Connecticut who opposed trade deals with the Fed and was critical of big trade deals. Yes, I’ve said it before. If we want to get deals out, we’ll have to get them. I have heard of something like the trade deal that resulted in a 60-year, 85-year trade surplus. So I don’t think I’ve got it all together and I don’t think I’ve got the courage to say this.
## PESTEL Analysis
I’ve also heard of things like trade deals that I’ve done a few times, maybe not a lot but many, like this one. I don’t think we realize much of what it means, but I’ve seen a lot of people tell us the TPP was disastrous. They said we couldn’t hold that amount. And that when we achieved that area, they would have more enforcement and they’d come to a deal “well done.” In many instances we saw them go after that amount. I think we will use these examples to drive our own argument to our own end. For example, we will use a case that was bad.
## PESTEL Analysis
EXAMPLE 8: Make this clear: I do not necessarily advocate an increase in the minimum wage, which I support. I do not respect or oppose any increase in the minimum wage. It is generally right. This was not a case in which the minimum wage had a problem, because, if the government did not fix it, it would come to a worse situation. Any more so. If the government does commit to implementing these measures it will probably drop them at the very least by several percentage points but we’ll find out. “But what they want to do is escalate to full-scale military action.
## Evaluation of Alternatives
They’re not going to do it. I’m going to judge it and say the same thing.” — Mrs. Warren, an advisor to President Obama when he was both Senate and House Speaker. If you saw what I said about the TPP and want to end it, then you would have to go to the very first steps to really think about what you are talking about, do you understand that? In fact, the new president said earlier this year he would make us put up<|endoftext|>
| 4.46875 |
1,727 |
July 21, 2024
Discover different methods, examples, and tools for finding the length of a triangle, such as step-by-step methods, formulaic methods, video tutorials, and real-world examples. This article explores common mistakes, practical applications, and interactive graphics to help readers better understand the topic.
I. Introduction
Triangles are among the most basic geometrical forms, yet they can be found in many aspects of everyday life. However, determining the length of a triangle can be challenging. Whether you are an architect designing a building or simply calculating the distance between two points, knowing how to find the length of a triangle is essential to problem-solving. This article provides a comprehensive guide on how to find the length of a triangle using different methods and tools.
II. Step-by-Step Method
One of the easiest ways of finding the length of a triangle is using the step-by-step method. Here are the basic steps to follow:
Identifying the Base and Height
A triangle has three sides, one of which is the base. The base is the longest side and is found at the bottom of the triangle. The height is a perpendicular line from the base to the opposite point or angle on the triangle.
Using the Pythagorean Theorem
The Pythagorean Theorem states that the sum of the squares of the two shorter sides of a right triangle equals the square of the length of the hypotenuse (the longest side of the triangle). This theorem can be used to find the length of the hypotenuse by taking the square root of the sum of the square of the two shorter sides.
Using Similar Triangles or Trigonometry
Another way to find the length of a triangle is using similar triangles. If two triangles have the same shape, their corresponding sides are proportional. This means that if you know the length of one side of one triangle and the corresponding side on the other triangle, you can find the length of the other side.
Alternatively, you can use trigonometry, specifically the sine, cosine, and tangent ratios. These ratios relate the lengths of the sides of a triangle to its angles and can be used to find the length of a side.
Offering Tips for Effectively Using This Method
It can be helpful to draw a diagram to visualize the triangle and label all sides and angles. Be sure to use the correct formula for the type of triangle you are working with. In addition, be aware of units of measurement if using real-world examples.
III. Real-World Examples
Real-world examples can help us better understand how to find the length of a triangle. Here are a few practical examples of finding triangle length:
Using a Map and Compass to Find Distance Between Two Points
The distance between two points on a map can be found by treating each point as a right angle (90-degree triangle). You can then use the Pythagorean Theorem to calculate the distance between the two points.
Using a Diagram of a Roof to Show How to Find the Length of a Sloped Line
When planning to build a roof, it is important to know the length of the sloped line. This length can be found using the Pythagorean Theorem by treating the roof and distance between the eaves as the two shorter sides and the sloped line as the hypotenuse.
Explaining How Real-World Examples Can Help Readers Better Understand the Concept
Real-world examples provide concrete applications of the concept and can help readers see how to apply the theory in practical scenarios. Such examples also make the concept more engaging and interesting.
IV. Interactive Graphics
Interactive graphics can be a useful tool for helping readers visualize the process of finding the length of a triangle. Here are a few tips for using interactive graphics:
Presenting the Idea of Using Interactive Graphics to Help Illustrate the Process
Interactive graphics can help readers understand the concept better by letting them drag and drop points or adjust measurements in real-time. Moreover, interactive graphics provide a more engaging experience for readers than static diagrams or pictures.
Suggesting Using Animations or Diagrams That Allow Readers to Drag and Drop Points
Animations can illustrate how the length is calculated step by step, while diagrams that allow readers to drag and drop points can help readers see the relationship between different parts of the triangle.
Highlighting the Benefits of Interactive Graphics for Engaging Readers and Promoting Understanding
Interactive graphics provide a more engaging experience for readers than static diagrams or pictures. Moreover, interactive graphics can explain concepts in a more dynamic way, allowing readers to see change in real-time. This in turn promotes better understanding of the concept.
V. Formulaic Method
In addition to the step-by-step method, there are other formulaic methods for finding the length of a triangle. Some of the most commonly used formulas are:
Pythagorean Theorem
The Pythagorean Theorem is applicable to right triangles only. It can be used to find the length of the hypotenuse.
Law of Sines
The Law of Sines can be used to find the length of a side when the length of two sides and the angle between them are known.
Law of Cosines
The Law of Cosines can be used to find the length of a side when the lengths of two sides and the angle opposite the unknown side are known. It is also applicable to non-right triangles.
Offering Advice on When Each Formula Is Most Appropriate to Use
The Pythagorean Theorem is used for right triangles only, while the Law of Sines and Law of Cosines are applicable to non-right triangles. The appropriate formula to use depends on the information you have about the triangle.
VI. Video Tutorials
Video tutorials are an excellent resource for visual learners who prefer to see and hear information. Here are some tips for creating effective tutorial videos:
Explaining the Advantages of Video Tutorials for Visual Learners
Video tutorials allow visual learners to see and hear the information being explained. They can replay the video, pause it to take notes, and learn at their own pace.
Offering Suggestions for Creating Effective Tutorial Videos
Effective tutorial videos should be clear, concise, and engaging. They should present the information visually and verbally, offer step-by-step instructions, and provide examples.
Providing an Example Tutorial Video That Demonstrates How to Find the Length of a Triangle
The following is an example tutorial video about how to find the length of a triangle using the Pythagorean Theorem:
VII. Common Mistakes
Here are a few common mistakes to avoid when finding the length of a triangle:
Common mistakes include using the wrong formula for the type of triangle, not using the correct units of measurement, and failing to accurately measure the sides and angles of the triangle.
Explaining How to Recognize and Avoid These Mistakes
To avoid these mistakes, double check the formula being used and ensure correct units of measurement are being used. Use a protractor and ruler to measure the sides and angles of the triangle accurately.
Offering Tips for Being More Accurate When Using the Step-by-Step Method or Formulas
It is important to have a clear diagram of the triangle and label all angles and sides accurately. Use a calculator to ensure accurate calculations, and double check all work to catch errors.
VIII. Applications of Length
Having the ability to find the length of a triangle has many practical applications in various fields. Here are a few examples:
Exploring Practical Applications of Finding the Length of a Triangle
Navigation, engineering, and architecture all use calculations of length on a regular basis. For example, navigators use the length of a triangle to determine travel distances, while engineers use length to design and build structures, bridges, and roads. Architects use length to ensure the safety and stability of their designs.
Discussing How Knowing the Length of a Triangle Could Be Useful
Knowing the length of a triangle can help you solve various problems, such as finding the distance between two points, calculating the slope of a roof, or determining the amount of material needed for construction.
IX. Conclusion
Knowing how to find the length of a triangle is an essential skill in various fields, such as navigation, engineering, and architecture. This article provided practical tips, formulas, and examples to help you solve problems involving triangle length. Remember to double check your work and use a clear and accurate diagram to ensure better accuracy. By using the various methods and tools provided, you can solve any problem involving triangle length and improve your problem-solving skills.
For further reading, check out resources such as Khan Academy or online calculators to continue your learning journey.<|endoftext|>
| 4.6875 |
642 |
## Relations and Functions
relation is a connection between the elements of a set or the elements of two sets.
e.g. "is greater than" and " is twice the size of " are examples of relations.
A function is a special type of relation.
### Representation
Relations can be represented in several ways:
• As a set of ordered pairs.
e.g. A = {(1, 2), (2, 3), (3, 4)}
The first value in each ordered pair is the x-value.
The second value in each ordered pair is the y-value.
• As an arrow graph or mapping.
e.g. The domainThe range
The set of x-values is called the domain. i.e. {1, 2, 3}
The set of y-values is called the range. i.e. {2, 3, 4}
• In table form
e.g.
Domain Range x y 1 2 2 3 3 4
• On a graph.
e.g.
• By a rule or formula.
e.g. y = x + 1
function is a relation where each of the members of the domain,(each x value) is connected to only one member of the range, (the y value). Most relations are also functions.
Exceptions are relations such as x 2 + y 2 = 9 which has a graph of a circle. This relation is not a function.
### Sequences
In mathematics, numbers are often arranged in a sequence. These sequences can be shown be a formula or function.
Often a sequence follows a rule or pattern.
Easy Sequence
Some sequences are obvious: 2, 4, 6, 8, 10...
This is the sequence of EVEN numbers.
The formula for this sequence is 2n where n stands for a counting number.
To find the next number in this sequence, calculate 2 × 6 = 12
Slightly Harder Sequence
The sequence 1, 4, 9, 16, 25... is the sequence of square numbers.
Comparing this sequence to the set of counting numbers {1, 2, 3, 4..}
1 2 3 4 5 ... 1 4 9 16 25 ...
The formula for this sequence is n2.
The next number in this sequence is 62 = 36
More Difficult Sequence
The number of diagonals in polygons forms a sequence.
Name of polygon Triangle Quadrilateral Pentagon Hexagon Heptagon Number of sides 3 4 5 6 7 Number of diagonals 0 2 5 9 14
The sequence for the number of diagonals is 0, 2, 5, 9, 14...
The pattern for this sequence is an increase of 2, 3, 4 between each term.
The formula for this sequence is (n2 + n − 2)/2
The next number in this sequence is 20<|endoftext|>
| 4.71875 |
1,624 |
## Kiss those Math Headaches GOODBYE!
### Multiplication Trick #5 — How to Multiply Two-Digit Numbers by 11
This is the fifth in my series on multiplication tricks. I suggest that you make mental math “tricks” a steady part of your math instruction. Benefits students will reap include:
— delight with the tricks themselves
— enhanced confidence in working with numbers
— students who otherwise don’t like math — or don’t like it much — often find the tricks irresistibly fun and interesting
TRICK #5:
WHAT THE TRICK LETS YOU DO: Multiply two-digit numbers by 11.
HOW YOU DO IT: To multiply a two-digit number by 11, first realize that the answer will have three digits. The first (left-most) digit of the answer is the first digit of the number; the last (right-most) digit of the answer is the last digit of the number; and the middle digit is the sum of the first and last digits.
But those are just words … here’s a living, breathing example …
Example: 11 x 25
Look at 25. The first digit is 2; the last digit is 5.
First digit of answer is 2, so thus far we know the answer looks like: 2 _ _
Last digit of answer is 5, so now we know the answer looks like: 2 _ 5
Middle digit is 7, since 2 + 5 = 7.
The answer is the three-digit number: 2 7 5, more casually known as 275.
It’s that easy!
ANOTHER EXAMPLE: 11 x 63
First digit of answer is 6, so thus far we know the answer looks like: 6 _ _
Last digit of answer is 3, so now we know the answer looks like: 6 _ 3
Middle digit is 9, since 6 + 3 = 9.
The answer is the three-digit number: 6 9 3, or just 693.
Try these for practice:
11 x 24
11 x 31
11 x 52
11 x 27
11 x 34
11 x 26
11 x 62
Answers:
11 x 24 = 264
11 x 31 = 341
11 x 52 = 572
11 x 27 = 297
11 x 34 = 374
11 x 26 = 286
11 x 62 = 682
NOTE: If you’re clever (and we’re sure that you are), you have probably realized that this trick, as described, works only when the digits add up to 9 or less. So what do you do when the digits add up to 10 or more? Some of you may figure this out on your own. For those who need a little help, the answer to this will be included in an upcoming blog post.
Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com Just click the links in the sidebar for more information!
Advertisements
### Multiplication Trick #2 — How to Multiply by 15 FAST!
Here’s the second in my set of multiplication tricks. (The first was a trick or multiplying by 5.)
TRICK #2:
WHAT THE TRICK LETS YOU DO: Multiply numbers by 15 — FAST!
HOW YOU DO IT: When multiplying a number by 15, simply multiply the number by 10, then add half.
EXAMPLE:15 x 6
6 x 10 = 60
Half of 60 is 30.
60 + 30 = 90
That’s the answer:15 x 6 = 90.
ANOTHER EXAMPLE:15 x 24
24 x 10 = 240
Half of 240 is 120.
240 + 120 = 360
That’s the answer:15 x 24 = 360.
EXAMPLE WITH AN ODD NUMBER:15 x 9
9 x 10 = 90
Half of 90 is 45.
90 + 45 = 135
That’s the answer:15 x 9 = 135.
EXAMPLE WITH A LARGER ODD NUMBER:23 x 15
23 x 10 = 230
Half of 230 is 115.
230 + 115= 345
That’s the answer:15 x 23 = 345.
PRACTICE Set:(Answers below)
15 x 4
15 x 5
15 x 8
15 x 12
15 x 17
15 x 20
15 x 28
ANSWERS Set:
15 x 4=60
15 x 5=75
15 x 8=120
15 x 12=180
15 x 17=255
15 x 20=300
15 x 28=420
Josh Rappaport is the author of five books on math, including the Parents Choice-award winning Algebra Survival Guide. If you like how Josh explains these problems, you’ll certainly like the Algebra Survival Guide and companion Workbook, both of which are available on Amazon.com Just click the links in the sidebar for more information!
### Multiplication Trick #4: Multiplying Teen Numbers
Ever wondered if there’s a quick way to multiply two number in the teens, problems like: 14 x 17, or 18 x 16?
Well, there is. And if you just stick around for two minutes, you’ll learn the trick. Let’s start wih 18 x 16.
First, add up the two digits in the one’s place. Here that’s the 8 and the 6. 8 + 6 = 14.
Next take that sum (14) and add 10 to it. 14 + 10 = 24.
Then tack a zero on at the end. 24 becomes 240.
Finally multiply the two numbers in the one’s place, and add the sum to the 240.
8 x 6 = 48, and 240 + 48 = 288.
That’s your answer. This may seem tricky at first, but it gets pretty easy if you try it a few times. Trust me …
O.K., don’t trust me. But just try it one more time, with 14 x 17, and see for yourself.
4 + 7 = 11. 11 + 10 = 21.
21 becomes 210.
4 x 7 = 28, and 210 + 28 = 238.
That’s all there is to it.
Now try these:
a) 13 x 16
b) 12 x 17
c) 14 x 19
d) 12 x 19
e) 13 x 14
f) 17 x 18
g) 19 x 17
h) 15 x 19
j) 16 x 17
k) 18 x 19
Answers:
a) 13 x 16 = 208
b) 12 x 17 = 204
c) 14 x 19 = 266
d) 12 x 19 = 228
e) 13 x 14 = 182
f) 17 x 18 = 306
g) 19 x 17 = 323
h) 15 x 19 = 285
j) 16 x 17 = 272
k) 18 x 19 = 342<|endoftext|>
| 4.84375 |
629 |
A new study from a team of scientists in Germany has revealed a previously undiscovered mechanism showing how caffeine can trigger the repair of heart muscles. The research, at this stage only involving mouse experiments, lends a newfound causal weight to the growing body of observational evidence suggesting caffeine has beneficial health effects.
Several recent large-scale observational studies have found that a moderate coffee intake may not only be safe, but also possibly beneficial to one's health. A massive umbrella study from 2017, collating data from 218 different meta-studies, concluded that coffee drinkers were 19 percent less likely to die from cardiovascular disease than non-coffee drinkers.
The problem with these observational studies is that, no matter how large the dataset, it's difficult to draw causal conclusions from the results. This new study, on the other hand, offers a vital clue that may help explain how coffee could confer protection against heart disease.
Prior research from the same German team established that caffeine improves the functional capacity of endothelial cells. These are important cells that line the inside of blood vessels, and when they become ineffective or dysfunctional it can lead to coronary heart disease, hypertension or diabetes.
This new study homes in further on the way caffeine affects these cells. It was discovered that caffeine induces the action of a protein called p27, which is responsible for promoting the migration of endothelial cells and boosting cell repair processes inside the heart.
"Our results indicate a new mode of action for caffeine, one that promotes protection and repair of heart muscle through the action of mitochondrial p27," says Judith Haendeler, one of the lead researchers on the project.
The researchers tested the protective heart benefits of caffeine in several mouse models, including pre-diabetic, obese and aged animals, and found positive effects across the board. The study suggests the optimal caffeine concentration to achieve these specific beneficial effects equates to something close to four to five cups of coffee a day for humans.
Other scientists examining this new research urge caution in how we may interpret these results. Kevin McConway, from The Open University, suggests that as these results have only been shown in mouse models, they don't necessarily mean humans should suddenly drink more coffee to protect their hearts.
"Even if these processes work the same way in human bodies as in mouse bodies and cell cultures – and I think that might be a big if – it's still not clear whether drinking coffee by aging humans, such as me, will work in this way to protect our heart health," says McConway.
While this newfound causal link between caffeine and heart health certainly needs more study before anyone would seriously recommend you drink five cups of coffee a day for your health, the research does point to a promising mechanism that could be more directly pharmacologically modulated in the future.
"…enhancing mitochondrial p27 could serve as a potential therapeutic strategy not only in cardiovascular diseases but also in improving healthspan," suggests Haendeler.
The new research was published in the journal PLOS.
Want a cleaner, faster loading and ad free reading experience?
Try New Atlas Plus. Learn more<|endoftext|>
| 3.765625 |
1,709 |
# Logic and expressions
The use and study of logic involves finding a new fact by analyzing whether some other facts together can prove to be true. Some facts, or conditions, when looked at together may prove another fact to be true, or maybe false.
If the temperature outside is below freezing and you don’t have a coat, you will feel cold. If you’re not sick, then you will feel well. If you can swim or ride in a boat in water, you will stay afloat. These are statements of fact that result from some condition being true.
## Truth statements
By taking some facts and putting them into a logical form, we can make an arithmetic that helps us analyze them and make a conclusion. Using the examples just mentioned, let’s turn them into some logical word equations:
• `Outside temperature is freezing` AND `I have no coat` = `I feel cold`
• NOT `sick` = `I feel well`
• `I can swim` OR `I'm in a boat` = `I'm floating`
You see the AND, NOT, and OR in the example word equations? These are our logical operators. Every day we make decisions when we think about one or more facts together using these operators. Sometimes, it’s necessary for all facts to be true in order for the conclusion to be true. This is the case when the AND operator is used. When analyzing facts with the OR operator, only on fact needs to be true for the conclusion to be true also.
Making a decision may require more than just one or two facts. When this happens, another operator is needed to combine the facts together to make a conclusion. In the last example word equation, you actually might not be floating if just those two condtions are true. To correctly prove that you’re actually floating, you need to state that you’re in water too.
• (`I can swim` OR `I'm in a boat`) AND `I'm in water` = `I'm floating`
To prove that you’re floating, the two facts that you can swim or you are in a boat must be made into a single fact that is combined with the other fact that you are also in water. Otherwise, if you’re able to swim but still on land, or in a boat that is on the beach, you’re not not floating.
## Boolean algebra
As a way to reduce the conditions, or facts as we’ve called them, into a form that is more compact, an algebra was invented. George Boole made a type of arithmetic (Boolean algebra) that uses symbols for the conditions, the operators, and the result. The conditions are considered as variables that have the value of either `true` or `false`. the operators like AND, OR, and NOT are single character symbols. If we want to change the statement “I’m happy when it’s sunny or when I’m eating a donut” into a Boolean equation, we could start by making the conditions into variables.
• Variable `A` = `"It's sunny"`
• Variable `B` = `"I've eaten a donut"`
The result then, is a variable called `Q` that is true when you’re happy and is a value of an operation of `A` with `B`. This operation is OR which is represented by the `+` symbol.
`Q` = `A + B`
The result of `Q` is `true` when either it’s sunny or you’ve had a donut. If other things make you happy, like being on vacation, you could add that to the equation.
• Variable `C` = `"I'm on vacation"`
`Q` = `A + B + C`
It could be that you’re easy to please and you just have to feel well to be happy. So, you’re happy when your NOT sick. We’ll use the `~` to mean NOT in our equation.
• Variable `A` = `"I'm sick"`
`Q` = `~A`
In the situation where all conditions must be true for the result to be true, the conditions use the AND operation. For the sun to shine on you, the sky must be clear and it has to be daytime. We put these two facts together with the AND symbol `·`.
• Variable `A` = `"The sky is clear"`
• Variable `B` = `"It's daytime"`
• Result `Q` = `"The sun is shining"`
`Q` = `A · B`
## Expressions
Sometimes different operations on the same conditions can make equivalent results. If we take the opposite case of the last example where the sun is not shining, the variables for that are:
• Variable `A` = `"The sky is clear"`
• Variable `B` = `"It's daytime"`
• Result `Q` = `"The sun is shining"`
• Result `~Q` = `"The sun is NOT shining"`
To make the opposite of `"the sun is shining"` we negate, use the NOT symbol, on both sides of the equation.
`~Q` = `~(A · B)`
Now, let’s think of the sun NOT shining due to negative conditions. If the sky isn’t clear OR it’s not daytime, then the sun isn’t shining. So, the NOT symbol is put in before the variables for each condition so that `"the sun is NOT shining"` has another equation like this:
`~Q` = `~A + ~B`
We see that the side with the `A` and `B` variables in both equations are equivalent to each other since they both equate to `~Q`:
`~(A · B)` = `~A + ~B`
The logic equation now doesn’t include the result variable `Q` but instead there are two expressions that are logically equivalent on each side.
#### De Morgan’s Thereom
That last equation, `~(A · B)` = `~A + ~B`, demonstrates an inportant property in Boolean algebra. It’s called De Morgan’s Thereom which says that the inverse (NOT) of a conjunction (AND) is logically equivalent to the disjunction (OR) of two inverses (NOT). Also, the inverse (NOT) of a disjunction (OR) is logically equivalent to the conjunction (AND) of two inverses (NOT).
This easier understood by seeing the Boolean equations for both cases:
`~(A · B)` = `~A + ~B`
– AND –
`~(A + B)` = `~A · ~B`
## Truth tables
A truth table is a way to see all possible condtions for the variables in a logical expression and to chart the results. Using the truth statement about when it’s freezing outside and you have no coat, here’s the truth table showing the possible conditions and their results:
It’s freezing I have no coat I feel cold
false false false
false true false
true false false
true true true
Because you feel cold only when both conditions are true, the statement becomes an AND expression in Boolean algebra.
• Variable `A` = `"it's freezing"`
• Variable `B` = `"I don't have a coat"`
`A · B` = `Q`
A truth table for the variables in the expression have the same values as the table for the truth statement (`true` and `false` are abbreviated to just `T` and `F`).
A B Q
F F F
F T F
T F F
T F T
What would happen if we changed the condition of `"I have no coat"` to `"I have a coat"`? How does that affect truth table about how cold you feel?
It’s freezing I have a coat I feel cold
false false false
false true false
true false true
true true false
A B Q
F F F
F T F
T F T
T F F
To write a Boolean equation for when you feel cold, we find the condtions in the table where `Q` is `true`. Here we see that you will feel cold only in one row, when condition `A` is `true` and condtion `B` is `false`. The Boolean equation for these conditions is this:
`A · ~B` = `Q`<|endoftext|>
| 4.5625 |
824 |
Grammar and Listening
To provide practice of the Passive via eliciting activities.
To provide specific information listening practice using a text about surveillance technology.
Procedure (25-50 minutes)
-Show them pictures about the subject - You are being watched - ask them; What comes to their minds? -Discuss it in pairs and then elicit their thoughts about it. -While showing the pictures ask them; What they are? Where can we see these? For what purpose we are using these? etc. -Elicit the meanings of vocabularies (surveillance | sɜːveɪləns | and CCTV-closed circuit televisions-| kləʊzd sɜːkɪt telɪvɪʒnz | )and answers of CCQs. -Then give them clear pronunciation of the words, and drill them. -Write the vocabularies on the board.
1A -Give clear instructions. -Work in pairs. -Give 2 minutes to think and discuss. -Show a picture and ask them to answer the questions that is on the worksheet. -Ask ICQs; What are you going to do? How you are going to this activity?In pairs in groups or individually?etc. -Elicit the ideas for each question 1B -Get their attention on the boxes in the picture and. What are you seeing? -Tell them there is some phrases in bold within them.Ask them to match them with the meanings. -If it is possible tell them to work in groups but if it is not possible tell them to do this activity individually. -Give 3 minutes. -Ask ICQs What are you going to do and how?In pairs or individually? How much time do you have?etc. -Elicit the answers. 1C -Ask it to whole class; Is this techniques being used in good or bad way?Give me at least one example. -Elicit their ideas.
2A -Ask them to listen the recording and taking very brief note to understand whether they like it nor not. 2B -Divide them into two as As and Bs. -Ask them to fill the box by listening the two people discussing types of surveillance technology. -As:make notes on the woman's opinion. Bs:make notes on the man's opinion. -SS can put + if they like it or - if they don't like it. -Ask CCQs. If they want to re-play the listening text. 2C -Ask them to work in pairs to exchange the information that they got from the listening track. 2D -Ask them; Which speaker do you agree with more? Did you opinions change after listening them?If yes,why?
-Introduce the topic: The Passive Passive Voice Song -Tell them they are going to listen a song about passive. -Play the song and want SS to join and try to sing the song together. -Elicit the structures of passive and write them on the board. 3A -Ask SS to underline the passive forms in the sentences. -SS work in pairs. -After they finished the exercise ask them; Why the passive is being used in each sentences? 3B -Ask SS to underline the correct one to complete the rules of Passive.(Individually) 4 -Tell them they are going to make another listening activity with grammar. -Tell them,if the words carrying the important information in the text they are usually stressed.Ask them to underline stressed words. -The first example can be done by teacher. -Use ICQs because it can be a little confusing for SS.
-Tell SS that they going to have a paper and the other pair going to have one too.But their texts are going to be a little different. -Ask them to exchange information by asking questions about blanks on their pages. -Elicit the answers from SS.<|endoftext|>
| 3.8125 |
1,699 |
# Factors of 110—with division and prime factorization
## Factors of 110
When you multiply two consecutive numbers 10 and 11, you get 110. For a better understanding, we will calculate the factors of 110, prime factors of 110, and factors of 110 in pairs along with solved examples.
Factors of 110: 1, 2, 5, 10, 11, 22, 55 and 110
Prime Factorization of 110: 110 = 2 × 5 × 11
## What are the Factors of 110?
The factors of 110 are the numbers multiplied in pairs to give the result of 110. Those factors that divide 110 without leaving any remainder are the factors of 110. Therefore, when you multiply two whole numbers with each other and get 110 as the answer, you can say that both those numbers are factors of 110.
Take note of the following numbers. When you multiply the given numbers, you get 110.
1 × 110 = 110
2 × 55 = 110
5 × 22 = 110
10 × 11 = 110
11 × 10 = 110
22 × 5 = 110
55 × 2 = 110
110 × 1 = 110
Therefore, we can say that “The factors of 110 are all the integers that can be divided by 110.”
## What are the Factors of -110?
When we multiply two numbers, and we get -110 as a product, that numbers are the factors of -110.
Here is the explanation:
-1 × 110 = -110
1 × -110 = -110
-2 × 55 = -110
2 × -55 = -110
-5 × 22 = -110
5 × -22 = -110
-10× 11 = -110
10 ×-11 = -110
We can get -110 by having (-1, 110), (-2, 55), (-5, 22), and (-10, 11) as a pair of factors. Similarly, we can also get -110 by having (1, -110), (2, -55), (5, -22), and (10, -11) as a pair of factors.
## How to Calculate the Factors of 110?
Start by calculating the factors of 110, starting with the smallest whole number, i.e., 1.
Divide 110 by this number. Is there a remainder of 0?
Definitely! This is what we get
110/1 = 110
110 × 1 = 110
2 is the next whole number
Divide 110 by this number.
110/2 = 55
2 × 55 = 110
In a similar fashion, we get
110 = 1 × 110 = 2 × 55 = 5 × 22 = 10 ×11
The factors of 110 are as follows:
Therefore, the factors of 110 are 1, 2, 5, 10, 11, 22, 55 and 110.
Using interactive examples and illustrations, explore factors.
Factors of 112—The factors of 112 are 1, 2, 4, 7, 8, 14, 16, 28, 56, 112
Factors of 108—The factors of 108 are 1, 2, 3 , 4, 6, 9, 12, 18, 27, 36, 54, 108
Factors of 50—The factors of 50 are 1, 2, 5, 10, 25, 50
Factors of 125—The factors of 125 are 1, 5, 25, 125
Factors of 15—The factors of 15 are 1, 3, 5, 15
Factors of 100—The factors of 100 are 1, 2, 4, 5, 10, 20, 25, 50, 100
## Factors of 110 by Prime Factorization
“Prime factorization” is the process of expressing a composite number as the product of its prime factors. We can determine the prime factorization of 110 by dividing it by its smallest prime factor, which is 2.
• 110/2 = 55
Using the smallest prime factor, 55 is divided by the quotient. This process continues until we get a quotient of 1.
Here is 110’s prime factorization:
Therefore, the prime factors of 110 are 2, 5, and 11.
## Factors of 110 in Pairs
Factor pairs of 110 are the pairs of numbers that give 110 when multiplied. Look at the following factors of 110 in pairs. According to the table above, no new factors are introduced after 10 × 11. Therefore, it is sufficient to find factors up until (10,11).
In the case of negative integers, both the numbers in the pair factors will be negative. Due to the fact that (- ve * – ve) = +ve, we can list the factor pairs of 110 as (-1, -110); (-2, -55); (-5, 22); and (-10, 11).
## Important Notes:
Since 110 ends with the digit 0, it has the factors 5 and 10. All numbers ending with 0 have these factors as well
There are no perfect squares in 110. Thus, it will have an even number of factors. This property applies to every non-perfect square number.
## Solved examples
1. Example 1:
Ali bought two rectangular rugs which had different dimensions, but had the same area of 110 inches². He wanted his friend Nadeem to guess the dimensions of the rugs. Could you help him with the two probable pairs other than (1,110) and (55, 2), that would result in the given area?
Solution:
Nadeem told that (5, 22) and (10,11) were the two factor pairs of 110 which could be the dimensions of the two rugs. Let us see how he explained it.
Since Area of a rectangle = length × breadth
For the first rug,
Area = 22 × 5 = 110 in²
For the second rug,
Area =10 × 11= 110 in²
Hence, the two factor pairs are (22, 5) and (10, 11).
Example 2:
Ali’s teacher told him that -55 is one of the factors of 110. Can you help him find the other factor?
Solution:
110 = Factor 1 × Factor 2
Therefore, 110 =(-55) × Factor 2
Factor 2 = 110/(-55) =(-2)
Thus, the other factor is -2
## FAQ’S
### What are the prime factors of 110?
There are three prime factors of 110: 2, 5, and 11.
### What is the GCF of 110 and 120?
The greatest common factor between two or more numbers is called the GCF. In the case of 110 and 120, the GCF will be 10.
### Which is the smallest prime factor of 110?
110 has a prime factor of 2 as its smallest.
### Which is the largest prime factor of 110?
11 is the largest prime factor of 110.
### What are the common factors of 110 and 140?
Factors of 110 include 1, 2, 5, 10, 11, 22, 55, and 110. There are 140 factors: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140.
Therefore, 1, 2, 5, and 10 are the common factors of 100 and 140.<|endoftext|>
| 4.8125 |
145 |
Write an essay of 750-1,000 words in which you:
1. Describe the structure and function of the electoral college.
- How and when was it created in the U.S.?
- Why was it created, and by whom?
2. Compare the electoral college to a popular vote approach for elections.
- How does the electoral college system operate/function?
- What are consequences of using an electoral college system versus a popular vote? Use the 2000 presidential election as an example.
3. Asses the value of an individual citizen’s vote under the electoral college system.
- Why does the U.S. still use the electoral college for presidential elections today?<|endoftext|>
| 3.71875 |
813 |
About This Lesson
This lesson is based on the National Park Service's brochure and Comprehensive Management and Use Plan for the Trail of Tears National Historic Trail, and the National Register of Historic Places files for the John Ross House (with photographs), Chieftains (with photographs), and Rattlesnake Springs. The lesson is also greatly indebted to John Ehle's Trail of Tears: The Rise and Fall of the Cherokee Nation and Carl Waldman's Atlas of the North American Indian. It was written by Kathleen A. Hunter, an educational consultant living in Hartford, Connecticut. It was edited by Marilyn Harper, a National Park Service consultant, and the Teaching with Historic Places Staff.
*Special note to teachers:
This lesson uses the terms commonly used during the 1830s to refer to men and women of English and European origin and to members of native tribes: "whites" and "Indians."
Where it fits into the curriculum
Topics: This lesson could be a part of a history unit on American Indians, Jacksonian America, Manifest Destiny, or westward expansion, a social studies unit on cultural diversity, or a geography unit on demography.
Time period: 1820s and 1830s
Relevant United States History Standards for Grades 5-12
Relevant Curriculum Standards for Social Studies
Find your state's social studies and history standards for grades Pre-K-12
Objectives for students
1) To identify the sources of conflict between American settlers and the Cherokee Nation.
2) To outline the events leading up to the forced relocation of the 1830s.
3) To describe the conflicts among the Cherokees and evaluate their effect on the relocation.
4) To research treaty agreements between the U.S. government and American Indian tribes in students' own region.
Materials for students
The materials listed below either can be used directly on the computer or can be printed out, photocopied, and distributed to students. The maps and images appear twice: in a smaller, low-resolution version with associated questions and alone in a larger version.
1) two maps showing the lands held by the Cherokee Nation and Cherokee removal routes;
2) three readings about the Cherokees, their leaders, and the relocation;
3) four photographs of historic places associated with the Cherokee removal;
4) one illustration showing a typical Cherokee homestead.
Visiting the site
The John Ross House is located in the town of Rossville in northern Georgia. It can be reached by taking I-75 to exit 350 (Battlefield Parkway). Go west on the parkway to Fort Oglethorp and turn right on U.S. Highway 27. In Rossville, turn left at the post office on Spring Street. It is open to the public during the summer months. For more information, contact the Walker County Chamber of Commerce, P. O. Box 430, Rock Springs, GA 30739.
"Chieftains," the Major Ridge House, is located in what is now Rome, Georgia. It can be reached by taking I-75 to exit 306 (State Road 140). Follow SR 140 to SR 53. Turn left to Highway 1 Loop to Riverside Road and follow signs. The house is open to the public Tuesday through Saturday. For more information contact the Chieftains Museum, P. O. Box 373, Rome, GA 30162.
Rattlesnake Springs is located on private property and is not open to the public.
The Trail of Tears National Historic Trail includes a variety of historic trail-related sites. An auto tour follows major highways that are close to the original trail route. The route is marked with the Trail of Tears National Historic Trail symbol. A map identifying the modern roads and their relationship to the historic trail is available on the Trail Web page. Guidebooks or local tourist agencies can provide directions to specific sites, some of which are National or state parks. Also included on the website is a printable travel guide.<|endoftext|>
| 4.21875 |
2,458 |
History remembers Eli Whitney’s cotton gin, invented in 1793, as the machine that notoriously revolutionized an industry. While this may be so, a bigger, more frightening machine became the vital cog in the Cotton Belt’s well-oiled industry by the end of the nineteenth century. The compress, gargantuan in size and vicious in task, carried the South to a new level of industrialization, beyond the span of Reconstruction-era policies, in the very market that had made it agrarian to begin with.
With the invention of the compress, continued railroad expansion, and the advent of electrical telegraphic communication, the cotton industry moved away from formerly monopolistic coastal cities, such as Houston, Galveston, and New Orleans, and moved inland to larger markets, now accessible by rail and interior water sources, such as St. Louis, Chicago, Memphis, Kansas City, and Dallas. Following the Civil War, farmers flocked to the frontier regions of Texas, which led all states in cotton production by 1889. By 1990, Texas gins accounted for thirty-four percent of the nation’s production. Compresses sprang up all across the Texas hinterland, particularly in the blackland prairies, to accommodate this inward push.
The compress’ technological innovation, in increasing a bale’s density to 22½ pounds per cubic foot and decreasing its physical size by half, radically altered shipping routes. The compress enabled a standard thirty-six rail car to hold up to 25,000 pounds of compressed cotton, more than double the former carrying capacity. This made railroad transport not only economically feasible but the dominant form of cotton shipment in Gilded-Age America. Furthermore, it signaled a change in gear, moving away from coastal shipping and towards inland rail shipping.
While the gin still initiated the process by separating fiber from seed, the compress served the pivotal role in a dynamic market process. It was at this “point of compression” where bales of cotton were compressed to half their original size, freeing up important railroad equipment (of which there was never enough to supply the demand), cotton was graded and priced based on quality, stored, purchased, and then rerouted to its designated mill. The lives of two very different but eloquently named crews–the screwmen and the spidermen—were tied to the fate of this behemoth.
Screwmen thrived during the era of coastal monopoly on the cotton industry. In port cities such as Galveston, Houston (Buffalo Bayou), Mobile, and New Orleans, these groups were tasked with stowing and packing bulky cotton bales in the holds of ships. Using screwjacks to do so—a device from which they derived their name—screwmen work crews could increase a ship’s bale capacity by ten to fifteen percent. These highly specialized stevedores—who were in effect the first compressors—thus ensured a profitable venture for the shipper.
Screwmen, however, struggled to keep up with increasingly efficient shipping methods developed through ongoing industrialization. A standard wooden sailing vessel could hold anywhere from eight hundred to one thousand bales, a feasible number of bales for screwmen to undertake. Later steamships, however, could hold up to twelve thousand compressed bales. Larger companies, which demanded quicker turnaround times at port, could not economically spare the time it took for local screwmen crews to handle such shipments, unprecedented in size.
Thus, a combination of larger, steel steam-powered ships, the ensuing acceleration of the shipping process, and the arrival of the compress all rendered cotton screwing an archaic practice. Once an honorable and highly-valued profession, screwmen found themselves increasingly out of work as the cotton industry moved further inland and new machines replaced manual labor.
The mechanical cotton compress increased the density of each bale, enabling a simultaneously larger and more rapid shipping process. The compress itself, however, required a cohort of workers to allow it to function smoothly. J.B. Coltharp, a retired engineer raised on a cotton farm in Coryell County, Texas, reminisced that the compress “with its awesome power was a fearful and fascinating thing to watch.” The machine was roughly forty feet in size and had an oversized steam cylinder that produced the energy necessary for the various gears, levers, platens, and other parts to compress each bale.
As loose, fluffy bales were compacted to half their size in the giant iron jaws of a compress, teams of “industrious spiders” worked in and around the machine to ensure its proper functioning. These workers came to be known as spidermen. Working in this environment required precise, coordinated movement given the constant danger working next to a machine of such power.
Each worker controlled one aspect of the larger process. The truckers would bring the bales from the front of the warehouse, at which point two dankeyman would load these bales on dollies and bring them to the compress. There, band snatchers would unhook the bands put in place at the gin, and setters would place the bare bales in between the press jaws. Once fitted inside the jaws, a set of teeth within the compress would close to compress the cotton, while steel bands were run through the bale and knotted, after which the bale was kicked out on a metal tongue. The leverman, boilerman, and several other workers ensured a proper press function, after which six tyers—three on each side—would tie the inserted steel bands to hold the compressed bale together. Two workers hauled away the finished bales.
The most important of the spidermen was the caller (singer). The caller ensured synchronized action through a choreographed call system, often sung in an old delta blues rhythm, to which all spidermen responded and performed their task. Clifford Blake, a lifetime caller working at the Natchitoches Warehouse & Compress in Natchitoches, Louisiana, exerted complete control in directing both man and machinery in the warehouse. The caller served as the glue that held together disparate workers and created one congruous unit.
Mr. Blake was the fearless leader of his crew. By singing his cadence, the caller not only kept the compress running smoothly and everyone safe, but addressed the emotional needs of his crew. Mr. Blake stated, “When I’d go to singing, regardless of how bad you feel, singing pulls your bad feelings away.” By expressing workers’ frustrations about their lives and dangerous work in his cadence lyrics, the caller could warm up the spidermen and give them “a mind to work.” His ability to make his men feel good and get them in the right mindset was the essential cog in the machine.
On top of holding together his crew of spidermen with resonant lyrics and commanding all activity on the floor, Mr. Blake displayed a lack of fear regarding the compress and inspired his men to brave the dangerous work conditions around the machine. Mr. Blake “rode it to the top of the building and down beneath the floor.” As the upper portion of the machine would descend, the platform on the floor would open up and the actual pressing would occur beneath ground level.
Mr. Blake would ride the press “down the hole,” with only the caller’s head showing above the floor when the machine was at its lowest point. Mr. Blake himself remarked, “Nobody does that anymore. I did it for forty years; when I got too slow, the press got me.” Mr. Blake was referring to February14, 1967, when he lost his footing while riding the press. The machine crushed his leg, ending his career. The dangers of working in and around such a beastly machine were constant.
The cotton industry’s transformation was gradual, and pockets of screwmen crews still existed well into the twentieth century. On a compressed bale, the exposed ends of the bound steel bands, buckles, and rivets were known as “spiders.” Screwmen wore a “handleather” to protect themselves when handling compressed bales. Nevertheless, injuries were still common. In 1910, the Screwmen’s Benevolent Association, a Galveston trade union of longshoremen, championed the passage of the “Spider” Bill. This legislature held cotton compress owners accountable for ensuring that bales were safely bound by their spidermen. This conflict between screwmen and spidermen illustrates a rare dynamic between specialized crews of different eras.
In 1900, Texas had approximately one hundred of the 269 compresses in the country. The Texas cotton belt was now blanketed with local compresses in northern and eastern towns, all becoming a “white sea of cotton” as sellers and buyers went to work in these interior markets. The localization of this new market system across the Texas landscape broadly opened its interior to new partnerships. New Orleans, continually looking to the Texas hinterland since Spanish colonial times as her “own economic province,” now vied for control of Texas trade with the newly prominent St. Louis cotton market.
With the new market system benefiting growers and railroads alike, smaller groups of tradesmen such as the screwmen could only watch as a macroeconomic trend transformed the lay of the land. The compress, a complex machine with a simplistically important task, dictated the fate of these tradesmen. It slowly eliminated the need for screwmen, but demanded new and dangerous work for spidermen. What these crews did not leave to fate, however, was the doggedness, ingenuity, and rectitude with which they performed their tasks.
Irony is not lost on the history of the compress. It has not received its due attention, as has the gin, in American history courses. Nevertheless, this historically-obscure machine produced equally impactful results in transforming both procedure and infrastructure of the cotton industry. The compress furthered industrialization processes in the South started by federal Reconstruction policies following the Civil War. The irony here is that the compress industrialized the signature product of a distinctly agrarian south. For all parties involved, this meant cotton was king once more, albeit a compressed one.
Interested in agrarian history? The North Texas Compress Company resided in Dension, Texas–birthplace of Dwight D. Eisenhower. Learn more about Eisenhower’s other hometown and the cattle trade here.
Expert caller Clifford Blake worked his compress in Natchitoches, Louisiana. Click here to learn about Natchitoches’ prehistoric history.
Clifford Farrington, Biracial Unions on Galveston’s Waterfront, 1865-1925, Dissertation, Austin: University of Texas, 2003.
J. B. Coltharp, “Reminiscences of Cotton Pickin’ Days,” in The Southwestern Historical Quarterly, vol. 73, no. 4 (Apr., 1970): 539-542.
L. Tuffly Ellis, “The Revolutionizing of the Texas Cotton Trade, 1865-1885,” in The Southwestern Historical Quarterly, vol. 73, no. 4 (Apr., 1970): 478-508.
L. Tuffly Ellis, “The Round Bale Cotton Controversy,” in The Southwestern Historical Quarterly, vol. 71, no. 2 (Oct., 1967): 194-225.
Lee A. Dew, “The Blytheville Case and Regulation of Arkansas Cotton Shipments,” in The Arkansas Historical Quarterly, vol. 38, no. 2 (Summer, 1979): 116-130.
Robert A. Calvert, “Nineteenth-Century Farmers, Cotton, and Prosperity,” in The Southwestern Historical Quarterly, vol. 73, no. 4 (Apr., 1970): 509-538.<|endoftext|>
| 3.703125 |
1,926 |
Become a math whiz with AI Tutoring, Practice Questions & more.
HotmathMath Homework. Do It Faster, Learn It Better.
# Different Bases
We tend to think it's perfectly natural to use $10$ symbols to write out numbers: $0,1,2,3,4,5,6,7,8,9$ . But the only reason we do this is because we grow up counting on our fingers, of which we happen to have ten. There's no real reason why ten is any better for math than another number, say $2,5,12$ or $16$ .
With one digit, we can count up to $9$ . Then we use place value to write larger numbers. " $10$ " means one ten and zero ones. The number $5723$ is really shorthand for:
$5723=\left(5×1000\right)+\left(7×100\right)+\left(2×10\right)+\left(3×1\right)$
The places stand for thousands, hundreds, tens, and ones. Notice that these are all powers of $10$ :
$5723=\left(5×{10}^{3}\right)+\left(7×{10}^{2}\right)+\left(2×{10}^{1}\right)+\left(3×{10}^{0}\right)$
## Example: Base $3$
What if we restricted ourselves to only three digits, $0$ , $1$ , and $2$ , and used powers of $3$ instead of powers of $10$ as the place values? Below we count up to $27$ in base $3$ .
BASE $3$ BASE $10$ $1$ $1$ $2$ $2$ $10$ $3$ $11$ $4$ $12$ $5$ $20$ $6$ $21$ $7$ $22$ $8$ $100$ $9$ $101$ $10$ $102$ $11$ $110$ $12$ $111$ $13$ $112$ $14$ $120$ $15$ $121$ $16$ $122$ $17$ $200$ $18$ $201$ $19$ $202$ $20$ $210$ $21$ $211$ $22$ $212$ $23$ $220$ $24$ $221$ $25$ $222$ $26$ $1000$ $27$
Notice that instead of the "tens", "hundreds", and "thousands" places, we have the "threes", "nines", and "twenty-sevens" places in the left column.
It may seem a little weird, but you can do math just as well in base $3$ as in base $10$ , or any other base. To illustrate, we'll do an addition problem (in base $3$ on the left, base $10$ on the right). Notice that we have to carry when we add $1+2$ !
$\begin{array}{ccccc}\begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\\ \underset{_}{+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2\text{\hspace{0.17em}}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\end{array}& & & & \begin{array}{l}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{0}\\ \underset{{_}}{{+}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{5}\text{\hspace{0.17em}}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{1}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{5}\end{array}\end{array}$
Historically, most but not all cultures have used base $10$ . The Yuki Indians of California used to use base $8$ , because they counted the spaces between their fingers rather than the fingers themselves. The Babylonians used base $60$ , and the Mayans used a mix of base $20$ and $18$ . Some old base $20$ terminology has even crept into the French and English languages. The French say "soixante et onze" for $71$ , which literally means "three twenties and eleven". And US President Abraham Lincoln's Gettysburg Address began, "Four score and seven", meaning $87$ .
Finally, in modern times, base $2$ ( binary ) and base $16$ ( hexadecimal ) are used frequently in computer science. (If you have ever played around with making a web page, you might know that HTML uses a $16$ -digit hexadecimal code to specify colors. The $16$ digits are $0,1,2,3,4,5,6,7,8,9,\text{A},\text{B},\text{C},\text{D},\text{E},\text{F}$ . The code for black is " $000000$ "; the code for white is " $\text{FFFFFF}$ "; " $9\text{B20DF}$ " is this sort of nice mellow purple color .)
There are also some people, like the Dozenal Society of America, who advocate changing the whole world over to a base $12$ system. They claim base $12$ is superior to base $10$ because it is divisible by more numbers... so it is easier to learn the multiplication tables!
;<|endoftext|>
| 4.71875 |
593 |
Plant Biology: Roots, Shoots, Stems, and Leaves
Your basic vascular plant parts are roots, shoots, stems, and leaves. Of course, there’s a wealth of variety within these types or parts, but it boils down to those four. Each part has distinct functions. Together, these parts reflect how vascular plants evolved to inhabit two distinct environments at the same time: the soil and the air. Why would plants do such a thing? The soil offers water and vital minerals. The air offers carbon dioxide and the energy of sunlight. To forge the successful lifestyles they enjoy today, plants evolved systems to tap into all these resources, both above and below the ground. In short, plants evolved roots and shoots. Shoots, in turn, can develop stems and leaves.
Roots are branched, underground structures that serve two major functions. First, somewhat obviously, roots firmly anchor the plant to a fixed spot. Once a plant takes root and begins to grow in an area with good access to moisture, soil nutrients, and light, it pays to stay. Second, roots serve as transport systems, allowing the plant to suck up water and dissolved nutrients from the soil to support the plant’s growth. Roots have specialized parts that develop from the three major types of plant tissue: ground, dermal, and vascular.
Shoots target the above-ground business of the plant. Very young plants may possess only simple, undeveloped shoots. As a plant grows, however, these tender shoots develop into stems and leaves. So, stems and leaves are really part of the shoot system. Stems and leaves are so different and specialized that it is worth considering them separately. Overall, the shoot system enables a plant to grow taller to gain access to energy-giving light, and allows the plant to convert that light energy into the chemical energy of sugar. Like roots, shoots develop from ground, dermal, and vascular tissues.
Stems are sturdy structures that grow in order to give a plant a fighting chance to spread its leaves in the sun. Stem growth can add to the plant’s height, broaden the area covered by the leaves, or even direct growth from a dark area toward one with more light. To provide mechanical support for a growing plant, stems need to be strong. To help move water and nutrients to the furthest reaches of the plant, stems are stuffed with little transport pipes in the form of xylem and phloem.
Leaves are the original solar panels, capturing energy from sunlight in a biochemical process called photosynthesis. The cells within leaf tissues are hectic with biochemistry, importing water and nutrients to support their frantic work, and exporting sugar to provide energy to the remainder of the plant. The import/export business conducted by the leaves is supported by xylem and phloem pipelines, which explains why leaves are so richly veined.<|endoftext|>
| 4.03125 |
858 |
Courses
Courses for Kids
Free study material
Offline Centres
More
Store
# A sum of money at simple interest amounts to Rs. 9440 in 3 years, if the rate of interest is increased by 25% the same sum amounts to Rs. 9800 at the same time. The original rate of interest is:${\text{A}}{\text{. 10% p}}{\text{.a}}{\text{.}} \\ {\text{B}}{\text{. 8% p}}{\text{.a}}{\text{.}} \\ {\text{C}}{\text{. 7}}{\text{.5% p}}{\text{.a}}{\text{.}} \\ {\text{D}}{\text{. 6% p}}{\text{.a}}{\text{.}} \\$
Last updated date: 14th Jun 2024
Total views: 411.6k
Views today: 4.11k
Verified
411.6k+ views
Hint: Simple interest is the accumulated interest on the principal amount while the total amount accumulated in a defined time period is the sum of the principal amount and the interest accumulated on the principal amount. The formula used for the calculation of the simple interest is $SI = \dfrac{{prt}}{{100}}$ where, $p$ is the principal amount, $r$ is the rate of interest annually (in percentage), and $t$ is the time for which the interest to be determined (in years). In this question, the rate of interest needed to be determined for two different total amounts. So, we need to get the ratio of the amounts by establishing a relation between the new interest rate and the original interest rate.
Complete step by step solution: Let the original rate of interest be $r\%$.
Then, the amount with the simple interest for the principal amount$(P)$ in the time period of 3 years will be given as:
$\dfrac{{{\text{Prt}}}}{{{\text{100}}}}{\text{ + P = 9440 - - - - - - (i)}}$
Now the rate of interest has been increased by 25% of the initial amount, we get:
$r' = r + 25\% {\text{of }}r \\ = 1.25r \\$
The amount with the simple interest for the principal amount$(P)$ in the time period of 3 years will be given as:
$\dfrac{{{\text{Pr't}}}}{{{\text{100}}}}{\text{ + P = 9800 - - - - - (ii)}}$
Dividing equation (i) and equation (ii) to determine the value of the original rate of interest as:
$\dfrac{{\left( {\dfrac{{{\text{Prt}}}}{{{\text{100}}}}{\text{ + P}}} \right)}}{{\left( {\dfrac{{{\text{Pr't}}}}{{{\text{100}}}}{\text{ + P}}} \right)}}{\text{ = }}\dfrac{{{\text{9440}}}}{{{\text{9800}}}} \\ \dfrac{{3r + 100}}{{3(1.25r) + 100}} = \dfrac{{944}}{{980}} \\ 980(3r + 100) = 944(3.75r + 100) \\ 2940r - 3540r = 94400 - 98000 \\ - 600r = - 3600 \\ r = \dfrac{{ - 3600}}{{ - 600}} \\ = 6\% {\text{ p}}{\text{.a}}{\text{.}} \\$
Hence, the value of the original rate of interest is 6% per annum.
Option D is correct.
Note: The candidates should not get confused with the term 25%, it is not the new interest rate but the increase in the rate of the interest from the original one. Moreover, the total amount should not be considered as the principal amount.<|endoftext|>
| 4.8125 |
1,384 |
Article by Nadine Stewart
What is Scoliosis?
Scoliosis is a medical condition where your spine is curved from side to side. The spine of an individual with typical scoliosis may look more like an "S" than a straight line. Approximately 2% of the population will have a scoliotic curvature in their spine and approximately 10% of these are severe.
A scoliosis is named according to where the apex of the curvature of the spine occurs. Most commonly these curves occur in the thoracic (mid-back) and thoracolumbar (junction between the thoracic and lumbar) areas of the spine. Scoliosis does not commonly occur in the neck.
Early adolescence, specifically between 11 and 14 years of age is the ideal time to screen your spine for scoliosis. Scoliosis can progress during adulthood if not treated during youth. Therefore, being checked and treated at any stage can greatly improve your comfort, muscular strength and mobility.
What Causes Scoliosis?
Several types of scoliosis exist:
Structural (Idiopathic) scoliosis has genetic roots. A family history of scoliosis, particularly along the female side can increase your likelihood of having it by up to 20%. Idiopathic scoliosis usually develops in early adolescence between the ages of 11-14, with a higher incidence occurring in females than males (10:1). Progression of scoliosis is also more common in females than males. If left undiagnosed and untreated in adolescence, the curvature of the spine can progress.
Functional scoliosis can develop in adulthood, often in response to an injury or repetitive practice of asymmetrical activities (i.e. tennis, golf swing etc). It is a curvature of the spine that has formed from overuse of muscles on one side and underuse of muscles on the reciprocal side. Since it is muscular based, it can reverse with appropriate treatment and exercise.
Pathology-related scoliosis can arise in people with a neuromuscular disease such as muscular dystrophy or in response to a severe injury to the spinal cord such as quadriplegia.
What are the Symptoms of Scoliosis?
Physical signs in children after the age of 8 that parents should suspect is scoliosis:
Contrary to common belief, scoliosis does not result from poor posture. However, it can progress to a more severe curve as a result of a poor posture of weak spinal muscles.
How is Scoliosis Diagnosed?
Checking for scoliosis is normally undertaken during a routine clinical examination by your physiotherapist or doctor.
Your physiotherapist will examine your spine, shoulders, rib cage, pelvis, legs and feet for abnormalities and asymmetry. If they suspect significant scoliosis, they will arrange for X-rays to confirm your cobb angle - or severity of scoliosis.
A significant curvature in the spine detected in adolescence will require a review from an orthopaedic spine specialist.
What are the Treatment Options for Scoliosis?
PHASE I - Pain Alleviation
While not all scoliosis sufferers experience pain or discomfort a percentage do. In these patients the provision of pain relief does assist with patient compliance with corrective or prevention exercises.
Pain relief can be achieved through a variety of techniques:
In this phase your physiotherapist may also introduce gentle exercises to maintain mobility in your spine as well as enhance your posture while your pain settles.
PHASE II - Rectifying Imbalances (Strengthening and Stretching!)
As your pain and inflammation settles, your physiotherapist will turn their attention to optimising the strength and flexibility of your muscles on either side of the scoliosis. They will also include adjacent areas such as the hip and shoulder region that may impact upon your spinal alignment.
The main treatment aims will include restoring normal spine range of motion, muscle length and resting tension, muscle strength, endurance and core stability.
Taping techniques may be applied until adequate strength and flexibility in the targeted muscles has been achieved.
PHASE III - Restoring Full Function
This scoliosis treatment phase is geared towards ensuring that you resume most of your normal daily activities, including sports and recreational activities without re-aggravation of your symptoms.
Depending on your chosen work, sport or activities of daily living, your physiotherapist will aim to restore your function to safely allow you to return to your desired activities.
Everyone has different demands for their body that will determine what specific treatment goals you need to achieve. For some it may be simply to walk around the block. Others may wish to run a marathon. Your physiotherapist will tailor your back rehabilitation to help you achieve your own functional goals.
PHASE IV - Preventing a Recurrence
Since scoliosis in many cases is a permanent structural change in the skeleton, ongoing self-management is paramount to preventing re-exacerbation of your symptoms. This will entail a routine of a few key exercises to maintain optimal strength, flexibility, core stability and postural support. Your physiotherapist will assist you in identifying which are the best exercises to continue in the long-term.
In addition to your muscle control, your physiotherapist will assess your hip biomechanics and determine if you would benefit from any exercises for adjacent muscles or some foot orthotics to address to correct for biomechanical faults. Some scoliosis results from an unequal leg length, which your therapist may address with a heel rise, shoe rise or a built-up foot orthotic.
Rectifying these deficits and learning self-management techniques is key to maintaining function and ongoing participation in your daily and sporting activities. Your physiotherapist will guide you.
What Results Can You Expect?
If you have mild to moderate scoliosis, you can expect a full return to normal daily, sporting and recreational activities. Your return to function is more promising when you are diagnosed and treated early.
Individuals with more moderate to severe spinal curvatures may need to be fitted for orthopaedic braces in order to halt curve progression. In some severe cases during adolescence, surgery is indicated. Both of these latter two pathways are overseen by an orthopaedic specialist who may require monitoring the progress of the curve with routine x-rays.
If you have any concerns or questions regarding your scoliosis, please ask your physiotherapist or doctor.
Common Scoliosis Treatments
Helpful Scoliosis Products
Nerve-related / Referred Pain
Share this page
Receive Special Offers and the Latest Injury Information
Enter Details Below to Signup:<|endoftext|>
| 4.125 |
699 |
A black hole may have been spotted in the process of formation for the first time in history – an event astronomers have long wanted to witness. There is also a chance that the phenomenon was a neutron star – a super dense corpse of a once-massive star – being formed, which in itself would be historic.
Astronomers at the ATLAS twin telescopes in Hawaii detected a massive burst of energy on June 17th, 2017, coming from a point 200 million light years away from Earth. After a dramatic flare, the source, seen in the constellation Hercules, quickly faded. Researchers designated the object AT2018cow, although most are affectionately calling it “The Cow.”
“We knew right away that this source went from inactive to peak luminosity within just a few days. That was enough to get everybody excited because it was so unusual and, by astronomical standards, it was very close by,” said Raffaella Margutti, astrophysicist at Northwestern University.
When massive stars die, they collapse into incredibly dense objects, including neutron stars or black holes. Most normal matter, including that which makes up stars, is composed of neutrons (having no electrical charge) and positively-charged protons in the nucleus, surrounded by clouds of negatively-charged electrons. Under the enormous pull of gravity in a collapsing massive star, the electron shells surrounding atoms are crushed, and the electrons are melded with protons, forming additional neutrons. The material is so dense that a small thimbleful of a neutron star would weigh as much as Mount Everest.
The escape velocity of an object (how fast an object needs to travel to escape its surface) depends on both the mass and diameter of the object. As bodies shrink while retaining their mass, the escape velocity rises. If a stellar corpse is dense enough, it can continue to collapse until the escape velocity from its surface is greater than the speed of light. Albert Einstein determined that no object can travel through space faster than the speed of light, meaning that nothing – not even light – could escape from the gravitational grip of a black hole.
The closest distance material can get to a black hole and still have a chance to escape is known as the event horizon. It is possible the large burst of energy seen from The Cow was matter – perhaps from a white dwarf star – swirling around this newly-formed frontier.
Astronomers have never before witnessed the moment when a neutron star or black hole formed until now. First thought to be a supernova, this event was shown to be between 10 and 100 times more powerful than these magnificent explosions. Particles from the burst spread apart at velocities of 30,000 kilometers per hour (over 18,600 MPH), about 10 percent of the speed of light. Within 10 days, almost all the energy had dissipated.
Study of the remains of the burst showed large quantities of hydrogen and helium, suggesting it was not caused by the collision of two dense, compact objects like neutron stars.
Although we are seeing the event now, the stellar collapse occurred 200 million years ago. At that time here on Earth, all the continents were still merged together into the super-continent of Pangaea, and a massive release of methane from the oceans (perhaps aided by tremendous volcanic eruptions as Pangaea ripped apart), wiped out much of the life on Earth, opening the way for the rise of the dinosaurs.<|endoftext|>
| 4.34375 |
564 |
Harmonics are additional pitches to the fundamental that are sounded within a signal. They are higher pitched and based on fractions or ratios of the original signal. For example a guitar string vibrates as a whole. It also vibrates as in halves (one half going one way while the other half does the opposite). In addition, it vibrates in thirds, fourths, fifths, and so on. Every time the length of the vibrating material or wave is affected the pitch changes. So, when a string vibrates in halves, it produces a pitch half the wavelength and twice the frequency (or one octave higher) of the original. The series of pitches produced as harmonics may seem familiar to you. They are the basis of western music, producing octaves, fifths, and thirds. The image on the right shows how different numbered harmonics fall into series with different notes of the scale. If you want to hear these for yourself, there is a simply test that you can do to help your ear notice the different harmonics.
Sit down at an acoustic piano and play a low “C” note. Hold it out and listen for higher pitches in the sound. Stop the note and play one of the harmonics (as listed in the staff such as “G”, “E”, or a higher “C”) immediately afterward. Listen for that harmonic as you play the low “C” note again. Repeat this process for the different harmonics. If you want to make the harmonics sound more apparent, hold down the sustain pedal while playing the fundamental. The harmonics of that note will cause the related note’s strings on the piano to resonate. This is called sympathetic vibration (when a specific item resonates at a specific frequency, other items tuned to the same frequency will begin to resonate as well). Now you should be able to hear many of the harmonics of that low “C” note. See how high in the harmonics series you can hear. You will notice that it gets more difficult as the pitches go higher. Listen to the richness of the sound that is created by having all of those notes sounding as harmonics with the one fundamental note. Finally, try different fundamental pitches and train your ear to listen for harmonics in other instruments as well.
The two above images show the difference between even and odd harmonics. Notice how the even harmonics are more stable ratios such as octaves while odd harmonics are more unstable, like thirds and sevenths. Even and odd harmonics have different characteristics. Even harmonics are commonly referred to as being more stable, smoother, and comforting. Odd harmonics are usually described as more jarring, unstable, and sometimes harsh.<|endoftext|>
| 4.03125 |
1,380 |
# Area Model
One of our models for multiplying whole numbers was an area model. For example, the product is the area (number of 1 × 1 squares) of a 23-by-37 rectangle:
So the product of two fractions, say, should also correspond to an area problem.
### Example (4/7 × 2/3)
Let us start with a segment of some length that we call 1 unit:
Now, build a square that has one unit on each side:
The area of the square, of course, is square unit.
Now, let us divide the segment on top into three equal-sized pieces. (So each piece is .) And we will divide the segment on the side into seven equal-sized pieces. (So each piece is .)
We can use those marks to divide the whole square into small, equal-sized rectangles. (Each rectangle has one side that measures and another side that measures .)
We can now mark off four sevenths on one side and two thirds on the other side.
The result of the multiplication should be the area of the rectangle with on one side and on the other. What is that area?
Remember, the whole square was one unit. That one-unit square is divided into 21 equal-sized pieces, and our rectangle (the one with sides and ) contains eight of those rectangles. Since the shaded area is the answer to our multiplication problem we conclude that
### Think / Pair / Share
1. Use are model to compute each of the following products. Draw the picture to see the answer clearly.
2. The area problem yielded a diagram with a total of 21 small rectangles. Explain why 21 appears as the total number of equal-sized rectangles.
3. The area problem yielded a diagram with 8 small shaded rectangles. Explain why 8 appears as the number of shaded rectangles.
### Problem 5
How can you extend the area model for fractions greater than 1? Try to draw a picture for each of these:
Work on the following exercises on your own or with a partner.
1. Compute the following products, simplifying each of the answers as much as possible. You do not need to draw pictures, but you may certainly choose to do so if it helps!
2. Compute the following products. (Do n0t work too hard!)
3. Try this one. Can you make use of the fraction rule to help you calculate? How?
### Think / Pair / Share
How are these two problems different? Draw a picture of each.
1. Pam had of a cake in her refrigerator, and she ate of it. How much total cake did she eat?
2. On Monday, Pam ate of a cake. On Tuesday, Pam ate of a cake. Both cakes were the same size. How much total cake did she eat?
When a problem includes a phrase like “ of …,” students are taught to treat “of” as multiplication, and to use that to solve the problem. As the above problems show, in some cases this makes sense, and in some cases it does not. It is important to read carefully and understand what a problem is asking, not memorize rules about “translating” word problems.
# Explaining the Rule
You probably simplified your work in the exercises above by using a multiplication rule like the following.
### Multiplying Fractions
Of course, you may then choose to simplify the final answer, but the answer is always equivalent to this one. Why? The area model can help us explain what is going on.
First, let us clearly write out how the area model says to multiply . We want to build a rectangle where one side has length and the other side has length . We start with a square, one unit on each side.
• Divide the top segment into equal-sized pieces. Shade of those pieces. (This will be the side of the rectangle with length .)
• Divide the left segment into equal-sized pieces. Shade of those pieces. (This will be the side of the rectangle with length .)
• Divide the whole rectangle according to the tick marks on the sides, making equal-sized rectangles.
If the answer is , that means there are total equal-sized pieces in the square, and of them are shaded. We can see from the model why this is the case:
• The top segment was divided into equal-sized pieces. So there are columns in the rectangle.
• The side segment was divided into equal-sized pieces. So there are rows in the rectangle.
• A rectangle with columns and rows has pieces. (The area model for whole-number multiplication!)
### Think / Pair / Share
Stick with the general multiplication rule
Write a clear explanation for why of the small rectangles will be shaded.
# Multiplying Fractions by Whole Numbers
Often, elementary students are taught to multiply fractions by whole numbers using the fraction rule.
### Example: Multiply Fractions
For example, to multiply , we think of “2” as , and compute this way
We can also think in terms of our original “Pies Per Child” model to answer questions like this.
### Example: Pies per Child
We know that means the amount of pie each child gets when 7 children evenly share 3 pies.
If we compute that means we double the amount of pie each kid gets. We can do this by doubling the number of pies. So the answer is the same as : the amount of pie each child gets when 7 children evenly share 6 pies.
Finally, we can think in terms of units and unitizing.
### Example: Units
The fraction means that I have 7 equal pieces (of something), and I take 3 of them.
So means do that twice. If I take 3 pieces and then 3 pieces again, I get a total of 6 pieces. There are still 7 equal pieces in the whole, so the answer is .
### Think / Pair / Share
1. Use all three methods to explain how to find each product:
2. Compare these different ways of thinking about fraction multiplication. Are any of them more natural to you? Does one make more sense than the others? Do the particular numbers in the problem affect your answer? Does your partner agree?
# Explaining the Key Fraction Rule
Roy says that the fraction rule
is “obvious” if you think in terms of multiplying fractions. He reasons as follows:
We know multiplying anything by 1 does not change a number:
So, in general,
Now, , so that means that
which means
By the same reasoning, , so that means that
which means
### Think / Pair / Share
What do you think about Roy’s reasoning? Does it make sense? How would Roy explain the general rule for positive whole numbers :<|endoftext|>
| 4.90625 |
611 |
Select Page
# Visualizing Proportional Reasoning: Working With Ratios and Proportions
## Solving Proportions With The Toronto Maple Leafs Win:Loss Ratio
As we began the Proportional Reasoning unit in my MFM1P Grade 9 Math Course, I was beginning to struggle with an idea to extend the idea of visualizing mathematics concepts into ratios, rates, and proportions. What I came up with was using a visual representation of Toronto Maple Leaf wins and losses in order to help scaffold students slowly from concrete examples to the algebraic representation utilizing a proportion of equivalent fractions. This could be a good way for you to understand the rankings and wins and losses of your fantasy hockey teams.
If you’re not into watching the video, you can see some screenshots below.
## Summary of the Working With Ratios and Proportions Video
The video begins with a formal definition of a ratio and heads straight into the scenario where the Toronto Maple Leafs win:loss ratio is 3:1.
Because this is a relatively basic example, many can find the unknown in the proportion using trial and error or even just logic without really understanding what they did to get there. I find it useful to demonstrate that we can see the same 3:1 ratio, 3 times on the right side of the proportion. Using terminology like “3 groups of 3 wins” and “3 groups of 1 loss” can help scaffold towards more difficult problems.
In the same visual fashion, we solve a second problem and represent the 3:1 ratio in an obvious fashion.
We then move on to another example. This time, we change the ratio from 3:1 to 3:2 in order to scaffold students towards better understanding the factor the original ratio has been multiplied by.
We focus on the unknown factor or multiplier represented by the question mark (?) and work to find that unknown value. I use the concept of opposite operations (since we see 3 times a variable, we must divide by 3 to isolate the variable) and follow through by performing the same operation on the other side.
We then use the factor/multiplier of 4 to determine the number of losses:
To confirm our work, we bring the visuals back in.
Finally, we look at an example that makes using visuals very time consuming and inefficient. This drives home why using proportional reasoning to solve problems with ratios is important.
As are all of my visual animations, they are my first attempt. Would appreciate any feedback (positive and negative) you can provide in the comments below.
## WANT TO LEARN HOW TO TEACH THROUGH TASK?
Download our Complete Guide to successfully implementing our Make Math Moments 3-Part Framework in your math class!
## Share With Your Learning Community:
I’m Kyle Pearce and I am a former high school math teacher. I’m now the K-12 Mathematics Consultant with the Greater Essex County District School Board, where I uncover creative ways to spark curiosity and fuel sense making in mathematics. Read more.<|endoftext|>
| 4.78125 |
1,045 |
<meta http-equiv="refresh" content="1; url=/nojavascript/">
You are viewing an older version of this Concept. Go to the latest version.
# Unit Conversions
## Converting between the English system and the metric system.
0%
Progress
Practice Unit Conversions
Progress
0%
Unit Conversions
Students will learn how to convert units from metric to english system and vice verse using dimensional analysis.
### Key Equations
$1\ \text{meter} & = 3.28\ \text{feet} && \\1\ \text{mile} & = 1.61 \text{~kilometers} && \\1\ \text{lb. (1\ pound)} & = 4.45\ \text{Newtons}$
Guidance
• The key to converting units is to multiply by a clever factor of one. You can always multiply by 1, because it does not change the number. Since 1 in. is equal to 2.54 cm, then $1 = {\frac{2.54 \;\mathrm{cm}}{1 \;\mathrm{in}}} = {\frac{1 \;\mathrm{in}}{2.54 \;\mathrm{cm}}}$ . Thus, one can multiply by this form of 1 in order to cancel units (see video below).
• Write out every step and show all your units cancelling as you go.
• When converting speeds from metric to American units, remember the following rule of thumb: a speed measured in mi/hr is about double the value measured in m/s ( i.e., $10 \mathrm{m/s}$ is equal to about 20 MPH). Remember that the speed itself hasn’t changed, just our representation of the speed in a certain set of units.
• When you’re not sure how to approach a problem, you can often get insight by considering how to obtain the units of the desired result by combining the units of the given variables. For instance, if you are given a distance (in meters) and a time (in hours), the only way to obtain units of speed (meters/hour) is to divide the distance by the time. This is a simple example of a method called dimensional analysis , which can be used to find equations that govern various physical situations without any knowledge of the phenomena themselves.
#### Example 1
Question : 20 m/s = ? mi/hr
Solution :
20 m/s (1 mi/1600 m) = .0125 mi/s
.0125 mi/s (60 s/min) = .75 mi/min
.75 mi/min (60 min/hr) = 45 mi/hr
### Time for Practice
1. Estimate or measure your height.
1. Convert your height from feet and inches to meters.
2. Convert your height from feet and inches to centimeters $(100 \;\mathrm{cm} = 1 \;\mathrm{m})$
2. Estimate or measure the amount of time that passes between breaths when you are sitting at rest.
1. Convert the time from seconds into hours
2. Convert the time from seconds into milliseconds $\;\mathrm{(ms)}$
3. Convert the French speed limit of $140 \;\mathrm{km/hr}$ into $\;\mathrm{mi/hr}$ .
4. Estimate or measure your weight.
1. Convert your weight in pounds into a mass in $kg$
2. Convert your mass from $kg$ into $\mu g$
3. Convert your weight into Newtons
5. Find the $SI$ unit for pressure.
6. An English lord says he weighs $12$ stone.
1. Convert his weight into pounds (you may have to do some research online)
2. Convert his weight in stones into a mass in kilograms
7. If the speed of your car increases by $10 \;\mathrm{mi/hr}$ every 2 seconds, how many $\;\mathrm{mi/hr}$ is the speed increasing every second? State your answer with the units $\;\mathrm{mi/hr/s}$ .
1. a. A person of height $5 \;\mathrm{ft}$ . $11 \;\mathrm{in}$ . is $1.80 \;\mathrm{m}$ tall b. The same person is $180 \;\mathrm{cm}$
2. a. $3 \;\mathrm{seconds} = 1/1200 \;\mathrm{hours}$ b. $3x10^3 \;\mathrm{ms}$
3. $87.5 \;\mathrm{mi/hr}$
4. If the person weighs $150 \;\mathrm{lb}$ then a. 67.9 kg (on Earth) b. 67.9 billion $\mu g$ c. this is equivalent to $668 \;\mathrm{N}$
5. Pascals (Pa), which equals $\;\mathrm{N/m}^2$
6. a. $168 \;\mathrm{lb}$ .,b. $76.2 \;\mathrm{kg}$
7. $5 \;\mathrm{mi/hr/s}$<|endoftext|>
| 4.8125 |
1,623 |
# Difference between variance-covariance and correlation matrix?
## Variance/Covariance
To start off, the sample variance formula is:
$s^2 = \frac{\sum_{i=1}^{n}(x_i - \overline{x})^2} {n - 1 }$
First of all, $x - \overline{x}$ is a deviation score (deviation from what? deviation from the mean). Summing the deviations will just get us zero so the deviations are squared and then added together. The numerator of this formula is then called the sum of squared deviations which is literally what it is. This is not yet what we refer to as the variance ($s^2$). We have to divide this by $n - 1$ which is the sample degrees of freedom.
If you have two variables, x and y, those two variables can covary. The formula is similar– instead of squaring the deviation scores, the product of the deviation scores of the two variables are used.
$cov(x, y) = \frac{\sum_{i=1}^{n}(x_i - \overline{x})(y_i - \overline{y})} {n - 1 }$
The numerator is also called the sum of cross products (which is what it is). Then dividing this by $n - 1$ is the covariance. The covariance of a variable with itself is also the variance which makes sense (instead of the cross product, you are multiplying the deviance with itself or just squaring it).
$cov(x, x) = s^2 = \frac{\sum_{i=1}^{n}(x_i - \overline{x})(x_i - \overline{x})} {n - 1 }$
That is pretty useful to know. However, the covariance though is not easy to interpret because it is dependent on the scale of your variables. For example, if you get the covariance of height and weight– one is measured in inches (or cm) and the other in pounds (or kg). Here’s an example (not height or weight):
x <- c(12, 15, 20, 25, 30)
y <- c(2, 6, 8, 10, 12)
mean(x)
## [1] 20.4
mean(y)
## [1] 7.6
var(x)
## [1] 53.3
var(y)
## [1] 14.8
cov(x, y) #gets one covariance at a time.
## [1] 27.2
We can put these two variables together in a data frame and estimate the covariance from there.
df <- data.frame(x, y)
cov(df) #same result
## x y
## x 53.3 27.2
## y 27.2 14.8
## Correlation
What then is the relationship with the correlation matrix? One way to think about it is that the covariance matrix is a bit hard to interpret (the covariances) because they are a mix of different units of measure. A way we get around that is standardizing the measures by converting them to z scores:
$z-scores = \frac{(x_i - \overline{x})} {SD_x }$
The scores then have a distribution with a M = 0 and SD = 1 (w/c also means a variance of 1). NOTE: how we can access variables in the data frame using the $sign. We can convert by using: zx <- ( (df$x) - mean(df$x) ) / sd(df$x)
zy <- ( (df$y) - mean(df$y) ) / sd(df$y) zx ## [1] -1.15057698 -0.73965663 -0.05478938 0.63007787 1.31494512 zy ## [1] -1.4556507 -0.4159002 0.1039750 0.6238503 1.1437255 NOTE: in R, a function to convert raw scores to z scores is the scale function. zx2 <- scale(df$x)
zy2 <- scale(df$y) zx2 ## [,1] ## [1,] -1.15057698 ## [2,] -0.73965663 ## [3,] -0.05478938 ## [4,] 0.63007787 ## [5,] 1.31494512 ## attr(,"scaled:center") ## [1] 20.4 ## attr(,"scaled:scale") ## [1] 7.300685 zy2 ## [,1] ## [1,] -1.4556507 ## [2,] -0.4159002 ## [3,] 0.1039750 ## [4,] 0.6238503 ## [5,] 1.1437255 ## attr(,"scaled:center") ## [1] 7.6 ## attr(,"scaled:scale") ## [1] 3.847077 If you want to scale the whole dataset: zdf <- scale(df) zdf ## x y ## [1,] -1.15057698 -1.4556507 ## [2,] -0.73965663 -0.4159002 ## [3,] -0.05478938 0.1039750 ## [4,] 0.63007787 0.6238503 ## [5,] 1.31494512 1.1437255 ## attr(,"scaled:center") ## x y ## 20.4 7.6 ## attr(,"scaled:scale") ## x y ## 7.300685 3.847077 NOTE: These variables now have a mean of 0 and sd of 1 (also a variance of 1). A one unit change is a one standard deviation change. NOTE: this is how you interpret standardized beta coefficients in regression. These new measures are now ‘unitless’. If you get the covariance of the two standardized scores, that will be the correlation (or r), cov(zx, zy) ## [1] 0.9684438 ### You can compare if we just get compute the correlation using the raw scores cov(zdf) ## x y ## x 1.0000000 0.9684438 ## y 0.9684438 1.0000000 cor(df) ## x y ## x 1.0000000 0.9684438 ## y 0.9684438 1.0000000 The result is the same. We can convert a covariance matrix into a correlation matrix. $cor(x, y) = \frac{cov(x, y)} {sd(x) \times sd(y) }$ You can take the variances from the covariance matrix (the diagonal) and then take the square root and those will be the standard deviations. #check cov(df) ## x y ## x 53.3 27.2 ## y 27.2 14.8 sqrt(53.3) #see diagonal ## [1] 7.300685 sd(df$x)
## [1] 7.300685
sd(df$y) ## [1] 3.847077 So to convert the covariance of 27.2, we divide it by the product of sd(x) and sd(y). 27.2 / (sd(df$x) * sd(df\$y))
## [1] 0.9684438
Think about it: Can you then convert a correlation matrix to a covariance matrix if all you had is the correlation matrix?
Previous
comments powered by Disqus<|endoftext|>
| 4.5 |
447 |
# 4.1 Introduction To Circles Essay
917 Words4 Pages
4.1 Introduction to Circles Name: ________________________ Math 2 Date: ________________________ Standard: Essential Question: Definition: A circle is the set of all points in a plane at a given distance from a given point. A. A circle is named by its center. The circle shown below has center C so it is called circle C. This is symbolized by writing [pic]C. B. Draw a line segment by connecting points C and D in the circle above. The segment you have drawn is called a radius. The plural of radius is radii. Name three other radii of the circle. Be sure to use the correct notation for a line segment: ______ , ______ , ______ Note: one endpoint of the radius is the center of the circle and the other endpoint is a point on the circle. Also, all radii of a circle are congruent. So, if AC = 2 cm, then CE = _____ cm. C. Now, make another segment by connecting points A and E. The segment you have drawn is called a chord. Name five other chords of the circle. ______ , ______ , ______ , ______ , ______ Note: both endpoints of a chord are points on the circle. Chords of a circle do not necessarily have the same length. D. If a chord passes through the center of a circle it is given a special name. It is called a diameter. Name the diameter pictured in [pic]C. _________ Now, draw that diameter. Note: a diameter is the longest chord of a circle. Its length is twice that of the radius. So, if AC = 2 cm, then AB = _____ cm. E. Next, draw a line passing through points B and F. The line you have drawn is called a tangent. A tangent lies in the plane of the circle and intersects the circle in only one point. The point of intersection is called the point of tangency. What is the point of tangency for [pic]? ________ F. Now, draw the line passing through points A and F. This line is called a secant. Any<|endoftext|>
| 4.75 |
1,535 |
This manual describes
the main functions and features of snow cover, and a procedure for snow survey along a
landscape profile. The activity includes location of survey sites, establishment of snow
exploring shaft, determination of snow layers according to a number of visual
characteristics, their measurement and description.
This field study has instructional video
featuring real students conducting the ecological field techniques in nature. Each video
illustrates the primary instructional outcomes and the major steps in accomplishing the
task including reporting the results.
One of the most important landscape characteristics in winter are the properties of
snow cover Ц its thickness and density, as well as depth of the frost zone at
different sites. It is well known that preservation of seeds and sprouts from winterkill
as well as the wintering success of many animal species depends on the depth of the soil
Depth of the seasonal frost zone is of great importance, as it influences
peculiarities of spring soil erosion causing destruction of soil structure. Unfortunately,
the study of permafrost is complicated by its laboriousness and use of specific equipment,
for example, the soil probe. Thus, this educational activity will be focused on snow Ц
i.e. peculiarities of its distribution along the relief forms and under different
vegetation types, structure of snow cover and study of snow's role in landscape function.
The activity is aimed at revealing the dependence of the thickness and structure
of snow cover upon relief forms and vegetation type. It is known that snow
cover is thinner and distributed unevenly in coniferous forests.
In contrast, the distribution of snow in deciduous forests is thicker and more evenly
distributed. In comparing a forest and an open site, it turns out that wind is much
lighter in forests than in open areas, so snow is not blown off the soil surface. Thus,
snow cover in forests is characterized by a more even distribution than in the field, for
instance, where wind differentiates thickness of the snow cover by stripping rises and
filling up relief depressions with snow.
In order to complete the task, students will require shovels (for unearthing snow),
rulers (it is recommended to use long rulers Ц for the full depth of the snow cover) or
measuring tapes, description forms (soil description forms can be used) or field logs,
compasses and some other materials at hand (sticks, blades and matches).
The teacher should explain the main functions and properties of the snow cover before
the activities, and students should be taught how to measure and describe the snow cover
prior to independent studies.
Main functions of snow cover
The role of snow in the function of an environment is especially obvious when air
temperature falls below zero degrees centigrade. In areas where there is no snow in cold
seasons, soil gets frozen through for many meters, and, for instance, in Yakutia, the
depth of permafrost makes up one and a half kilometer.
The preservative function of snow or thermal insulation is, perhaps, the most
important one. It is known that snow cover has loose structure due to the different shapes
of snowflakes. Interstices among snowflakes are filled up with air that is characterized
by low heat conductivity, so we owe such a wonderful property of snow to
air. The air, as we sometimes (but incorrectly) say, Уwarms up well.Ф
Due to the low heat conductivity of snow, day-to-day temperature variation penetrates
into the snow cover only for a depth of 24 centimeters on average. As specific research
has shown, if the amplitude of temperature fluctuation reaches 30 degrees at the snow
surface, then at 5 cm depth it amounts to only 16 degrees, at 24 cm depth it makes up 3
degrees and at 44 cm depth the amplitude is insignificant Ц 0.8 degrees.
Not just soils are protected from frost penetration due to snow as mentioned above, but
plants as well, which can stay green through the winter. They often serve as the only food
for animals, and harbor seeds, which serve as security for renewal of the vegetation cover
However, snow is known not only for its heat insulating function. As snow is water in
solid aggregative state, it accumulates in large quantities and remains until
spring in order to water the earth and allow plants to start growing when it becomes warm.
Besides heat insulating and accumulating functions, snow cover also exerts influence
upon a climate.
Everyone knows that air masses move around the earth's surface. Coming from remote
areas, they bring along characteristics of the area they originated from Ц mainly air humidity
and temperature. As they move above the surface continuously, they change slightly;
however, if they stagnate in one area for a certain time, they acquire temperature and
moisture characteristics of the given region.
It is also known that air cannot be heated directly by the sun. Sunlight is absorbed by
a surface and then the surface gives heat back to the air in the form of infrared
radiation. Some solar radiation is reflected by the surface in the form of light and is
not transformed into heat. The less is the surface reflection power and the higher its
absorption capacity, the warmer the surface becomes. Thus, surface temperature depends on
its reflection properties, or albedo.
The albedo of a black body is equal to 0%, whereas the albedo of a white body is equal
to 100%. Fresh-fallen snow is very close to 100% according to its albedo. Correspondingly,
when the ground is covered with snow, the ground cannot warm up air ...
This was only the first page from the manual and its full version you can see in the
Ecological Field Studies 4CD Set:
It is possible to purchase the complete set of 40 seasonal Ecological Field
Study Materials (video in mpg + manuals in pdf formats) in an attractive 4 compact disk set.
These compact disks are compatible with Mac and PC computers.
The teacher background information and manuals can be printed out for easy reference.
The videos are suitable for individual student or whole class instruction. To purchase the complete 4CD set
write to [email protected] in a free form.
Some of these manuals you can also purchase in the form of applications for Android devices on
Ecological Field Studies Demo Disk:
We also have a free and interesting demonstration disk that explains our ecological field studies approach.
The demo disk has short excerpts from all the seasonal field study videos as well as sample text from all the teacher manuals.
The disk has an entertaining automatic walk through which describes the field study approach and explains how field studies meet education standards.
You can also download the Demo Disc from ecosystema.ru/eng/eftm/CD_Demo.iso.
This is a virtual hybrid (for PC and Mac computers) CD-ROM image (one 563 Mb file "CD_Demo.iso").
You can write this image to the CD and use it in your computer in ordinary way.
You also can use emulator software of virtual CD-ROM drive to play the disk directly from your hard disk.
Other Ecological Field Studies Instructive<|endoftext|>
| 3.671875 |
1,014 |
Celebrations are under way this week in Indonesia. The country is looking back on 64 years since founding president Sukarno and vice-president Mohammed Hatta proclaimed independence from the Netherlands.
The 1945 declaration took place in extraordinary circumstances. It was the first breakout from the colonial system since the Philippines’ short-lived independence from 1898 to 1901.
Japan had occupied the Dutch East Indies in 1942 – but now the Japanese had surrendered and the Dutch were in no position to immediately re-establish their colonial control.
Nationalist leaders Sukarno and Hatta were kidnapped by militant youths desperate to seize the hour and persuade them to strike while the iron was hot.
The declaration of independence was read with very little ceremony and no fanfare.
Word spread quickly across the islands of Java and Sumatra and a tremendous surge of anti-colonial nationalist sentiment took place. The Republic of Indonesia was certain to have the backing of the masses here, on the two largest of the thousands of islands that made up the new country.
What the militants could not have known was that decisions were being taken elsewhere that would very soon impact on the republic.
In Ceylon (now Sri Lanka), then part of the British Empire, the Southeast Asia Command under Admiral Louis Mountbatten was preparing an intervention in Indonesia.
The plan had two immediate purposes. First, to disarm and demobilise the thousands of Japanese occupation troops left in Indonesia at the surrender.
Second, to locate and secure the large number of prisoners of war and civilian internees in the Japanese camps in Java and Sumatra, where they had been held for more than three years in dreadful conditions.
The British had been planning a return to their own colonial possessions in southeast Asia, principally Burma, Malaya, the Straits Settlements (Singapore, Penang, Malacca) and British North Borneo (now Sabah, Malaysia), since the Japanese had overrun them early in 1942.
Plans had been drawn up for re-entry under the umbrella of a British Military Authority (BMA). Nothing was to be allowed to stand in its way, least of all nationalist aspirations.
The rubber and tin of colonial Malaya were so vital to Britain’s economy that the British would stop at nothing to restore these territories to its control.
The BMA landed in Singapore in the first week of September 1945 and Mountbatten moved down from Ceylon.
Under his direction a Royal Navy vessel, HMS Cumberland, arrived in Jakarta Bay. Before long British and Indian troops were disembarking in Java.
The BMA refused to recognise the authority of the Republican government. For its part the new government appealed to the newly constituted United Nations for recognition.
What makes this situation particularly unusual is that the so-called “founding fathers” of Indonesia had no concrete plans for an army.
It is thus a bitter irony of post‑independence Indonesia that the armed forces should grow to be such a powerful and sinister influence, most notably in the mass killings that took place in the anti-leftist bloodbath of 1965-66.
To begin with there was no obvious sign that the British intended to restore Dutch colonial rule.
But the British and Indian troops found themselves confronting the masses in Java and Sumatra and the Dutch could play piggy-back as they prepared to attempt restoration of colonial rule.
Dutch prisoners ignored Mountbatten’s appeals to remain in the prison camps until British and Indian troops arrived to secure them, and began to appear on the streets of Java’s big cities – Jakarta, Bandung, Surabaya and Semarang in particular – as well as Medan in Sumatra.
Very soon violent confrontations took place between the returning colonialists and the Indonesian nationalists – particularly the irregular youth militias that had begun acquiring Japanese arms by one means or another.
The BMA’s determination not to deal with the Republican government hardly facilitated the task of reaching the camps.
The camps came under siege from the militias, thus putting the lives of the more than 150,000 prisoners, of various nationalities, under a much greater threat.
The broad context is simple to grasp. The colonial powers were hell-bent on restoring a broad arc of rule in southeast Asia and for them the colonised peoples were to have no say in the matter.
The colonised peoples had different ideas, and in Indonesia this meant 14 months of bitter fighting in a conflict which took the lives of more than 700 on the British side and thousands of Indonesians.
By the end of 1949 the colonialists were forced to accept defeat and recognise Sukarno’s government. Unfortunately the imperial powers continued to meddle in Indonesian affairs after this.
David Jardine lives in Bogor, West Java and is the author of Foreign Fields Forever: Britain’s Forgotten War In Indonesia 1945-46, which is available from [email protected]<|endoftext|>
| 3.765625 |
402 |
stORytime: every day. everywhere.
It's built on the understanding that parents are a child's first teacher and they can grow the skills needed to become successful in school through these simple, every day activities:
(and we would throw in WRITING)
When parents interact with their children in fun and meaningful ways, learning happens!
I like to compare the act of raising a reader with growing a flower. 4 basic needs must be met if the flower is going to grow:
Think of the SUN as TALKING. If you surround your child with the warm glow of words, stories and conversations, their vocabularies will bloom, reaching always higher and higher.
Think of the WATER as SINGING. If you feed a melodious stream of song to your child, they will grow to respond to the rhythms and sounds that make up our language.
Think of the AIR as WRITING. If you draw and engage in fingerplays with your child, they will build the motor skills needed to put their own thoughts into the shape of the written word.
Think of the SOIL as READING. If you plant your child firmly in a ground of reading, they will grow rooted to a world of books and learning.
There is one last ingredient that keeps flowers blooming year in, year out: BEES!
Think of the BEES as PLAYING. Through a steady pollination of play, children go from being potential readers to actual readers. Children must enjoy reading and see its benefits firsthand. It is critical that we keep all early learning activities buzzing with fun!
If you follow these 5 basic practices in your daily interactions with your children, you will help them blossom into beautiful readers and eager learners!
We encourage you to learn more about Oregon's stORytime initiative by visiting them on Facebook, Twitter and YouTube.
Also, check out this activity sheet for some fun ideas and download this bookmark.<|endoftext|>
| 3.765625 |
965 |
# Arithmetic: answer key to the exercises in the final year of high school in PDF.
Exercise of arithmetic in terminale S.
1-Position d = gcd(a,b)
We have if d divides a and d divides b then d divides b and d divides (a-bq)
Reciprocally: if d divides b and d divides (a-bq) then d divides ( a – bq ) +bq = a
2- it is the previous relationship with .
Demonstrate a property, exercise of mathematics in terminale S on arithmetic.
Let δ = PGCD(a ;b) and µ = PPCM(a ;b).
Then we have a= δa’ and b = δb’ with a’ and b’ prime between them.
So we have PPCM(a’ ;b’) = a’b’
µ = PPCM(δa’; δb’) = δ×PPCM(a’; b’) = δ×a’×b’
Thus δµ = δ²×a’×b’ = δ×a’× δ×b’ = ab
System of equations and arithmetic.
From the relation PPCM(a;b)×PGCD(a;b) = ab,
we deduce from the second equation of the system :
3×PPCM(a ;b) = PGCD(a ;b)×PPCM(a ;b)
So PGCD(a;b) = 3
Then there are integers a’ and b’ (prime to each other) such that a = 3a’ and b = 3b’.
Carrying over into the first equation, we get :
9a’² – 9b’² = 405
Let (a’ + b’)(a’ – b’) = 45
Now 45 = 3²×5
We can therefore have the following 3 systems:
a’ = 23 and b’ = 22 a’ = 9 and b’ = 6 a’ = 7 and b’ = 2
Hence (a ;b) = (66 ;69) is not suitable because hence (a ;b) = (42 ;6)
9 and 6 are not prime to each other
The solution pairs of the system are therefore (42 ;6) and (66 ;69).
Prime numbers .
1)
2)
because a and b are greater than or equal to 2.
is the product of two integers greater than 1.
So is not a prime number.
Exercise:
Exercise:
Exercise:
Exercise:
Exercise:
Exercise:
Exercise:
Exercise:
Exercise:
Exercise:
Exercise:
## The exercises in the final year
After having consulted the answer key of this exercise arithmetic in senior year, you can return to the exercises in senior year
Cette publication est également disponible en : Français (French) Español (Spanish) العربية (Arabic)
You have the possibility to download then print this document for free «arithmetic: answer key to the exercises in the final year of high school in PDF.» in PDF format.
## Other forms similar to arithmetic: answer key to the exercises in the final year of high school in PDF..
• 93
The answer key to the math exercises in 1st grade on the barycenter n weighted points. Use the stability and associativity properties of the barycenter in first grade. Exercise 1: It is up to you to make these constructions knowing that the barycenter is necessarily aligned with points A and…
• 91
The answer key to the math exercises on vectors in 2nd grade. Know how to use the Chasles relation and demonstrate that vectors are collinear in second grade. Exercise 1: Are the points P, Q and R aligned? Yes they are aligned, show that the vectors and are collinear .…
• 89
Exercise of mathematics in class of sixth (6eme) on the decimal numbers.Determine the positions of a number and conversions of units of magnitude. Exercise 1: Place spaces in the following numbers: a. 1 512 b. 63 829 c. 468 803 576 Exercise 2: a. In 13, the number of units…
Les dernières fiches mises à jour.
Voici les dernières ressources similaires à arithmetic: answer key to the exercises in the final year of high school in PDF. mis à jour sur Mathovore (des cours, exercices, des contrôles et autres), rédigées par notre équipe d'enseignants.
On Mathovore, there is 13 703 967 math lessons and exercises downloaded in PDF.
Categories Uncategorized
Mathovore
FREE
VIEW<|endoftext|>
| 4.71875 |
5,873 |
-->
# Section notes: Linear regression¶
Kate Shulgina and June Shin (10/26/2018)
In this section, we will cover first some explanation of linear regression and fitting, and then go through some code examples in Python.
### Linear regression as a predictive model¶
In the lecture notes, Sean covers linear regression from the perspective of a generative probabilistic model. While this is a good framework for figuring out how to write out the assumptions of the model explicitly, there are also other perspectives.
One perspective that helped me personally understand linear regression is the perspective of linear regression as a predictive model. Think of linear regression as the process of building a model for the purposes of using later to make predictions.
Let's pretend that we make a model that can predict what grade a student will get on an exam. We have various information on the past students in the class, like how many hours they studied total, how many hours they slept the night before, their age, etc etc, alongside known exam scores.
As a first step, it might make sense to see if there is a predictive relationship between the number of hours studied and the exam score. This is what the relationship looks like:
Our goal is to build a predictive model such that if we get a new student, and know the number of hours they slept, we could predict what grade they would get. One way might be to fit a line. From high school algebra, the equation of a line is $y=mx+b$. In this case, the equation is $y=\beta_1 x + \beta_0$, where $y=\text{exam grade}$, $x=\text{hours studied}$, $\beta_1$ is the slope and $\beta_0$ is the y-intercept.
We could imagine various different fits to the data-- some good and some not so good. We get these different fits by varying the parameter $\beta_1$ ($\beta_0$ is held at 0 to keep this example simple)
How can we compare the different fits and pick the best one? One common method is to compare the sum of squared residuals (RSS). A residual ($e_i$) can be viewed as the prediction error-- for a given student, how far off was the prediction from the fitted line? Good fits will have low prediction error and a low RSS.
\begin{aligned} e_i &= y_i - \hat y_i \\ &= y_i - (\beta_1 x_i + \beta_0) \end{aligned}$$RSS = \sum_{i=1}^n (y_i - \beta_1 x_i - \beta_0)^2$$
Let's try to visualize how the RSS changes for the different values of $\beta_1$ plotted above. You could imagine scanning across all possible values of $\beta_1$ and getting a sense of how the RSS changes with the parameter.
The value of $\beta_1$ that will give us the best fit is the one that has the lowest RSS, so we are trying to minimize the RSS, also known as the Least Squares Error. The function being minimizing has a lot of names in statistics and machine learning-- you might hear it called the objective function, the error function, the loss function, etc. They are all the same thing with different jargon-y names. We can use an optimizer like the scipy.optimize.minimize function to find the value of $\beta_1$ that gives the lowest value for the RSS, and therefore the best predictive line.
### Multiple regression¶
Suppose we find the best fit line, and realize that using number of hours studied just isn't that good at predicting the exam score. We have other data at our disposal, like the number hours slept the night before the exam, that we could use in our model to improve our predictions.
Let's expand to two predictor variables: $x_1 = \text{hours studied}$ and $x_2 = \text{hours slept}$
And we are still predicting one variable: $y=\text{exam grade}$
Now we can imagine fitting a plane that given some values for $x_1, x_2$ can predict the student's exam score.
A plane fit to this data would have the equation $\hat y = \beta_2 x_2 + \beta_1 x_1 + \beta_0$. In the machine learning literature, they like to write things in compact matrix notation, so don't get scared if you see something like this $\hat y = \vec \beta \vec x$. Here, $\vec \beta = [ \beta_0, \beta_1, \beta_2 ]$ and $\vec x = [ 1, x_1, x_2 ]$. There is an extra 1 in $\vec x$ such that if you take the dot product of the two vectors, you get the expanded out equation that we started with.
Just like in the case with one predictor $x$, we can compute the prediction error (the residuals) by subtracting the observed $y$ from the one predicted by the plane.
$$RSS = \sum_{i=1}^n (y_i - \beta_2 x_{i2} - \beta_1 x_{i1} - \beta_0)^2$$
If we plugged the RSS into an optimizer, it would scan over the 3D range of possible $\beta_0, \beta_1, \beta_2$ values and find the ones that minimize the error.
You can imagine expanding this out to include any number of predictor variables. You can have predictor variables that are columns of data that you have, or even functions of data columns like $x_i = \sqrt{x_1}$ (in machine learning speak, these are called basis functions).
However, one thing to watch out for when adding more and more predictor variables is overfitting. This is when your model fits the training data so well, minimizes the errors so nicely, that it actually can't generalize and performs poorly on data was hasn't been previously seen. If it can't generalize, then it's a bad predictor.
There's a large amount of thought on how to address this issue, and many rely on evaluating how good the model is on some held out test data.
### Probabilistic framework for linear regression¶
The greatest strength of the probabilitic framework for linear regression that Sean brought up in class is that in defining the probability model, you are explicitly stating many of the assumptions that are unspoken in least squares fitting. Let's review it briefly.
In this way of thinking about linear regression, we are proposing that there is some absolute true linear relationship between $x$ and $y$, but the problem is that we make observations from the real world which is noisy so the observed points don't exactly fall on the line, but scatter around it with some noise.
$$y_i = \beta_1 x_i + \beta_0 + e_i$$
The "noise" in the observations is the error between the true line and the observations, which are the residuals. We model the noise as a random variable, usually Normally distributed. Our goal is to infer the parameters $\vec \beta$ of the original line that make the points scatted by noise as least unlikely as possible.
A common modelling decision is to model the residuals as being distributed $e_i \sim \mathcal{N}(0, \sigma^2)$, a Normal distribution with the same variance across all points. Most of the time, this assumption isn't even considered or questioned! You should always ask yourself if these are reasonable assumptions for your data? You'll see in the homework that this is a critical mistake that Moriarty makes. In the lecture notes, Sean shows that the maximum likelihood estimate for the parameters with this assumption is equivalent to least squares fitting-- so Normally-distributed noise is implicitly assumed in least squares!
Since $y_i$ is a function of a random variable ($e_i$), it is also a random variable with a Normal distirbution centered around the true line.
$$y_i | x_i, \vec \beta, \sigma^2 \sim \mathcal{N}(\beta_1 x_i + \beta_0, \sigma^2)$$
The likelihood of some particular values of the parameters $\vec \beta$ and $\sigma$ is the probability of all of the observed data under that model. We use the Normal PDF to get the probability of the $y_i$s
\begin{aligned} \text{likelihood}(\vec \beta, \sigma^2) &= P(y_1, ..., y_n | \vec \beta, \sigma^2) \\ &= \prod_{i=1}^n P(y_i | x_i, \vec \beta, \sigma^2) \\ &= \prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2}} e^{\frac{(y_i - \beta_1x_i -\beta_0)^2}{2\sigma^2}} \end{aligned}
Now we can compare how well different lines fit the data by evaluating the likelihood of those parameters. To find the best values, we can feed this equation into an optimizer function.
Optimizers have been standardized to be minimizers instead of maximizers, so we need to feed it the negative log-likelihood instead. In this case, it would be
$$\text{NLL} = - \sum_{i=1}^n \bigg( \text{log} \big( \frac{1}{\sqrt{2\pi\sigma^2}} \big) + \frac{(y_i - \beta_1x_i -\beta_0)^2}{2\sigma^2} \bigg)$$
I think one of the nice parts of this model, is that you can naturally get a "confidence interval" that's interpretable as a probability, directly from the Normal distribution of each $y_i$. You can imagine overlaying onto the linear regression line a band representing where you would expect 95% of your observed points to scatter around the line.
### Example demonstrating the flexibility of the probabilistic approach¶
Let's walk through one last example together that hopefully will help you think through the homework assignment.
Here is our dataset.
What we have plotted here is income of various people as a function of their age. What we want to do is fit a predictive line to this data-- maybe we are an advertising company and we want to make better targetted ads, for instance.
What is unusual about this dataset? The spread of the income increases with age (this might make sense-- not a lot of 25 year olds have had the time to become CEOs). This effect goes by the jargon-y term heteroskedasticity.
What does this trend mean with regards to least squares fitting? What are the assumptions implicit in least squares fitting? Does this data fit those assumptions?
What could we do about this issue? There are several options, including transforming the $y_i$s into a different space, or estimating the individual $\sigma^2_i$s separately.
In our case, let's suppose the variance is directly proportional to age, so we can define a relationship $\sigma_i^2 = a x_i \sigma^2$.
How would we write the distribution of the $y_i$s now?
$$y_i | x_i, \vec \beta, \sigma \sim \mathcal{N}(\beta_1 x_i + \beta_0, a x_i \sigma^2)$$
Then the likelihood would be the same, except of the sigma term.
$$\text{likelihood}(\vec \beta, \sigma) = \prod_{i=1}^n \frac{1}{\sqrt{2\pi a x_i \sigma^2}} e^{\frac{(y_i - \beta_1x_i -\beta_0)^2}{2 a x_i \sigma^2}}$$
See how easy it was to incorporate a new assumption of the model into the probabilistic framework for linear regression? Now we can easily find the best fit linear regression line by minimizing the negative log-likelihood with respect to the $\vec \beta$ and $\sigma$ using an optimizer. If we were trying to modify least squares regression, it would not be as clear how to correctly modify the RSS function to incorporate some variation in how the residuals are distributed.
### Now let's code it up!¶
In [1]:
import numpy as np
import pandas as pd
import scipy.stats as stats
import scipy.optimize as optimize
import matplotlib.pyplot as plt
% matplotlib inline
##### We are going to start by doing a basic linear regression fit:¶
$$y = mx + b$$
In [2]:
x = [1.52, 3.91, 2.23, 4.99, 5.81, 6.79, 6.94, 8.81, 9.02, 9.78]
y = [11.04, 8.86, 7.52, 8.43, 6.01, 5.71, 3.95, 3.66, 1.21, 0.47]
In [3]:
plt.plot(x, y, 'o')
Out[3]:
[<matplotlib.lines.Line2D at 0x11aba5400>]
In [4]:
def predict_yhat(x, m, b):
yhat = m*x + b
return yhat
##### We're going to use the L2 norm to minimize and fine the line of best fit: this is also known as the sum of square residuals¶
$$\Sigma_i (\hat{y_i} - y_i)^2$$
##### We can calculate these residuals directly and minimize this sum¶
In [5]:
def calculate_L2norm(params, x, y):
L2 = 0
for i in range(len(x)):
yhat = predict_yhat(x[i], params[0], params[1])
residual = yhat - y[i]
L2 += residual ** 2
return L2
In [6]:
guess=np.array([1.,1.])
In [7]:
minimization = optimize.minimize(calculate_L2norm,guess,(x,y))
m = minimization.x[0]
b = minimization.x[1]
x_pred = np.arange(0, 10., 0.1)
y_pred = predict_yhat(x_pred, m, b)
In [8]:
plt.plot(x, y, 'o')
plt.plot(x_pred, y_pred, '-')
Out[8]:
[<matplotlib.lines.Line2D at 0x11a8e7940>]
##### We can also calculate the likelihood of the data given the predicted parameters by calculating the probability of the residual from a normal distribution¶
In [9]:
def calculate_nll(params, x, y):
nll=0.
# Loop through each time point to get a total negative log likelihood for the gene.
for i in range(len(x)):
yhat = predict_yhat(x[i], params[0], params[1])
resid = yhat - y[i]
nll -= stats.norm.logpdf(resid, loc=0, scale=1)
return nll
In [10]:
minimization = optimize.minimize(calculate_nll,guess,(x,y))
m = minimization.x[0]
b = minimization.x[1]
x_pred = np.arange(0, 10., 0.1)
y_pred = predict_yhat(x_pred, m, b)
In [11]:
plt.plot(x, y, 'o')
plt.plot(x_pred, y_pred, '-')
Out[11]:
[<matplotlib.lines.Line2D at 0x11ab65518>]
##### We can do the same, but now pull from a skewnormal distribution (weighted more by direction above or below line) or a lognormal distribution (weighted by amount of variance with respect to the mean)¶
In [12]:
def calculate_nll_skewnormal(params, x, y):
nll=0.
# Loop through each time point to get a total negative log likelihood for the gene.
for i in range(len(x)):
yhat = predict_yhat(x[i], params[0], params[1])
resid = yhat - y[i]
nll -= stats.skewnorm.logpdf(resid, a=2, loc=0, scale=1)
return nll
In [13]:
minimization = optimize.minimize(calculate_nll_skewnormal,guess,(x,y))
m = minimization.x[0]
b = minimization.x[1]
x_pred = np.arange(0, 10., 0.1)
y_pred = predict_yhat(x_pred, m, b)
In [14]:
plt.plot(x, y, 'o')
plt.plot(x_pred, y_pred, '-')
Out[14]:
[<matplotlib.lines.Line2D at 0x11adfc550>]
In [15]:
def calculate_nll_lognormal(params, x, y):
nll=0.
# Loop through each time point to get a total negative log likelihood for the gene.
for i in range(len(x)):
yhat = predict_yhat(x[i], params[0], params[1])
resid = np.log(yhat) - np.log(y[i])
nll -= stats.norm.logpdf(resid, loc=0, scale=1)
return nll
In [16]:
minimization = optimize.minimize(calculate_nll_lognormal,guess,(x,y))
m = minimization.x[0]
b = minimization.x[1]
x_pred = np.arange(0, 10., 0.1)
y_pred = predict_yhat(x_pred, m, b)
In [17]:
plt.plot(x, y, 'o')
plt.plot(x_pred, y_pred, '-')
Out[17]:
[<matplotlib.lines.Line2D at 0x11af380f0>]
##### Make it Bayesian (add a prior)!¶
In [18]:
def calculate_nll(params, x, y):
nll=0.
# Loop through each time point to get a total negative log likelihood for the gene.
for i in range(len(x)):
yhat = predict_yhat(x[i], params[0], params[1])
resid = yhat - y[i]
nll -= stats.norm.logpdf(resid, loc=0, scale=1) + stats.norm.logpdf(params[0], loc=-1, scale=1) + stats.norm.logpdf(params[1], loc=10, scale=5)
return nll
In [19]:
minimization = optimize.minimize(calculate_nll,guess,(x,y))
m = minimization.x[0]
b = minimization.x[1]
x_pred = np.arange(0, 10., 0.1)
y_pred = predict_yhat(x_pred, m, b)
In [20]:
plt.plot(x, y, 'o')
plt.plot(x_pred, y_pred, '-')
Out[20]:
[<matplotlib.lines.Line2D at 0x11ae99d68>]
##### We can make the functions more complicated and easily get polynomial regression¶
In [21]:
def predict_yhat(x, m1, m2, m3, m4, b):
yhat = m1*x + m2*(x**2) + m3*(x**3) + m4*(x**4) + b
return yhat
In [22]:
def calculate_nll(params, x, y):
nll=0.
# Loop through each time point to get a total negative log likelihood for the gene.
for i in range(len(x)):
yhat = predict_yhat(x[i], params[0], params[1], params[2], params[3], params[4])
resid = yhat - y[i]
nll -= stats.norm.logpdf(resid, loc=0, scale=1)
return nll
In [23]:
guess=np.array([1.,1.,1.,1.,1.])
In [24]:
minimization = optimize.minimize(calculate_nll,guess,(x,y))
m1 = minimization.x[0]
m2 = minimization.x[1]
m3 = minimization.x[2]
m4 = minimization.x[3]
b = minimization.x[4]
x_pred = np.arange(0, 10., 0.1)
y_pred = predict_yhat(x_pred, m1, m2, m3, m4, b)
In [25]:
plt.plot(x, y, 'o')
plt.plot(x_pred, y_pred, '-')
Out[25]:
[<matplotlib.lines.Line2D at 0x11b1f3518>]
##### But be careful of overfitting!¶
In [26]:
def predict_yhat(x, m):
yhat = 0
for i in range(len(m)):
yhat += m[i]*(x**i)
return yhat
In [27]:
def calculate_nll(params, x, y):
nll=0.
# Loop through each time point to get a total negative log likelihood for the gene.
for i in range(len(x)):
yhat = predict_yhat(x[i], params)
resid = yhat - y[i]
nll -= stats.norm.logpdf(resid, loc=0, scale=1)
return nll
In [28]:
guess=np.array([1.]*6)
In [29]:
minimization = optimize.minimize(calculate_nll,guess,(x,y))
m = minimization.x
x_pred = np.arange(0, 10., 0.1)
y_pred = predict_yhat(x_pred, m)
In [30]:
plt.plot(x, y, 'o')
plt.plot(x_pred, y_pred, '-')
Out[30]:
[<matplotlib.lines.Line2D at 0x11b3e53c8>]
##### We can also do multivariate linear regression as well: if our predictor variable has more than one dimension:¶
In [31]:
x = [[1.52, 7.64, 10.87], [3.91, 6.64, 5.04], [2.23, 5.22, 3.23],
[4.99, 6.75, 5.91], [5.81, 4.07, 2.27], [6.79, 4.17, 1.47],
[6.94, 4.05, 0.44], [8.81, 3.97, -0.21], [9.02, 2.21, -1.04], [9.78, 2.07, -2.33]]
y = [11.04, 8.86, 7.52, 8.43, 6.01, 5.71, 3.95, 3.66, 1.21, 0.47]
In [32]:
def predict_yhat(x, m0, m1, m2, b):
yhat = x[0]*m0 + x[1]*m1 + x[2]*m2 + b
return yhat
In [33]:
def calculate_nll(params, x, y):
nll=0.
# Loop through each time point to get a total negative log likelihood for the gene.
for i in range(len(x)):
yhat = predict_yhat(x[i], params[0], params[1], params[2], params[3])
resid = yhat - y[i]
nll -= stats.norm.logpdf(resid, loc=0, scale=1)
return nll
In [34]:
guess=np.array([1.,1.,1.,1.])
In [35]:
minimization = optimize.minimize(calculate_nll,guess,(x,y))
m0 = minimization.x[0]
m1 = minimization.x[1]
m2 = minimization.x[2]
b = minimization.x[3]
y_pred = [predict_yhat(x_pred, m0, m1, m2, b) for x_pred in x]
In [36]:
plt.plot(y, y_pred, 'o')
Out[36]:
[<matplotlib.lines.Line2D at 0x11b4c9908>]
##### We can make the predictions categorical! This is called logistic regression, one way linear regression can be extended to generalized linear models¶
In [37]:
x = [1.52, 3.91, 2.23, 4.99, 5.81, 6.79, 6.94, 8.81, 9.02, 9.78]
y = [0, 0, 0, 0, 0, 1, 1, 1, 1, 1]
In [38]:
plt.plot(x, y, 'o')
Out[38]:
[<matplotlib.lines.Line2D at 0x11b6d2128>]
In [39]:
def predict_yhat(x, m, b):
yhat = 1/(1 + np.exp(-(m*x + b)))
return yhat
In [40]:
def calculate_crossent(params, x, y):
cross_entropy_logit = 0
for i in range(len(x)):
yhat = predict_yhat(x[i], params[0], params[1])
#print(yhat)
cross_entropy_logit += (-1 * y[i]) * np.log(yhat) - (1 - y[i]) * np.log(1 - yhat)
return cross_entropy_logit
In [41]:
minimization = optimize.minimize(calculate_crossent,guess,(x,y))
m = minimization.x[0]
b = minimization.x[1]
x_pred = np.arange(0, 10., 0.1)
y_pred = predict_yhat(x_pred, m, b)
/Users/jung-eunshin/anaconda/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: divide by zero encountered in log
/Users/jung-eunshin/anaconda/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in double_scalars
/Users/jung-eunshin/anaconda/lib/python3.6/site-packages/ipykernel_launcher.py:2: RuntimeWarning: overflow encountered in exp
/Users/jung-eunshin/anaconda/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: divide by zero encountered in log
/Users/jung-eunshin/anaconda/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in double_scalars
/Users/jung-eunshin/anaconda/lib/python3.6/site-packages/ipykernel_launcher.py:2: RuntimeWarning: overflow encountered in exp
/Users/jung-eunshin/anaconda/lib/python3.6/site-packages/ipykernel_launcher.py:2: RuntimeWarning: overflow encountered in exp
/Users/jung-eunshin/anaconda/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: divide by zero encountered in log
/Users/jung-eunshin/anaconda/lib/python3.6/site-packages/ipykernel_launcher.py:6: RuntimeWarning: invalid value encountered in double_scalars
In [42]:
plt.plot(x, y, 'o')
plt.plot(x_pred, y_pred, '-')
Out[42]:
[<matplotlib.lines.Line2D at 0x11b63c240>]<|endoftext|>
| 4.65625 |
4,265 |
Many of the provisions of our constitution has been borrowed from the Government of India Act of 1935 as well as from the constitution of various other countries that includes USSR, France, Japan, Germany and many more. The fundamental rights as described in Articles 12 – 35 of Constitution of India constitute the philosophical part of the constitution and are inspired from the American constitution. These rights are the basic human rights and apply to every citizen of India irrespective of religion, colour, sex, birth place, race or caste. They guarantee development of human personality.
In this Part, unless the context otherwise requires, “the State” includes the Government and Parliament of India and the Government and the Legislature of each of the States and all local or other authorities within the territory of India or under the control of the Government of India.
Laws inconsistent with or in derogation of the fundamental rights
1. All laws in force in the territory of India immediately before the commencement of this Constitution, in so far as they are inconsistent with the provisions of this Part, shall, to the extent of such inconsistency, be void.
2. The State shall not make any law which takes away or abridges the rights conferred by this Part and any law made in contravention of this clause shall, to the extent of the contravention, be void.
3. In this article, unless the context otherwise requires,-
a. “law” includes any Ordinance, order, bye-law, rule, regulation, notification, custom or usage having in the territory of India the force of law;
b. “laws in force” includes laws passed or made by a Legislature or other competent authority in the territory of India before the commencement of this Constitution and not previously repealed, notwithstanding that any such law or any part thereof may not be then in operation either at all or in particular areas.
4. Nothing in this article shall apply to any amendment of this Constitution made under article 368.
Right to Equality
Equality before law
The State shall not deny to any person equality before the law or the equal protection of the laws within the territory of India.
Prohibition of discrimination on grounds of religion, race, caste, sex or place of birth
1. The State shall not discriminate against any citizen on grounds only of religion, race, caste, sex, place of birth or any of them.
2. No citizen shall, on grounds only of religion, race, caste, sex, place of birth or any of them, be subject to any disability, liability, restriction or condition with regard to-
a. access to shops, public restaurants, hotels and places of public entertainment; or
b. the use of wells, tanks, bathing ghats, roads and places of public resort maintained wholly or partly out of State funds or dedicated to the use of the general public.
3. Nothing in this article shall prevent the State from making any special provision for women and children.
Equality of opportunity in matters of public employment
1. There shall be equality of opportunity for all citizens in matters relating to employment or appointment to any office under the State.
2. No citizen shall, on grounds only of religion, race, caste, sex, descent, place of birth, residence or any of them, be ineligible for, or discriminated against in respect of, any employment or office under the State.
3. Nothing in this article shall prevent Parliament from making any law prescribing, in regard to a class or classes of employment or appointment to an office [under the Government of, or any local or other authority within, a State or Union territory, any requirement as to residence within that State or Union territory] prior to such employment or appointment.
4. Nothing in this article shall prevent the State from making any provision for reservation in matters of promotion or appointments or posts in favour of any backward class of citizens which, in the opinion of the state, is not adequately represented in the service under the state.
(4A) Nothing in this article shall prevent the State from making any provision for reservation in matters of promotion to any class or classes of posts in the services under the State in favour of the Scheduled Castes and the Scheduled Tribes which, in the opinion of the State, are not adequately represented in the services under the State.]
(4B) Nothing in this article shall prevent the State from considering any unfilled vacancies of a year which are reserved for being filled up in that year in accordance with any provision for reservation made under clause (4) or clause (4A) as a separate class of vacancies to be filled up in any succeeding year or years and such class of vacancies shall not be considered together with the vacancies of the year in which they are being filled up for determining the ceiling of fifty per cent. reservation on total number of vacancies of that year.]
5. Nothing in this article shall affect the operation of any law which provides that the incumbent of an office in connection with the affairs of any religious or denominational institution or any member of the governing body thereof shall be a person professing a particular religion or belonging to a particular denomination.
Abolition of Untouchability
“Untouchability” is abolished and its practice in any form is forbidden. The enforcement of any disability rising out of “Untouchability” shall be an offence punishable in accordance with law.
Abolition of titles
1. No title, not being a military or academic distinction, shall be conferred by the State.
2. No citizen of India shall accept any title from any foreign State.
3. No person who is not a citizen of India shall, while he holds any office of profit or trust under the State, accept without the consent of the President any title from any foreign State.
4. No person holding any office of profit or trust under the State shall, without the consent of the President, accept any present, emolument, or office of any kind from or under any foreign State.
Right to Freedom
Protection of certain rights regarding freedom of speech, etc.
1. All citizens shall have the right-
a. to freedom of speech and expression;
b. to assemble peaceably and without arms;
c. to form associations or unions;
d. to move freely throughout the territory of India;
e. to reside and settle in any part of the territory of India;
f. this clause has been omitted by section 2(a)(ii), by the constitution
g. to practise any profession, or to carry on any occupation, trade or business.
2. Nothing in sub-clause (a) of clause (1) shall affect the operation of any existing law, or prevent the State from making any law, in so far as such law imposes reasonable restrictions on the exercise of the right conferred by the said sub-clause in the interests of the sovereignty and integrity of India, the security of the State, friendly relations with foreign States, public order, decency or morality, or in relation to contempt of court, defamation or incitement to an offence.
3. Nothing in sub-clause (b) of the said clause shall affect the operation of any existing law in so far as it imposes, or prevent the State from making any law imposing, in the interests of the sovereignty and integrity of India or public order, reasonable restrictions on the exercise of the right conferred by the said sub-clause.
4. Nothing in sub-clause (c) of the said clause shall affect the operation of any existing law in so far as it imposes, or prevent the State from making any law imposing, in the interests of the sovereignty and integrity of India or public order or morality, reasonable restrictions on the exercise of the right conferred by the said sub-clause.
5. Nothing in sub-clauses (d) and (e) of the said clause shall affect the operation of any existing law in so far as it imposes, or prevent the State from making any law imposing, reasonable restrictions on the exercise of any of the rights conferred by the said sub-clauses either in the interests of the general public or for the protection of the interests of any Scheduled Tribe.
6. Nothing in sub-clause (g) of the said clause shall affect the operation of any existing law in so far as it imposes, or prevent the State from making any law imposing, in the interests of the general public, reasonable restrictions on the exercise of the right conferred by the said sub-clause, and, in particular, nothing in the said sub-clause shall affect the operation of any existing law in so far as it relates to, or prevent the State from making any law relating to:
i. the professional or technical qualifications necessary for practicing any profession or carrying on any occupation, trade or business, or
ii. the carrying on by the State, or by a corporation owned or controlled by the State, of any trade, business, industry or service, whether to the exclusion, complete or partial, of citizens or otherwise.
Protection in respect of conviction for offences
1. No person shall be convicted of any offence except for violation of a law in force at the time of the commission of the Act charged as an offence, nor be subjected to a penalty greater than that which might have been inflicted under the law in force at the time of the commission of the offence.
2. No person shall be prosecuted and punished for the same offence more than once.
3. No person accused of any offence shall be compelled to be a witness against himself.
Protection of life and personal liberty
No person shall be deprived of his life or personal liberty except according to procedure established by law.
Protection against arrest and detention in certain cases
1. No person who is arrested shall be detained in custody without being informed, as soon as may be, of the grounds for such arrest nor shall he be denied the right to consult, and to be defended by, a legal practitioner of his choice.
2. Every person who is arrested and detained in custody shall be produced before the nearest magistrate within a period of twenty-four hours of such arrest excluding the time necessary for the journey from the place of arrest to the court of the magistrate and no such person shall be detained in custody beyond the said period without the authority of a magistrate.
3. Nothing in clauses (1) and (2) shall apply:
(a) to any person who for the time being is an enemy alien; or
(b) to any person who is arrested or detained under any law providing for preventive detention.
4. No law providing for preventive detention shall authorise the detention of a person for a longer period than three months unless-
(a) an Advisory Board consisting of persons who are, or have been, or are qualified to be appointed as, Judges of a High Court has reported before the expiration of the said period of three months that there is in its opinion sufficient cause for such detention:
Provided that nothing in this sub-clause shall authorise the detention of any person beyond the maximum period prescribed by any law made by Parliament under sub-clause (b) of clause (7); or
(b) such person is detained in accordance with the provisions of any law made by Parliament under sub-clauses (a) and (b) of clause (7).
5. When any person is detained in pursuance of an order made under any law providing for preventive detention, the authority making the order shall, as soon as may be, communicate to such person the grounds on which the order has been made and shall afford him the earliest opportunity of making a representation against the order.
6. Nothing in clause (5) shall require the authority making any such order as is referred to in that clause to disclose facts which such authority considers to be against the public interest to disclose.
7. Parliament may by law prescribe-
a. the circumstances under which, and the class or classes of cases in which, a person may be detained for a period longer than three months under any law providing for preventive detention without obtaining the opinion of an Advisory Board in accordance with the provisions of sub-clause (a) of clause (4);
b. the maximum period for which any person may in any class or classes of cases be detained under any law providing for preventive detention;
c. the procedure to be followed by an Advisory Board in an inquiry under sub-clause (a) of clause (4).
Right against Exploitation
Prohibition of traffic in human beings and forced labour
1. Traffic in human beings and beggar and other similar forms of forced labour are prohibited and any contravention of this provision shall be an offence punishable in accordance with law.
2. Nothing in this article shall prevent the State from imposing compulsory service for public purposes, and in imposing such service the State shall not make any discrimination on grounds only of religion, race, caste or class or any of them.
Prohibition of employment of children in factories, etc.
No child below the age of fourteen years shall be employed to work in any factory or mine or engaged in any other hazardous employment.
Right to Freedom of Religion
Freedom of conscience and free profession, practice and propagation of religion
1. Subject to public order, morality and health and to the other provisions of this Part, all persons are equally entitled to freedom of conscience and the right freely to profess, practice and propagate religion.
2. Nothing in this article shall affect the operation of any existing law or prevent the State from making any law:
a. regulating or restricting any economic, financial, political or other secular activity which may be associated with religious practice;
b. providing for social welfare and reform or the throwing open of Hindu religious institutions of a public character to all classes and sections of Hindus.
Explanation I: The wearing and carrying of kirpans shall be deemed to be included in the profession of the Sikh religion.
Explanation II: In sub-clause (b) of clause (2), the reference to Hindus shall be construed as including a reference to persons professing the Sikh, Jaina or Buddhist religion, and the reference to Hindu religious institutions shall be construed accordingly.
Freedom to manage religious affairs
Subject to public order, morality and health, every religious denomination or any section thereof shall have the right:
a. to establish and maintain institutions for religious and charitable purposes;
b. to manage its own affairs in matters of religion;
c. to own and acquire movable and immovable property; and
d. to administer such property in accordance with law.
Freedom as to payment of taxes for promotion of any particular religion
No person shall be compelled to pay any taxes, the proceeds of which are specifically appropriated in payment of expenses for the promotion or maintenance of any particular religion or religious denomination.
Freedom as to attendance at religious instruction or religious worship in certain educational institutions
1. No religious instruction shall be provided in any educational institution wholly maintained out of State funds.
2. Nothing in clause (1) shall apply to an educational institution which is administered by the State but has been established under any endowment or trust which requires that religious instruction shall be imparted in such institution.
3. No person attending any educational institution recognised by the State or receiving aid out of State funds shall be required to take part in any religious instruction that may be imparted in such institution or to attend any religious worship that may be conducted in such institution or in any premises attached thereto unless such person or, if such person is a minor, his guardian has given his consent thereto.
Cultural and Educational Rights
Protection of interests of minorities
1. Any section of the citizens residing in the territory of India or any part thereof having a distinct language, script or culture of its own shall have the right to conserve the same.
2. No citizen shall be denied admission into any educational institution maintained by the State or receiving aid out of State funds on grounds only of religion, race, caste, language or any of them.
Right of minorities to establish and administer educational institutions
1. All minorities, whether based on religion or language, shall have the right to establish and administer educational institutions of their choice.
1A. In making any law providing for the compulsory acquisition of any property of any educational institution established and administered by a minority, referred to in clause (1), the State shall ensure that the amount fixed by or determined under such law for the acquisition of such property is such as would not restrict or abrogate the right guaranteed under that clause.
2. The State shall not, in granting aid to educational institutions, discriminate against any educational institution on the ground that it is under the management of a minority, whether based on religion or language.
Compulsory acquisition of property
Repealed by the Constitution (Forty-fourth Amendment) Act, 1978, s. 6 (w.e.f. 20-6-1979).
Right to Constitutional Remedies
Remedies for enforcement of rights conferred by this Part
1. The right to move the Supreme Court by appropriate proceedings for the enforcement of the rights conferred by this Part is guaranteed.
2. The Supreme Court shall have power to issue directions or orders or writs, including writs in the nature of habeas corpus, mandamus, prohibition, quo warranto and certiorari, whichever may be appropriate, for the enforcement of any of
3. Without prejudice to the powers conferred on the Supreme Court by clauses (1) and (2), Parliament may by law empower any other court to exercise within the local limits of its jurisdiction all or any of the powers exercisable by the Supreme Court under clause (2).
4. The right guaranteed by this article shall not be suspended except as otherwise provided for by this Constitution.
Constitutional validity of State laws not to be considered in proceedings under article 32
Repealed by the Constitution (Forty-third Amendment) Act, 1977, s. 3 (w.e.f. 13-4-1978).
Power of Parliament to modify the rights conferred by this Part in their application to Forces, etc.
Parliament may, by law, determine to what extent any of the rights conferred by this Part shall, in their application to,-
a. the members of the Armed Forces; or
b. the members of the Forces charged with the maintenance of public order; or
c. persons employed in any bureau or other organisation established by the State for purposes of intelligence or counter intelligence; or
d. persons employed in, or in connection with, the telecommunication systems set up for the purposes of any Force, bureau or organisation referred to in clauses (a) to (c),
be restricted or abrogated so as to ensure the proper discharge of their duties and the maintenance of discipline among them.
Restriction on rights conferred by this Part while martial law is in force in any area
Notwithstanding anything in the foregoing provisions of this Part, Parliament may by law indemnify any person in the service of the Union or of a State or any other person in respect of any act done by him in connection with the maintenance or restoration of order in any area within the territory of India where martial law was in force or validate any sentence passed, punishment inflicted, forfeiture ordered or other act done under martial law in such area.
Legislation to give effect to the provisions of this Part.- Notwithstanding anything in this Constitution area
Notwithstanding anything in the Constitution:
a. Parliament shall have, and the legislature of a state shall not have, power to make laws:
i. With respect to any of the matters which under clause (3) of article 16, clause (3) of article 32, article 33 and article 34 may be provided for by law made by parliament; and
ii. For prescribing punishment for those acts which are declared to be offences under this part;
And parliament shall, as soon as may be after the commencement of this constitution, make laws for prescribing punishment for the acts referred to in sub-clause (ii);
b. Any law in force immediately before the commencement of this constitution in the territory of India w.r.t. any of the matters referred to in sub-clause (i) of clause (a) or providing for punishment for any act referred to in sub-clause (ii) of that clause shall, subject to the terms thereof and to any adaptations and modifications that may be made therein under article 372, continue in force until altered or repealed or amended by parliament.
Explanation: in this article, the expression ‘law in force’ has the same meaning as in article 372.<|endoftext|>
| 3.71875 |
637 |
COMP1011 Assignment 2 - 05s2
Computing 1A 05s2 Last updated Thu 13 Oct 2005 15:53
Mail [email protected]
# Mathematical concepts
## Scalars
Scalar is simply a mathematical term for a value that is not a compound value. e.g. 3. Often mathematicians will talk of multiplying a vector by a scalar. This simply means that each component of the vector is multipled by the scalar.
## Vector multiplication
Vectors can be multiplied in two different ways. Refer the the following figure in the following two sections.
## Dot Product
The dot product of a pair of vectors u = (ux, uy,uz) and v = (vx, vy,vz) is defined to be:
u·v = uxvx + uyvy + uzvz
This is not a vector, it is a scalar. It is also the case that:
u·v = |u||v|cos &theta
where &theta is the angle between the vectors and |w| is the magnitude of a vector w.
In the case of unit vectors which have a magnitude of one one can see clearly that u·v = cos &theta.
It is thus, quite easy to find the angle between two vectors using the dot product and you will see that we have done so in module `Physics`.
## Cross product
Unlike the dot product, the cross product of two vectors is another vector. This vector points in a direction that is perpendicular to the other two. Obviously there are two ways for this to occur. If we view the two vectors as being in the same plane the resulting vector could point out of either the front or the back face.
Which direction it does point depends on the relative directions of the two vectors. If the second vector is clockwise with respect to the first in the plane then the vector points towards us. Otherwise it points away from us. This is illustrated in the following figure (where u x v = w):
In the left diagram, v is clockwise with respect to u and so w points towards us. On the right, v is anti-clockwise with respect to u and so w points away from us.
Now for the definition of the cross product:
u x v = |u||v|sin &theta n
where &theta is the angle between u and v and n is a unit vector in the direction perpendicular to both u and v subject to the figure above.
It is also the case that:
u x v = (uyvz - uzvy, uzvx - uxvz, uxvy - uyvx)
## Planes and rotations
Another way to think of n is as the vector that defines the plane in which u and v lie. This is particularly useful if you wish to rotate a vector within a plane. We have provided a function called `rotateInPlane` in module (you guessed it) `Physics` which takes a unit vector defining a plane, a vector to be rotated, and an angle and rotates the vector in that plane. See the module for more details.<|endoftext|>
| 4.65625 |
1,274 |
By Dr. Martha Nizinski, NOAA Office of Science and Technology, National Systematics Laboratory
Submarine canyons are major geologic features of continental margins that link the upper continental shelf to the abyssal plain. Results of the most recent surveys estimate approximately 9,000 canyons worldwide. Even with increased research activities in recent years, most canyons remain poorly known. These geologically and morphologically diverse environments support a wide variety of habitats. Some findings suggest that increased habitat heterogeneity in canyons is responsible for enhancing benthic biodiversity and creating biomass hotspots. Patterns of benthic community structure and productivity have been studied in relatively few submarine canyons.
We are finding that canyons support deepwater coral communities, as well as a number of other sessile (or immobile) filter feeders, in addition to standard slope fauna. Taxonomic richness is also found to be higher in areas of exposed hard substrate. Therefore, canyons are unique in that they provide habitat for a variety of sessile taxa that are not found in other slope environments. These images illustrate the diverse array of habitats and faunal assemblages that we have observed in the Northeast canyons.
Numerous major and minor canyons exist along the continental margin off the U.S. East Coast. The Northeast and Mid-Atlantic canyons were the subject of several recent scientific initiatives. Canyons in this region were a high priority for federal and state agencies tasked with research and management responsibilities, particularly because of deep-sea corals. Several research teams, using a variety of tools, explored many canyons in this region.
In 2012, Atlantic Canyons Undersea Mapping Expedition (ACUMEN) brought together stakeholders and resources to produce a focused, efficient initiative to gather information needed to protect and conserve canyon habitats, in particular, those where deep-sea corals are found. The ACUMEN mapping blitz laid the groundwork for the NOAA Ship Okeanos Explorer Northeast U.S. Canyons expedition, as well as other canyon research in the region.
The Northeast Regional Deep Sea Coral Initiative (2011-2015), funded primarily by NOAA’s Deep Sea Coral Research and Technology Program, used a broad-scale approach, collecting contemporary data in multiple canyons. Canyons were prioritized and sampling locations selected based on data needs of the Mid-Atlantic and New England Fishery Management Councils. Twenty-four canyons were surveyed using a towed-camera system to gather data on coral diversity, abundance and distribution.
The Okeanos Explorer Northeast U.S. Canyons expedition (2013), a partnership between NOAA's Office of Ocean Exploration and Research (OER) and the Northeast Regional Deep Sea Coral Initiative, explored 10 additional canyons in the region. Through live telepresence coverage, the canyons, corals, associated organisms, slope habitats, and conservation issues of the Northeast became an interest of a wider community of scientists, managers, environmentalists, and citizens.
Additionally, the Bureau of Ocean Energy Management (BOEM), in partnership with NOAA OER and U.S. Geological Survey (USGS), funded work (2011-2015) by other researchers to conduct intensive, focused sampling in Norfolk and Baltimore canyons. Results from this initiative not only increased our knowledge of sensitive habitats in the region, but also served as an environmental assessment to inform legislators and managers interested in oil and gas exploration in that region.
Coral diversity, abundance, and distribution data are used to better inform the councils and assist them in their decision-making process, as they work to satisfy all stakeholders in the region, as well as to protect deep-sea corals and their habitats. Both councils are proactively working to protect deep-sea coral habitat. The Mid-Atlantic council has recommended closures of discrete and broad-zone coral protection areas. The New England council is reviewing information and deliberating over potential coral protection zones.
Interest in canyons and coral habitats remains high and the geographic region of exploration and research is moving southward to include areas from North Carolina to Georgia. NOAA, USGS, and BOEM are all interested in deepwater habitats in this region.
The Exploring Carolina Canyons expedition aboard NOAA Ship Pisces will focus on three canyons off the coast of North Carolina, specifically Keller, Pamlico, and Hatteras canyons. Historically, work has been conducted in the Pamlico/Hatteras canyon system. Here, the species composition within the canyon was very different from that along the continental slope north and south of the canyon.
The proposed area of operations is extremely dynamic. Perhaps most noteworthy of this region is the known transition area between the faunas of the Mid-Atlantic Bight and those of the South Atlantic Bight. The Hatteras Slope area, referred to as “The Point,” is hydrographically unique, because it is an area influenced by the convergence of three major currents: the Gulf Stream, Western Boundary Undercurrent, and the Virginia Current. Additionally, the Hatteras slope is highly unusual in that is has the highest densities of sediment-dwelling fauna along the U.S. East Coast continental slope, unusual communities of megafaunal invertebrates and benthic fishes, high sedimentation rates, and high flux of organic carbon.
The anomalous—or unusual—conditions recorded from this region likely influence the biological and physical characteristics of the canyons found in this area. Exploring these canyons is a logical follow-on that will complement our previous work. Preliminary results suggest there are regional differences between Mid-Atlantic canyons and those canyons further north, and results from previous studies suggest that a partial zoogeographic barrier exists on the slope off North Carolina. Our surveys in the Carolina canyons will help to identify and characterize the community structure and geological characteristics of these southern canyons. This new data will allow us to make detailed latitudinal comparisons between the geology and biology of canyons from off the North Carolina coast to those extending northward to the Canadian border.<|endoftext|>
| 3.6875 |
359 |
This article on new research from the Oxford Martin Centre Programme on the Future of Food, University of Oxford suggests that by 2050 the limits of the earth’s capacity to provide sufficient food will be exceeded unless globally coordinated changes in food production, consumption and waste are implemented. The study is not simply centred on climate change effects of of agricultural activity, but also on the consequences of the spread of western-style diets combined with the expected additional growth of human population of well over a billion extra mouths to feed.
The global food system has a lot to answer for. It is a major driver of climate change, thanks to everything from deforestation to cows burping. Food production also transforms biodiverse landscapes into fields inhabited by a single crop or animal. It depletes valuable freshwater resources, and even pollutes ecosystems when fertilisers and manure washed into streams and rivers.
The planet can only take so much of this stress. Staying within its environmental limits will require a global shift towards healthy and more plant-based diets, halving food loss and waste, and improving farming practices and technologies. That’s what a team of international researchers and I found in a new study published in the journal Nature
Without concerted action, we estimated that the environmental pressure of the food system could increase by 50-90% by 2050 as a result of population growth and the continued Westernisation of diets. At that point, those environmental pressures would exceed key planetary boundaries that define a safe operating space for humanity.
SUFFICIENTARIANISM – Sufficientarianism is a theory of distributive justice. Rather than being concerned with inequalities as such or with making the situation of the least well off as good as possible, sufficientarian justice aims at making sure that each of us has enough.<|endoftext|>
| 3.703125 |
1,244 |
24 June 2022 2:47
# Calculate time to reach investment goals given starting balance?
## How do you calculate an investment time?
The rule is a shortcut, or back-of-the-envelope, calculation to determine the amount of time for an investment to double in value. The simple calculation is dividing 72 by the annual interest rate.
## How do you calculate investment growth over time?
ROI is calculated by subtracting the initial value of the investment from the final value of the investment (which equals the net return), then dividing this new number (the net return) by the cost of the investment, and, finally, multiplying it by 100.
## How do you calculate Rule of 72?
The rule says that to find the number of years required to double your money at a given interest rate, you just divide the interest rate into 72. For example, if you want to know how long it will take to double your money at eight percent interest, divide 8 into 72 and get 9 years.
## How do you calculate future value time?
The future value formula is FV=PV(1+i)n, where the present value PV increases for each period into the future by a factor of 1 + i. The future value calculator uses multiple variables in the FV calculation: The present value sum. Number of time periods, typically years.
## What is the investment formula?
Investment problems usually involve simple annual interest (as opposed to compounded interest), using the interest formula I = Prt, where I stands for the interest on the original investment, P stands for the amount of the original investment (called the “principal”), r is the interest rate (expressed in decimal form),
## How do you calculate number of years in time value of money?
Time Value of Money Formula
1. FV = the future value of money.
2. PV = the present value.
3. i = the interest rate or other return that can be earned on the money.
4. t = the number of years to take into consideration.
5. n = the number of compounding periods of interest per year.
## How do I calculate investment growth in Excel?
= PV * (1 + i/n)
STEP 1: The Present Value of investment is provided in cell B3. STEP 2: The annual interest rate is in cell B4 and the interest is compounded monthly so the interest will be divided by the compounding frequency 12 (in cell B6).
## How do you calculate initial investment?
Initial investment is the amount required to start a business or a project. It is also called initial investment outlay or simply initial outlay. It equals capital expenditures plus working capital requirement plus after-tax proceeds from assets disposed off or available for use elsewhere.
## How do we calculate growth rate?
Calculate growth rate FAQs
To calculate the percentage growth rate, use the basic growth rate formula: subtract the original from the new value and divide the results by the original value. To turn that into a percent increase, multiply the results by 100.
## How do you calculate time?
The formula for time is given as [Time = Distance ÷ Speed].
## What is the future value of \$1000 in 5 years at 8?
Answer and Explanation: The future value of a \$1000 investment today at 8 percent annual interest compounded semiannually for 5 years is \$1,480.24. See full answer below.
## How do you calculate time in compound interest?
Compound Interest Formulas and Calculations:
1. Calculate Accrued Amount (Principal + Interest) A = P(1 + r)t
2. Calculate Principal Amount, solve for P. P = A / (1 + r)t
3. Calculate rate of interest in decimal, solve for r. r = (A/P)1/t – 1.
4. Calculate rate of interest in percent. R = r * 100.
5. Calculate time, solve for t.
## What is an investment calculator?
The Investment Calculator can be used to calculate a specific parameter for an investment plan. The tabs represent the desired parameter to be found. For example, to calculate the return rate needed to reach an investment goal with particular inputs, click the ‘Return Rate’ tab.
## How do you calculate ROI manually?
ROI is calculated by subtracting the beginning value from the current value and then dividing the number by the beginning value. It can be calculated by hand or via excel.
## What is formula planning?
Formula plans consist of the basic rules and regulations for purchasing and selling investments. Formula plans enable the investors to estimate the total amount that he has to spend on purchase of securities. Investors may become emotional and they may not act rationally while making investments.
## What are the four commonly used formula plans?
Different Types of Formula Plans are given below:
• Constant-Rupee-Value Plan: The constant rupee value plan specifies that the rupee value of the stock portion of the portfolio will remain constant. …
• Constant Ratio Plan: …
• Variable Ratio Plan:
## What is an optimal portfolio?
An optimal portfolio is one designed with a perfect balance of risk and return. The optimal portfolio looks to balance securities that offer the greatest possible returns with acceptable risk or the securities with the lowest risk given a certain return.
## How do you measure portfolio risk?
The most common risk measure is standard deviation. Standard deviation is an absolute form of risk measure; it is not measured in relation to other assets or market returns. Standard deviation measures the spread of returns around the average return.
Absolute Risk Measures.
US Equity Fund 12.26%
Multiple Asset Fund 9.23%
## What is the formula for calculating risk?
risk = probability x loss
What does it mean? Many authors refer to risk as the probability of loss multiplied by the amount of loss (in monetary terms).
## How do you calculate portfolio risk in Excel?
Quote: So the variance for a portfolio is equal to the percentage you put in a squared. Times the variance of a plus the percentage you put in b. Squared times the variance of b.<|endoftext|>
| 4.5 |
190 |
Lesson organisation varies according to the subject or topic being taught. During the week children take part in whole class, group and individual work. Our staff are experts in deciding upon the most effective teaching strategies for the children in their class in order to achieve the objectives of the lesson. These include:
- teaching the whole class;
- working in groups determined by ability or friendship
- “rotating” groups, where children move between a range of activities requiring different levels of teacher support
- individual work, when a child needs additional teacher/adult support to complete a task or extension work to develop particular skills or knowledge
Childrens’ understanding is continually assessed by the class teacher. This may be done through written work, drawing, speaking, observation, play and discussion.
These “formative” (informal) assessments help the teachers to plan follow-up, extension or consolidation activities for the children.<|endoftext|>
| 3.71875 |
899 |
One-Way Tables
David M. Lane
Prerequisites
Chi Square Distribution, Basic Concepts of Probability, Significance Testing
Learning Objectives
1. Describe what it means for there to be theoretically-expected frequencies
2. Compute expected frequencies
3. Compute Chi Square
4. Determine the degrees of freedom
The data in Table 1 were obtained by rolling a six-sided die 36 times. If the die were a fair die, then the probability of of any given outcome on a single roll would be 1/6. However, as can be seen in Table 1, some outcomes occurred more frequently than others. For example a "3" came up nine times whereas a "4" came up only two times. Are these data consistent with the hypothesis that the die is a fair die? Naturally, we do not expect the sample frequencies of the six possible outcomes throws to be the same since chance differences will occur. So, the finding that the frequencies differ does not mean that the die is not fair. One way to test whether the die is fair is to conduct a significance test. The null hypothesis is that the die is fair. This hypothesis is tested by computing the probability of obtaining frequencies as discrepant or more discrepant from a uniform distribution of frequencies as obtained in the sample. If this probability is sufficiently low, then the null hypothesis that the die is fair can be rejected.
Table 1. Outcome Frequencies
from a Six-Sided Die
Outcome Frequency 1 2 3 4 5 6 8 5 9 2 7 5
The first step in conducting the significance test is to compute the expected frequency for each outcome given that the null hypothesis is true. For example, the expected frequency of a "1" is 6 since the probability of a "1" coming up is 1/6 and there were a total of 36 rolls of the die.
Expected frequency = (1/6)(36) = 6
Note that the expected frequencies are expected only in a theoretical sense. We do not really "expect" the observed frequencies to match the "expected frequencies" exactly.
The calculation continues as follows. Letting E be the expected frequency of an outcome and O be the observed frequency of that outcome, compute
for each outcome. Table 2 shows these calculations.
Next we add up all the values in Column 4 of Table 2.
This sampling distribution of
is approximately distributed as Chi Square on k-1 degrees of freedom where k is the number of categories. Therefore, for this problem the test statistic is
which means the value of Chi Square with 5 degrees of freedom is 5.333.
From a Chi Square calculator it can be determined that the probability of a Chi Square of 5.333 or larger is 0.377. Therefore, the null hypothesis that the die is fair cannot be rejected.
This Chi Square test can also be used to test other deviations between expected and observed frequencies. The following example shows a test of whether the variable "University GPA" in the SAT and College GPA case study is normally distributed.
The second column of Table 3 shows the proportions of a normal distribution falling between various limits. The expected frequencies (E) are calculated by multiplying the number of scores (105) by the proportion. The final column shows the observed number of scores in each range. It is clear that the observed frequencies vary greatly from the expected frequencies. Note that if the distribution were normal then there would have been only about 35 scores between -1 and 0 whereas 60 were observed.
Table 3. Expected and Observed Scores for 105 University GPA Scores.
Range Proportion E O Above 1 0.159 16.695 19 0 to 1 0.341 35.805 17 -1 to 0 0.341 35.805 60 Below -1 0.159 16.695 9
The test of whether the observed scores deviate significantly from the expected is computed using the familiar calculation.
The subscript "3" means there are three degrees of freedom. As before, the degrees of freedom is the number of outcomes, which is four in this example. The Chi Square distribution calculator shows that p < 0.001 for this Chi Square. Therefore, the null hypothesis that the scores are normally distributed can be rejected.<|endoftext|>
| 4.625 |
495 |
# Which system of linear inequalities is represented by the graph? y > x – 2 and x – 2y < 4 y > x + 2 and x + 2y < 4 y > x – 2 and x + 2y < 4 y > x – 2 and x + 2y < –4
Students were asked to answer a question at institution and to mention what is most important for them to succeed. One which response stood out from the rest was practice. People who commonly are successful do not become successful by being born. They work hard and determination their lives to succeeding. If you want to reach your goals, keep this in mind! followed below some question and answer examples that you can easily use to elevate your knowledge and gain insight that will guide you to maintain your school studies.
## Question:
Which system of linear inequalities is represented by the graph? y > x – 2 and x – 2y < 4 y > x + 2 and x + 2y < 4 y > x – 2 and x + 2y < 4 y > x – 2 and x + 2y < –4
Answer: Four graphs matching the inequalities are attached.
You will have to choose. Compare the graph you have with the ones in this answer. Look carefully at the slopes and the intercepts.
Step-by-step explanation:
The inequalites you have to choose from are a bit complicated, as the second one in each pair is not in slope intercept form. When you rewrite them, it is much easier to see the amount and direction of the slope and find the y-intercept. The 2y results in a slope of 1/2 when you divide both sides by 2
Remember, as in the first set, if you divide by a negative number, switch the direction of the inequality sign.
Please do not expect Brainly participants always to do this much work for you! (I’d appreciate a ‘Brainliest’ for my efforts on this one.)
From the answer and question examples above, hopefully, they can guide the student answer the question they had been looking for and remember of every detail declared in the answer above. You would possibly then have a discussion with your classmate and continue the school learning by studying the subject to one another.
READ MORE Which sentence is the best example of an objective summary?<|endoftext|>
| 4.625 |
174 |
## Algebra 1
$r = -\frac{1}{2}$
$7r = -\frac{7}{2}$ We start by isolating the variable $r$ by dividing through by $7$. In division operations involving fractions, we usually multiply the divided by the reciprocal of the divisor. In this case, the dividend is $-\frac{7}{2}$ and the divisor is $7$. The reciprocal of $7$ is $\frac{1}{7}$. Therefore, the operation becomes: $7r \times \frac{1}{7} = -\frac{7}{2} \times \frac{1}{7}$ Simplifying, we have: $r = -\frac{1}{2}$ Therefore, the equation has a possible solution at $r = -\frac{1}{2}$<|endoftext|>
| 4.4375 |
238 |
Gum disease (Periodontal Disease) is responsible for about 70 percent of adult tooth loss. It is characterized by swollen, inflamed gums surrounding the teeth. Plaque, a sticky substance that forms in the mouth from food, saliva and bacteria gets inside the space between the gum line and the tooth. If not removed, plaque hardens into a substance called calculus or tartar that is very difficult to remove. Eventually, the bacteria in the plaque and tartar eat away at the fibers that hold the gums to the teeth, creating deep pockets. As bacteria spread, the pockets become deeper until the bacteria finally eat away the bone that holds the tooth in place.
Think of it as if bugs are eating away at the soil around a tree trunk. Eventually, they eat away all of the soil and part of the tree’s roots, causing the tree to collapse.
Gum disease is diagnosed through a process that measures the depth of the pockets around each tooth. Pockets that are greater than 3 millimeters in depth are considered hazardous and will generally require treatment.
The early detection and prevention of gum disease is another reason to see your dentist regularly.<|endoftext|>
| 3.703125 |
914 |
Courses
Courses for Kids
Free study material
Offline Centres
More
Last updated date: 05th Dec 2023
Total views: 381.9k
Views today: 9.81k
# The base of an isosceles triangle is $\dfrac{4}{3}\text{cm}$. The perimeter of the triangle is $\dfrac{62}{15}\text{cm}$. What is the length of either of the remaining equal sides?
Verified
381.9k+ views
Hint: In this question, the perimeter and length of the base are given. Take the sum of the sides of the isosceles triangle and equate it to the perimeter. Then substitute the given values to find the length of the two remaining sides.
Here, we are given that the base of the isosceles triangle is $\dfrac{4}{3}\text{cm}$. The perimeter of the triangle is $\dfrac{62}{15}\text{cm}$. We need to find the length of either of the remaining sides.
Before proceeding with this question, we must know what an isosceles triangle is. An isosceles triangle is a triangle that has two sides of equal length. The two sides of triangles are called legs while the third side is called the base of the triangle.
Let us consider an isosceles triangle ABC.
In the above isosceles triangle, sides AB = AC and side BC is the third side that is the base of the given triangle.
Now, we know that the perimeter of any polygon is given by the sum of its sides. So, we get,
Perimeter of the triangle = Sum of its 3 sides
Therefore, for the given triangle, we get,
Perimeter of $\Delta \text{ABC = AB + AC + BC}$
We know that AB = AC.
So, let us substitute AB = AC = x. So, we get,
Perimeter of $\Delta \text{ABC = 2x + BC}$
Also, we know that BC is the base of the triangle. So, we get,
Perimeter of isosceles $\Delta \text{ABC = 2x + Base of }\Delta \text{ABC}$
So, for any general isosceles triangle, we get,
Perimeter of isosceles triangle = 2 (Length of either of the equal sides) + (length of the base)
We are given that the base of the isosceles triangle $=\dfrac{4}{3}\text{cm}$ and its perimeter $=\dfrac{62}{15}\text{cm}$. By substituting these values in the above equation, we get,
$\dfrac{62}{15}\text{cm = 2 }\left( \text{length of either of equal sides} \right)+\text{ }\dfrac{4}{3}\text{cm}$
By subtracting $\dfrac{4}{3}$ from both the sides of the above equation, we get,
$\dfrac{62}{15}-\dfrac{4}{3}=2\left( \text{length of either of equal sides} \right)$
$\Rightarrow \dfrac{62-20}{15}=2\left( \text{length of either of equal sides} \right)$
Or, $2\left( \text{length of either of equal sides} \right)=\dfrac{42}{15}$
Now by dividing 2 on both the sides, we get,
Length of either of equal side $=\dfrac{7}{5}\text{cm}\approx \text{1}\text{.4cm}$
Hence, we get the length of each equal side of the isosceles triangle as $\dfrac{7}{5}$ or 1.4 cm.
Note:
Students must note the triangles are divided into 3 categories based on length of sides that are scalene triangles, isosceles triangles and equilateral triangles. In scalene triangles, all sides are unequal, in isosceles triangles, at least 2 out of 3 sides are equal while in the equilateral triangle, all sides are equal. Students can also cross-check their answer by adding the length of all the sides and equating it to the given perimeter and checking if LHS = RHS or not.<|endoftext|>
| 4.78125 |
304 |
How to draw a ball python
1) Draw a rectangle that will define the conditional proportions and boundaries of the chosen drawing.
2) From the middle of the rectangle, draw one vertical and one horizontal line equally dividing the shape.
3) Draw another horizontal line equally dividing the upper half of the rectangle. Similarly, draw a horizontal line equally dividing the bottom half of the rectangle.
4) Draw a vertical line equally dividing the left half of the rectangle. Similarly, draw a vertical line equally dividing the right half of the rectangle.
Mark off the width and height of the picture. Draw an oval for the ball python’s head. Draw a guideline for a body of the snake.
With smooth lines, define the general shape of the body.
Outline the back and belly of the ball python. Mark the mouth with a short line.
Add more lines to indicate the shape of the body. Draw the head and eyes.
Delineate the lower jaw. Work on the ball python, paying special attention to details.
Contour the ball python, trying to vary the thickness and blackness of the line. Add more details and the ground. Erase all guidelines.
Some rights reserved. This work is licensed under a Creative Commons Attribution-Share Alike 4.0 License. You are free to share or adapt it for any purpose, even commercially under the following terms: you must give a link to this page and indicate the author's name and the license.<|endoftext|>
| 4.34375 |
837 |
Now we will learn what decimal numbers are, let’s learn how to use them in basic arithmetic operations.
So, the basic principles behind the addition of real numbers are pretty much the same as the ones with the whole numbers. And the decimal point actually doesn’t make as much of a difference as we might think. Let’s take a look at this example:
The most important thing to remember is that the numbers on the right side of the decimal point add up the same way as the numbers on the left side of the decimal point. That means that we add the tenths with the tenths, the hundredths with the hundredths, etc. When we add the numbers, write one beneath the other in a way that their decimal points align. This will help us keep track of which parts we’re adding.
As in any other addition, we start by adding the rightmost numerals of both numbers, and keep going from right to left. If the sum of two added numerals adds up to more than ten, we will just increase the sum of the first numerals to the left by one. We can add as many zeroes as we want behind the rightmost numeral of our number. That’s basically the same thing as adding zeroes in front of the leftmost numeral to the left of the decimal point – it won’t change the value of the number at all, but it will make calculations easier.
And the same principles apply to subtraction. We start by subtracting the rightmost numeral behind the decimal point of the subtrahend from the rightmost numeral of the minuend, and work our way to their respective leftmost numerals. Sounds complicated? Well, it’s easier than it seems. Follow the basic rules for subtraction, and then apply what you learned about the addition of real numbers.
Let’s do a couple of examples together – an addition and a subtraction – to start things off. We’ll do it step by step to make things as clear as possible. And as the first example, let’s add the numbers $107.445$ and $224.581$.
We begin by adding the last decimals of both numbers together. Number $5$ and number $1$ is $6$. Now we add together the next two decimals, $4$ and $8$. Since $4$ and $8$ add up to $12$, and $12$ is greater than $10$ by $1$, we’ll write down the $2$ and remember $1$ as well.
Now we add together the next two decimals, $4$ and $5$, and increase their sum by the number $1$ we remembered earlier. Since $4 + 5 + 1$ is $10$, we write down the $0$, and remember $1$ once more.
Now that we’re past the decimal point, we’re in familiar territory. Now it’s time to add together the number $7$ and the $4$, and increase their sum by $1$, also we remembered that from the previous addition. Since sum $7 + 4 + 1$ is equal to $12$, which is bigger than $10$, you can remember $0$, if it makes it easier to keep track, and we will add the $0$ in the next addition.
After that, we just write down the results of our additions to their appropriate places. Therefore, the final result of our addition is decimal number $2.971$.
Subtraction works in a similar fashion. The biggest difference is that, instead of “remembering the $1$”, we “borrow the $1$”.
Well, that’s it for the addition and subtraction of decimal numbers. If you would like to practice a bit more and get a better hang of it, try solving the examples from the worksheets below.<|endoftext|>
| 4.5 |
729 |
## How do you factor algebraic equations?
So, if, in your equation, your b value is twice the square root of your c value, your equation can be factored to (x + (sqrt(c)))2. For example, the equation x2 + 6x + 9 fits this form. 32 is 9 and 3 × 2 is 6. So, we know that the factored form of this equation is (x + 3)(x + 3), or (x + 3)2.
## What are the 4 methods of factoring?
The following factoring methods will be used in this lesson:Factoring out the GCF.The sum-product pattern.The grouping method.The perfect square trinomial pattern.The difference of squares pattern.
## What are the 6 types of factoring?
The lesson will include the following six types of factoring:Group #1: Greatest Common Factor.Group #2: Grouping.Group #3: Difference in Two Squares.Group #4: Sum or Difference in Two Cubes.Group #5: Trinomials.Group #6: General Trinomials.
## What are the 4 ways to solve quadratic equations?
The four methods of solving a quadratic equation are factoring, using the square roots, completing the square and the quadratic formula.
## What are factoring methods?
Factoring is the process by which we go about determining what we multiplied to get the given quantity. A common method of factoring numbers is to completely factor the number into positive prime factors. A prime number is a number whose only positive factors are 1 and itself.
## Where is factoring used in real life?
Factoring is a useful skill in real life. Common applications include: dividing something into equal pieces, exchanging money, comparing prices, understanding time and making calculations during travel.
## What is factorisation method?
Factorisation is the process of reducing the bracket of a quadractic equation, instead of expanding the bracket and converting the equation to a product of factors which cannot be reduced further. For example, factorising (x²+5x+6) to (x+2) (x+3). These factors can be either variable, integers or algebraic expressions.
## What is the first rule of factoring?
RULE # 1: The First Rule of Factoring: Always see if you can factor something out of ALL the terms. This often occurs along with another type of factoring.
## Why do we use factoring in math?
Factoring is an important process that helps us understand more about our equations. Through factoring, we rewrite our polynomials in a simpler form, and when we apply the principles of factoring to equations, we yield a lot of useful information.
### Releated
#### Equation for decibels
How do you calculate dB? Find the logarithm of the power ratio. log (100) = log (102) = 2 Multiply this result by 10 to find the number of decibels. decibels = 10 × 2 = 20 dB If we put all these steps together into a single equation, we once again have the definition […]
#### Convert to an exponential equation
How do you convert a logarithmic equation to exponential form? How To: Given an equation in logarithmic form logb(x)=y l o g b ( x ) = y , convert it to exponential form. Examine the equation y=logbx y = l o g b x and identify b, y, and x. Rewrite logbx=y l o […]<|endoftext|>
| 4.9375 |
1,227 |
A dock (from Dutch dok) is either the area of water between or next to a human-made structure or group of structures involved in the handling of boats or ships, usually on or close to a shore, or the structures themselves. The exact meaning varies among different variants of the English language. "Dock" may also refer to a dockyard or shipyard where the loading, unloading, building, or repairing of ships occurs.
The earliest known docks were those discovered in Wadi al-Jarf, an ancient Egyptian harbor dating from 2500 BCE located on the Red Sea coast. Archaeologists also discovered anchors and storage jars near the site. A dock from Lothal in India dates from 2400 BCE and was located away from the main current to avoid deposition of silt. Modern oceanographers have observed that the Harappans must have possessed great knowledge relating to tides in order to build such a dock on the ever-shifting course of the Sabarmati, as well as exemplary hydrography and maritime engineering. This was the earliest known dock found in the world, equipped to berth and service ships. It is speculated that Lothal engineers studied tidal movements, and their effects on brick-built structures, since the walls are of kiln-burnt bricks. This knowledge also enabled them to select Lothal's location in the first place, as the Gulf of Khambhat has the highest tidal amplitude and ships can be sluiced through flow tides in the river estuary. The engineers built a trapezoidal structure, with north-south arms of average 21.8 metres (71.5 ft), and east-west arms of 37 metres (121 ft).
In British English, a dock is an enclosed area of water used for loading, unloading, building or repairing ships. Such a dock may be created by building enclosing harbour walls into an existing natural water space, or by excavation within what would otherwise be dry land.
There are specific types of dock structure where the water level is controlled:
- A wet dock or impounded dock is a variant in which the water is impounded either by dock gates or by a lock, thus allowing ships to remain afloat at low tide in places with high tidal ranges. The level of water in the dock is maintained despite the raising and lowering of the tide. This makes transfer of cargo easier. It works like a lock which controls the water level and allows passage of ships. The world's first enclosed wet dock with lock gates to maintain a constant water level irrespective of tidal conditions was the Howland Great Dock on the River Thames, built in 1703. The dock was merely a haven surrounded by trees, with no unloading facilities. The world's first commercial enclosed wet dock, with quays and unloading warehouses, was the Old Dock at Liverpool, built in 1715 and held up to 100 ships. The dock reduced ship waiting giving quick turn arounds, greatly improving the throughput of cargo.
- A drydock is another variant, also with dock gates, which can be emptied of water to allow investigation and maintenance of the underwater parts of ships.
A dockyard (or "shipyard") consists of one or more docks, usually with other structures.
In American English, a dock is technically synonymous with pier or wharf—any human-made structure in the water intended for people to be on. However, in modern use, pier is generally used to refer to structures originally intended for industrial use, such as seafood processing or shipping, and more recently for cruise ships, and dock is used for most everything else, often with a qualifier, such as ferry dock, swimming dock, ore dock and others. However, pier is also commonly used to refer to wooden or metal structures that extend into the ocean from beaches and are used, for the most part, to accommodate fishing in the ocean without using a boat.
In the cottage country of Canada and the United States, a dock is a wooden platform built over water, with one end secured to the shore. The platform is used for the boarding and offloading of small boats.
- Dry dock: a narrow basin that can be flooded and drained to allow a load to
come to rest on a dry platform
- Ferry slip: a specialized docking facility that receives a ferryboat
- Floating dock (impounded)
- Floating dock (jetty): a walkway over water, made buoyant with pontoons
- Mole (architecture)
- Pier: a raised walkway over water, supported by widely spread pilings or pillars
- Pontoon (boat): a buoyant device, used to support docks or floating bridges
- Slipway: a ramp on the shore by which ships or boats can be moved to and from the water
- Wharf: a fixed platform, commonly on pilings, where ships are loaded and unloaded
- Boyle, Alan (15 April 2013). "4,500-year-old harbor structures and papyrus texts unearthed in Egypt". NBC.
- Marouard, Gregory; Tallet, Pierre (2012). "Wadi al-Jarf - An early pharaonic harbour on the Red Sea coast". Egyptian Archaeology. 40: 40–43. Retrieved 18 April 2013.
- Tallet, Pierre (2012). "Ayn Sukhna and Wadi el-Jarf: Two newly discovered pharaonic harbours on the Suez Gulf" (PDF). British Museum Studies in Ancient Egypt and Sudan. 18: 147–68. ISSN 2049-5021. Retrieved 21 April 2013.
- Rao, pages 27–28
- Rao, pages 28–29
- Rao, S. R. (1985). Lothal, a Harappan Port Town (1955–62). New Delhi: Archaeological Survey of India. OCLC 60370124.
|Wikimedia Commons has media related to Docks.|
|Look up dock in Wiktionary, the free dictionary.|<|endoftext|>
| 3.765625 |
1,510 |
# Measures of shape
## Definition
Measures of shape describe the distribution (or pattern) of the data within a dataset.
The distribution shape of quantitative data can be described as there is a logical order to the values, and the 'low' and 'high' end values on the x-axis of the histogram are able to be identified.
The distribution shape of a qualitative data cannot be described as the data are not numeric.
## Shapes of a dataset
A distribution of data item values may be symmetrical or asymmetrical. Two common examples of symmetry and asymmetry are the 'normal distribution' and the 'skewed distribution'.
### Symmetrical distribution
In a symmetrical distribution the two sides of the distribution are a mirror image of each other.
A normal distribution is a true symmetric distribution of observed values.
When a histogram is constructed on values that are normally distributed, the shape of columns form a symmetrical bell shape. This is why this distribution is also known as a 'normal curve' or 'bell curve'.
The following graph is an example of a normal distribution.
### Normal distribution: Height of students
Histogram graph showing the frequency of student's height.
• 114cm - 1
• 115cm - 1
• 116cm - 2
• 117cm - 2
• 118cm - 3
• 119cm - 5
• 120cm - 7
• 121cm - 8
• 122cm - 8
• 123cm - 7
• 124cm - 5
• 125cm - 3
• 126cm - 2
• 127cm - 2
• 128cm - 1
• 129cm - 1
If represented as a 'normal curve' (or bell curve) the graph would take the following shape (where µ = mean, and σ = standard deviation):
### Bell curve example
Example of a bell curve showing how around 68% of values lie within one standard deviation away from the mean, about 95% of the values lie within two standard deviations and about 99.7% are within three standard deviations.
+3 standard deviations - 49.3%
+2 standard deviations - 47.7%
+1 standard deviation - 34.1%
mean
-1 standard deviation - 34.1%
-2 standard deviations - 47.7%
-3 standard deviations - 49.3%
Key features of the normal distribution:
• symmetrical shape
• mode, median and mean are the same and are together in the centre of the curve
• there can only be one mode (i.e. there is only one value which is most frequently observed)
• most of the data are clustered around the centre, while the more extreme values on either side of the centre become less rare as the distance from the centre increases (i.e. About 68% of values lie within one standard deviation (σ) away from the mean; about 95% of the values lie within two standard deviations; and about 99.7% are within three standard deviations. This is known as the empirical rule or the 3-sigma rule.
### Asymmetrical distribution
In an asymmetrical distribution the two sides will not be mirror images of each other.
Skewness is the tendency for the values to be more frequent around the high or low ends of the x-axis.
When a histogram is constructed for skewed data it is possible to identify skewness by looking at the shape of the distribution.
For example:
A distribution is said to be positively skewed when the tail on the right side of the histogram is longer than the left side. Most of the values tend to cluster toward the left side of the x-axis (i.e. the smaller values) with increasingly fewer values at the right side of the x-axis (i.e. the larger values).
### Positively skewed distribution: Height of students
Histogram graph showing the frequency of student's height.
114cm - 3
115cm - 5
116cm - 8
117cm - 8
118cm - 7
119cm - 5
120cm - 4
121cm - 4
122cm - 3
123cm - 3
124cm - 2
125cm - 2
126cm - 2
127cm - 1
128cm - 1
129cm - 1
130cm - 1
A distribution is said to be negatively skewed when the tail on the left side of the histogram is longer than the right side. Most of the values tend to cluster toward the right side of the x-axis (i.e. the larger values), with increasingly less values on the left side of the x-axis (i.e. the smaller values).
### Negatively skewed distribution: Height of students
Histogram graph showing the frequency of student's height.
114cm - 1
115cm - 1
116cm - 1
117cm - 1
118cm - 2
119cm - 2
120cm - 2
121cm - 3
122cm - 3
123cm - 4
124cm - 4
125cm - 5
126cm - 6
127cm - 8
128cm - 8
129cm - 5
130cm - 3
Key features of the skewed distribution:
• asymmetrical shape
• mean and median have different values and do not all lie at the centre of the curve
• there can be more than one mode
• the distribution of the data tends towards the high or low end of the dataset
## Other possible distribution shapes
Other distributions include uni-modal, bi-modal, or multimodal.
A uni-modal distribution occurs if there is only one 'peak' (a highest point) in the distribution, as seen in the previous histograms. This means there is one mode (a value that occurs more frequently than any other) for the data item (variable).
The distribution shape of the data in the histogram below is bi-modal because there are two modes (two values that occur more frequently than any other) for the data item (variable).
### Bi-modal distribution: Height of students
Histogram graph showing the frequency of student's height.
114cm - 3
115cm - 4
116cm - 5
117cm - 6
118cm - 7
119cm - 7
120cm - 6
121cm - 5
122cm - 4
123cm - 4
124cm - 5
125cm - 6
126cm - 7
127cm - 7
128cm - 6
129cm - 5
130cm - 4
131cm - 3
## Uses of measure of shape
The shape of the distribution can assist with identifying other descriptive statistics, such as which measure of central tendency is appropriate to use.
If the data are normally distributed, the mean, median and mode are all equal, and therefore are all appropriate measure of centre central tendency.
If data are skewed, the median may be a more appropriate measure of central tendency.<|endoftext|>
| 4.40625 |
1,683 |
Chapter 1. An Introduction to the Human Body
By the end of this section, you will be able to:
- Explain the importance of organization to the function of the human organism
- Distinguish between metabolism, anabolism, and catabolism
- Provide at least two examples of human responsiveness and human movement
- Compare and contrast growth, differentiation, and reproduction
The different organ systems each have different functions and therefore unique roles to perform in physiology. These many functions can be summarized in terms of a few that we might consider definitive of human life: organization, metabolism, responsiveness, movement, development, and reproduction.
A human body consists of trillions of cells organized in a way that maintains distinct internal compartments. These compartments keep body cells separated from external environmental threats and keep the cells moist and nourished. They also separate internal body fluids from the countless microorganisms that grow on body surfaces, including the lining of certain tracts, or passageways. The intestinal tract, for example, is home to even more bacteria cells than the total of all human cells in the body, yet these bacteria are outside the body and cannot be allowed to circulate freely inside the body.
Cells, for example, have a cell membrane (also referred to as the plasma membrane) that keeps the intracellular environment—the fluids and organelles—separate from the extracellular environment. Blood vessels keep blood inside a closed circulatory system, and nerves and muscles are wrapped in connective tissue sheaths that separate them from surrounding structures. In the chest and abdomen, a variety of internal membranes keep major organs such as the lungs, heart, and kidneys separate from others.
The body’s largest organ system is the integumentary system, which includes the skin and its associated structures, such as hair and nails. The surface tissue of skin is a barrier that protects internal structures and fluids from potentially harmful microorganisms and other toxins.
The first law of thermodynamics holds that energy can neither be created nor destroyed—it can only change form. Your basic function as an organism is to consume (ingest) energy and molecules in the foods you eat, convert some of it into fuel for movement, sustain your body functions, and build and maintain your body structures. There are two types of reactions that accomplish this: anabolism and catabolism.
- Anabolism is the process whereby smaller, simpler molecules are combined into larger, more complex substances. Your body can assemble, by utilizing energy, the complex chemicals it needs by combining small molecules derived from the foods you eat
- Catabolism is the process by which larger more complex substances are broken down into smaller simpler molecules. Catabolism releases energy. The complex molecules found in foods are broken down so the body can use their parts to assemble the structures and substances needed for life.
Taken together, these two processes are called metabolism. Metabolism is the sum of all anabolic and catabolic reactions that take place in the body (Figure 1). Both anabolism and catabolism occur simultaneously and continuously to keep you alive.
Every cell in your body makes use of a chemical compound, adenosine triphosphate (ATP), to store and release energy. The cell stores energy in the synthesis (anabolism) of ATP, then moves the ATP molecules to the location where energy is needed to fuel cellular activities. Then the ATP is broken down (catabolism) and a controlled amount of energy is released, which is used by the cell to perform a particular job.
Responsiveness is the ability of an organism to adjust to changes in its internal and external environments. An example of responsiveness to external stimuli could include moving toward sources of food and water and away from perceived dangers. Changes in an organism’s internal environment, such as increased body temperature, can cause the responses of sweating and the dilation of blood vessels in the skin in order to decrease body temperature, as shown by the runners in Figure 2.
Human movement includes not only actions at the joints of the body, but also the motion of individual organs and even individual cells. As you read these words, red and white blood cells are moving throughout your body, muscle cells are contracting and relaxing to maintain your posture and to focus your vision, and glands are secreting chemicals to regulate body functions. Your body is coordinating the action of entire muscle groups to enable you to move air into and out of your lungs, to push blood throughout your body, and to propel the food you have eaten through your digestive tract. Consciously, of course, you contract your skeletal muscles to move the bones of your skeleton to get from one place to another (as the runners are doing in Figure 2), and to carry out all of the activities of your daily life.
Development, growth and reproduction
Development is all of the changes the body goes through in life. Development includes the process of differentiation, in which unspecialized cells become specialized in structure and function to perform certain tasks in the body. Development also includes the processes of growth and repair, both of which involve cell differentiation.
Growth is the increase in body size. Humans, like all multicellular organisms, grow by increasing the number of existing cells, increasing the amount of non-cellular material around cells (such as mineral deposits in bone), and, within very narrow limits, increasing the size of existing cells.
Reproduction is the formation of a new organism from parent organisms. In humans, reproduction is carried out by the male and female reproductive systems. Because death will come to all complex organisms, without reproduction, the line of organisms would end.
Most processes that occur in the human body are not consciously controlled. They occur continuously to build, maintain, and sustain life. These processes include: organization, in terms of the maintenance of essential body boundaries; metabolism, including energy transfer via anabolic and catabolic reactions; responsiveness; movement; and growth, differentiation, reproduction, and renewal.
Interactive Link Questions
View this animation to learn more about metabolic processes. What kind of catabolism occurs in the heart?
Fatty acid catabolism.
1. Metabolism can be defined as the ________.
- adjustment by an organism to external or internal changes
- process whereby all unspecialized cells become specialized to perform distinct functions
- process whereby new cells are formed to replace worn-out cells
- sum of all chemical reactions in an organism
2. Adenosine triphosphate (ATP) is an important molecule because it ________.
- is the result of catabolism
- release energy in uncontrolled bursts
- stores energy for use by body cells
- All of the above
3. Cancer cells can be characterized as “generic” cells that perform no specialized body function. Thus cancer cells lack ________.
- both reproduction and responsiveness
Critical Thinking Questions
1. Explain why the smell of smoke when you are sitting at a campfire does not trigger alarm, but the smell of smoke in your residence hall does.
2. Identify three different ways that growth can occur in the human body.
- assembly of more complex molecules from simpler molecules
- breaking down of more complex molecules into simpler molecules
- changes an organism goes through during its life
- process by which unspecialized cells become specialized in structure and function
- process of increasing in size
- sum of all of the body’s chemical reactions
- process by which worn-out cells are replaced
- process by which new organisms are generated
- ability of an organisms or a system to adjust to changes in conditions
Answers for Review Questions
Answers for Critical Thinking Questions
- When you are sitting at a campfire, your sense of smell adapts to the smell of smoke. Only if that smell were to suddenly and dramatically intensify would you be likely to notice and respond. In contrast, the smell of even a trace of smoke would be new and highly unusual in your residence hall, and would be perceived as danger.
- Growth can occur by increasing the number of existing cells, increasing the size of existing cells, or increasing the amount of non-cellular material around cells.<|endoftext|>
| 3.8125 |
1,627 |
# Proving properties of matrix
• Mar 30th 2010, 09:09 PM
arze
Proving properties of matrix
Let A be the matrix $\left(\begin{array}{cc}a&b\\c&d\end{array}\right)$, where no one of a,b,c,d is zero. It is required to find a non-zero 2x2 matrix X such that AX+XA=0, where 0 is the zero 2x2 matrix. Prove that either
(a) a+d=0, in which case the general solution for X depends on two parameters, or
(b) ad-bc=0, in which case the general solution for X depends on one parameter.
I don't know where to begin other than naming X= $\left(\begin{array}{cc}\\x_{11}&x_{12}\\x_{21}&x_{ 22}\end{array}\right)$ and then find the matrices AX and XA.
Thanks
• Mar 31st 2010, 02:40 AM
tonio
Quote:
Originally Posted by arze
Let A be the matrix $\left(\begin{array}{cc}a&b\\c&d\end{array}\right)$, where no one of a,b,c,d is zero. It is required to find a non-zero 2x2 matrix X such that AX+XA=0, where 0 is the zero 2x2 matrix. Prove that either
(a) a+d=0, in which case the general solution for X depends on two parameters, or
(b) ad-bc=0, in which case the general solution for X depends on one parameter.
I don't know where to begin other than naming X= $\left(\begin{array}{cc}\\x_{11}&x_{12}\\x_{21}&x_{ 22}\end{array}\right)$ and then find the matrices AX and XA.
Thanks
Exactly, that's what you have to do...and then solve $AX+XA=0$ , where the zero in the right is, of course, the zero 2x2 matrix.
Tonio
• Mar 31st 2010, 02:58 AM
arze
Quote:
Originally Posted by tonio
Exactly, that's what you have to do...and then solve $AX+XA=0$ , where the zero in the right is, of course, the zero 2x2 matrix.
Tonio
Ok, i did that and this is what I got
$AX=\left(\begin{array}{cc}{ax_{11}+bx_{21}}&{ax_{1 2}+bx_{22}}\\{cx_{11}+dx_{21}}&{cx_{12}+dx_{22}}\e nd{array}\right)$
$XA=\left(\begin{array}{cc}{ax_{11}+cx_{12}}&{bx_{1 1}+dx_{12}}\\{ax_{21}+cx_{22}}&{bx_{21}+dx_{22}}\e nd{array}\right)$
equating it with the zero matrix I get four equations.
$2ax_{11}+x_{12}(b+c)=0$ __1
$b(x_{22}+x_{11})+x_{12}(a+d)=0$ __2
$x_{21}(a+d)+c(x_{11}+x_{22})=0$ __3
$2dx_{22}+cx_{12}+bx_{21}=0$ __4
Now what do I do next?
Thanks
• Mar 31st 2010, 03:37 AM
tonio
[quote=arze;484206]Ok, i did that and this is what I got
$AX=\left(\begin{array}{cc}{ax_{11}+bx_{21}}&{ax_{1 2}+bx_{22}}\\{cx_{11}+dx_{21}}&{cx_{12}+dx_{22}}\e nd{array}\right)$
$XA=\left(\begin{array}{cc}{ax_{11}+cx_{12}}&{bx_{1 1}+dx_{12}}\\{ax_{21}+cx_{22}}&{bx_{21}+dx_{22}}\e nd{array}\right)$
equating it with the zero matrix I get four equations.
$2ax_{11}+x_{12}(b+c)=0$ __1
$b(x_{22}+x_{11})+x_{12}(a+d)=0$ __2
$x_{21}(a+d)+c(x_{11}+x_{22})=0$ __3
$2dx_{22}+cx_{12}+bx_{21}=0$ __4
Now what do I do next?
Solve the system of equations, what else?! But first fix the first one: it must be
$2ax_{11}+bx_{21}+cx_{12}=0$
Tonio
• Mar 31st 2010, 03:44 AM
arze
I have tried but can't seem to get them.
• Mar 31st 2010, 04:49 AM
tonio
Quote:
Originally Posted by arze
I have tried but can't seem to get them.
You have to work harder and make a bigger effort. The matrix of coefficients of the system ( in the unknonws $x_{11},x_{12},x_{21},x_{22}$ ) is:
$\begin{pmatrix}2a&c&b&0\\b&a+d&0&b\\c&0&a+d&c\\0&c &b&2d\end{pmatrix}$ . Divide now the first row by $2a$ (why can you?) and substract from each of the 2nd, 3rd rows a corresponding multiple of the first one to obtain:
$\begin{pmatrix}1&c\slash 2a&b\slash 2a&0\\0&a+d-bc\slash 2a&-b^2\slash 2a&b\\0&-x^2\slash 2a&a+d-bc\slash 2a&c\\0&c&b&2d\end{pmatrix}$ . Interchange now the 2nd and 4th rows, divide the new 2nd row by $c$ (again, why can you) and ...etc. This is
just the Gauss-Jordan reduction method of a matrix to echelon form!:
$\begin{pmatrix}1&c\slash 2a&b\slash 2a&0\\0&1&b\slash c&2d\slash c\\0&0&-\frac{b}{c}(a+d)&\left(\frac{b}{a}-\frac{2d}{c}\right)(a+d)\\0&0&a+d&\frac{c}{a}(a+d) \end{pmatrix}$ ...or something like this (check this carefully: I didn't!) ...and etc.
Tonio
• Mar 31st 2010, 05:01 AM
arze
Quote:
Originally Posted by tonio
$\begin{pmatrix}2a&c&b&0\\b&a+d&0&b\\c&0&a+d&c\\0&c &b&2d\end{pmatrix}$ .
I don't understand this part. How did you get that?
Thanks<|endoftext|>
| 4.4375 |
304 |
The Federalist Papers, a series of 85 essays designed to encourage ratification of the United States Constitution, provide important insight on the history of U.S. federal taxation. Published in late 1787 and early 1788, they illuminate key ideas behind America's constitutional creation, including the federal taxing power.
Written by Alexander Hamilton, James Madison, and John Jay, the Federalist Papers served a dual rhetorical purpose. On the one hand, they outlined the need for a strong central government. Essays stressed the weakness of the Articles of Confederation and explained the strength of the proposed Constitution. Simultaneously, however, the essays were intended to allay fears that the new federal government would become too powerful; while arguing that the Confederation government was entirely too weak, the authors insisted that its replacement would never grow too strong.
Federal revenue was a key concern for the authors of the federalist Papers, looming large in their critique of the Articles and their defense of the Constitution. At various points, they expounded on the efficacy and fairness of consumption taxes, specifically customs duties. They insisted, however, that the federal government be granted unlimited taxing powers, including the authority to assess domestic excise taxes. Debates over "direct" vs. "indirect" taxation received considerable attention, as did the constitutional requirement for tax uniformity.
The Tax History Project has reproduced the full text of nine of the Federalist Papers, including all substantive discussion of federal revenue powers. The links above will take you to the relevant documents.<|endoftext|>
| 3.921875 |
3,340 |
### 20120229: What's Wrong with this Solution?
Refer to the set of solution written in the board.
Identify as many error that you can spot and pen them in the comments.
After discussion:
### 20120224: Submission of REPLIES to Australian Mathematics Competition 2012
The following have not submitted the form:
• Alexandre Shing
• Farruq Daniel B Humardany
• Lau Guan Heng Jose
• Seah Gui Cong Aloysius
Please submit by next week.
### Submission of REPLIES to Australian Mathematics Competition 2012
The following have not submitted the returns.
Please do so, latest by tomorrow 24 February 2012 (Friday).
• Achutha d/o Karuppiah
• Chiam Chuen
• Khor Ethan
• Koh Guo Feng
• Kota Kiran Chand
• Ng Keen Yung
• Tan Shi Jie
• Tham Chun Leong
### Commutative Law & Associative Law of Addition & Multiplication
Commutative Law of Addition
Given that 4 + 5 = 9 and 5 + 4 = 9
Therefore, 4 + 5 = 5 + 4
The order of adding any two numbers does not affect the result.
This is the commutative law of addition.
Associative Law of Addition
We can add three numbers together in two different ways:
1st way: 2 + 3 + 5
= 2 + (3 + 5)
= 2 + 8
= 10
2nd way: 2 + 3 + 5
= (2 + 3) + 5
= 5 + 5
= 10
The order of grouping the numbers together does not affect the answer.
This property is called the associative law of addition,
Commutative Law of Multiplication
Given that 2 x 6 = 12 and 6 x 2 = 12
Therefore, 2 x 6 = 6 x 2
The multiplication of numbers follows the commutative law.
Associative Law of Multiplication
Since 2 x (3 x 4) = 2 x 12, resulting 24
and (2 x 3) x 4 = 6 x 4, resulting 24
Multiplication is therefore associative.
### Real Numbers (Activity) Which Tribe do I belong to?
The solution is available at the GoogleSite > Mathematics > Class page
### Homework not submitted (updated 18 Feb)
Chapter 2: Real Numbers
The following has not submitted Homework 1a:
• Lim Zhongzhi
• Ng Keen Yung
### Chap 2: Real Numbers - Story Marathon at FaceBook
"Numbers" is part of our everyday life! It comes in different forms - Positive & Negative; Integers, Decimals, Fractions, and they can be "Irrational"!
In this story marathon, you will let your imagination run wild! Using Real Numbers, continue the story where the last person has commented. You may comment more than once. Each time, real numbers must be used, in one way or another. Do not limit yourself to only positive integers.
It is going to take place in Facebook > Group: Mathematics in Real Life
You must be a member to be able to comment.
Look for the post on 2012 S1-04 The Princess & REAL NUMBERS
This story marathon will run till Term 1 Week 10 before we consolidate it as a full story to share with others :)
### Homework (17 Feb)
Attempt the following on the topic: Real Numbers
• Homework 1b
• Worksheet 1
For those of you who have not watched the videoclips on "Real Numbers in Operation",
please watch them to understand the concepts behind the Addition and Subtraction of Integers.
Then you link it back to the technique we discuss in class:
(1) By looking at the numerical value, identify the larger one.
(2) Adopt the sign of this number (+ or -)
(3) If both numbers have the same sign, we'll add the numerical values. If both numbers have different signs, then we'll subtract the numerical values.
### Chap 2: Real Numbers - Investigative Activity on Recurring Decimals
Dear Students
The NUMBERS file for the activity has been uploaded to the GoogleSite (in your class page).
You may download to check against what you have done in the class.
During lesson, you should have indicated in your notebook:
(1) Which are the fractions that will result a recurring decimal
(2) How to write a recurring decimal
Points to Ponder:
Are the following recurring decimals?
If they are recurring decimals, how would you write them?
• 0.0212112111211112111112111111...
• 1.2031203120312031...
• 30.0029292929292...
### Chap 2: Real Numbers - Introduction of the Relationships
In the previous lessons, we discussed the relationship between the different types of numbers, and came up with a Venn Diagram to represent their relationships :)
We have also started exploring the "Recurring Decimals" using NUMBERS.
What is your observation amongst the fractions? 1/2, 1/3, 1/4, ... 1/50
Are there some characteristics of the denominators that you could tell if the number is going to be a terminating decimal, recurring decimal, or a non-terminating (& non-recurring) decimal?
What's happening next...
• Concept behind the addition and subtraction of Real Numbers - Introduction of Zero Pairs
• http://sst2012-s104maths.blogspot.com/p/real-numbers-in-operation.html
### Gentle Reminder... Level Test on 13 Feb 2012 (Monday)
Please remember to bring along with you a working calculator.
As pointed out in class, no homework is assigned over the weekend as you should be preparing for the level tests with the hand-outs that were returned to you.
As advised during the lesson, you should have done the corrections to the Homework copies, by making reference to the solutions put up in the GoogleSite.
In your file, you should have the following for your revision:
• Diagnostic Test (Factors & Multiples)
• Worksheet 1
• Homework 1(a)
• Homework 1(b)
• Worksheet 2
• Homework 2(a)
• Homework 2(b)
• Homework 2(c)
• Worksheet 3
• Homework 3(a)
• Homework 3(b)
• Homework 3(c)
• Formative Assessment
### Homework not submitted (updated 7 Feb 2012)
The following have not submitted homework:
Homework 3(b) Chiam Chuen
Homework 3(c): Tan Shi Jie
Homework 2(b): Chiam Chuen, Kota Kiran
Homework 2(c): Chiam Chuen, Kota Kiran
### Maths Level Test 13 February (Monday)
Duration: 1 hour
Chapter 1: Factors and Multiples
- page 1 to page 22 (entire chapter)
- Make reference to all the Worksheets and Homework assigned under this chapter
Chapter 2: Real Numbers - Calculator use only
- page 47 to page 49
Calculator is allowed.
### 6 AM Quiz: Factors & Multiples
Answers to QUIZ Word Problems #1 to #5
1. 14 tables in each row
• Use square root
2. 15625 cubes
• 1. Convert the lengths to common unit (i.e. cm)
• 2. Find the number of small cubes that can be lined up along the side.
• 150/6 = 25 cubes per side
• No. of cubes = 25^3 = 15625
3. 14352 cubes
• 1. Find HCF, hence the common factor of no. of cubes could be lined up along the sides.
• 2. Calculate the no. of cubes for each side by dividing the length by the 'length of the smaller cube'.
• 3. Hence the total number of cubes.
• HCF of 312, 184, 128 is 8
No. of cubes by the length = 39
No. of cubes by the breadth = 23
No. of cubes by the height = 16
Total no. of cubes = 14352
4. 64 m
• 1. Find area of the triangle
• 2. Using the area, find the length of square
• 3. Find perimeter
• Area of triangle = 256
Length of square = sqrt(256) = 16
Perimeter = 64
5. Date: 15 April 2011
• 1. Find LCM
• 2. Count from the day after 1 Jan 2011
• LCM of 8, 13 = 104Jan = 31-1 = 30 days (reason being both planes took off on 1 Jan, so we start counting from 2 Jan); Feb = 28 days; Mar = 31 daysTotal = 89 days (short of 15 days)
### Chap 2: Real Numbers - Investigative Activity (I) Addition of Numbers
In this investigative activity, you are going to add integers of different combinations:
• positive and negative integers
• negative and negative integers
Observe patterns amongst the answers generated.
Draw up a 'rule' that would guide us when we add integers (especially when it involves negative integers).
Vocabulary List: Numerical Part. Sign. Positive. Negative. Difference. Sum. Same.
You are going to work in pairs or threes.
1. One of you will download the file "Chapter 2 Investigative Activity - Addition and Product.numbers" (from GoogleSite: 01 Mathematics). Another will display the blog post so that you could enter your discussion into the Comments.
2. Refer to the worksheet "Addition of Numbers":
• (a) Enter formula to find the value of "a + b" of the first set of integers (i.e. 10 + (-1)). The spreadsheet will compute the value automatically for you.
• (b) Copy the formula to the rest of the cells below; also to the next column.
3. Discuss the patterns you observe between the addition of integers:
• (a) What happens when a positive integer + a negative integer (given that the positive number has a greater numerical value)?
• (b) What happens when a positive integer + a negative integer (given that the positive number has a smaller numerical value)?
• (c) What happens when a negative integer + a negative integer? How similar or different it is when compared the sum of two positive numbers?
4. Pen the observations down in the Comments. Suggest a rule that would help one when adding two integers.
5. Sign off the Comments with the name(s) of both of you.
### Chap 2: Real Numbers - Investigative Activity (II) Product of Integers
In this investigative activity, you are going to multiply integers of different combinations:
• positive and positive integers
• positive and negative integers
• negative and negative integers
Observe patterns amongst the answers generated.
Draw up a 'rule' that would guide us when we multiply integers (especially when it involves negative integers).
Vocabulary List: Numerical Part. Sign. Positive. Negative. Product.
You are going to work in pairs or threes.
1. One of you will download the file "Chapter 2 Investigative Activity - Addition and Product.numbers" (from GoogleSite: 01 Mathematics). Another will display the blog post so that you could enter your discussion into the Comments.
2. Refer to the worksheet "Product of Numbers":
• (a) Enter formula to find the value of "a x b" of the first set of numbers (i.e. (-4) x (-4)).
• Press ENTER and the spreadsheet will compute the value automatically.
• (b) We are going to tell the spreadsheet to use the row of numbers in Row 2 and Column B for computation.
• At the formula bar, click at C2 and select (C\$2) absolute row. The spreadsheet will use the numbers in Row 2 for calculation.
• At the formula bar, click at B3 and select (\$B3) absolute column. The spreadsheet will use the numbers in Column B for calculation.
• (c) To copy the formula, select cell B3, place the cursor at the bottom right corner. It will turn to a cross-hair. Now drag it to cover the entire area to be computed.
3. Discuss the patterns you observe between the product of integers:
• (a) What happens when a positive integer x a negative integer (given that the positive number has a greater numerical value)?
• (b) What happens when a positive integer x a negative integer (given that the positive number has a smaller numerical value)?
• (c) What happens when a negative integer x a negative integer? How similar or different it is when compared the product two positive numbers?
4. Pen the observations down in the Comments. Suggest a rule that would help one when adding two numbers.
5. Sign off the Comments with the name(s) of both of you.
### Maths Olympiad Selection Test
Information pertaining to the Selection Test is now available at the Student Blog.
### Homework (2 Feb)
Worksheet 3, on Square Roots & Cube Roots is issued today.
This is part of your preparation for the Level Test in Week 7.
Please try them and we shall discuss this on next Tuesday when we meet.
A 6AM Quiz has been scheduled this Saturday, as part of your preparation for the level test. This is optional; however, it's a good opportunity to test your own understanding :)
Remember to explore NUMBERS as it will be used in the Investigative activity next Tuesday.
### eTextbook: Activation Code (updated 6 Feb)
Dear Students
A digital copy of the Maths textbook is available.
For those of you who bought the learning device through NCS, it has been already been installed in your device, except those who bought Macbook Air*.
Use Spotlight to search for iBook.
You will be prompted to enter the serial number. Enter 22169 (which is compulsory) and the second number is either 31718 OR 19607
• For those who bought the Macbook Air and did not buy the learning device through NCS, please see Mr Lim at the HelpDesk to help install the softcopy of the textbook.
• Suggestion: Make an appointment in the mornng with Mr Lim before morning assembly or during recess.
Note: Some of you pointed out that there's problem accessing the eTextbook. We are checking out with the vendor. Will update all once we get the update on the access and the serial number.
### Homework (1 Feb)
There is no homework for the day.
Please remember to submit:
- WORKSHEET 2 in the last Maths lesson of this week
- WORKSHEET 3 in the first Maths lesson of next week
### Group 1
BY SHIJIE, JIAWEN,QIANZHE,ACHUTHA and ZHONGZHI
### Chap 2: Number Puzzle
You are going to arrange the number cards in ascending order.
Take a picture of the final arrangement.
One person in the Group will post it in the blog with
• Subject Title: Chap 2: Number Puzzle (by Group.....)
• Insert a label: Real Numbers
After it is posted, each member of the group will go to "Comment", to describe how the group solved the problem, such that it can serve as an advice to anyone given the same puzzle.
### Chap 2: Real Numbers - Negative Numbers
Group 1:
http://www.blogger.com/img/blank.gif
Group 2:
http://www.blogger.com/img/blank.gifhttp://www.blogger.com/img/blank.gif
Group 3:
Group 4:
Group 5:<|endoftext|>
| 4.40625 |
635 |
# SAT Practice Test 4, Section 2: Questions 1 - 5
Related Topics:
More Lessons for SAT math
More Resources for SAT
Math Worksheets
This is for SAT in Jan 2016 or before.
The following are worked solutions for the questions in the math sections of the SAT Practice Tests found in the The Official SAT Study Guide Second Edition.
It would be best that you go through the SAT practice test questions in the Study Guide first and then look at the worked solutions for the questions that you might need assistance in. Due to copyright issues, we are not able to reproduce the questions, but we hope that the worked solutions will be helpful.
Given:
1 package contains 12 rolls
Other packages contain 8 rolls
She bought 5 packages
To find:
The number of rolls purchased
Solution:
She bought 5 packages: 1 package with 12 rolls and 4 packages with 8 rolls.
1 × 12 + 4 × 8 = 12 + 32 = 44
Given:
A, B and C are points on a line in that order.
AB = 30
BC is 20 more than AB
To find:
AC
Solution:
AB = 30.
BC is 20 more than AB; BC = 20 + 30 = 50
AC= AB + BC = 30 + 50 = 80
Given:
x + 3 = a
To find:
2x + 6
Solution:
2x + 6 = 2(x + 3) = 2a
Given:
The graph
To find:
The student whose change from Test I to Test II is the greatest
Solution:
The x-coordinate of a point represents the score on Test I.
The y-coordinate of a point represents the score on Test II.
We can calculate the difference between the x and y coordinates to determine the change in scores.
Point A is (40, 70), 70 – 40 = 30
Point B is (40, 60), 60 – 40 = 20
Point C is (60, 70), 70 – 60 = 10
Point D is (80, 80), 80 – 80 = 0
Point E is (80, 60), 80 – 60 = 20
The point with the greatest difference is A.
Given:
The graph
To find:
Average of the scores of the 5 students on Test II
Solution:
Topic(s): Statistics
To get the average of the scores of 5 students on test II, add up all the scores of Test II (which is the y-coordinates of the points) and divide by 5
Rotate to landscape screen format on a mobile phone or small tablet to use the Mathway widget, a free math problem solver that answers your questions with step-by-step explanations.
You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.<|endoftext|>
| 4.78125 |
572 |
#### Is there a formula to add a sequence of cubes?
By examining the first five sums a remarkable discovery is suggested:
13 = 1
13+23=9
13+23+33=36
13+23+33+43=100
13+23+33+43+53=225
It seems that the sum is always square, but what is even more remarkable is that the sum of the first n cubes, 13+23+...+n3 = (n(n+1)/2)2, which is the square of the nth triangle number.
For example, 13+23+...+103=(10×11/2)2=552 = 3025.
Using a similar method used to prove that formula for the Sum of Squares, we shall prove this result deductively; it is hoped that it will offer some insight into how further the series of powers may be found.
Proof
n ∑ r=1
r4–(r–1)4 = n4–(n–1)4 + (n–1)4–(n–2)4 + ... + 34–24 + 24–14 + 14–04
= n4
But r4–(r–1)4 = r4 – (r4–4r3+6r2–4r+1) = 4r3–6r2+4r–1.
∴ ∑ 4r3–6r2+4r–1 = 4∑ r3 – 6∑ r2 + 4∑ r – ∑1 = 4∑ r3 – 6n(n+1)(2n+1)/6 + 4n(n+1)/2 – n = 4∑ r3 – n(n+1)(2n+1) + 2n(n+1) – n = n4.
∴ 4∑ r3 = n4 + n(n+1)(2n+1) – 2n(n+1) + n = n(n3 + (n+1)(2n+1) – 2(n+1) + 1 = n(n3 + 2n2+3n+1 – 2n–2 + 1) = n(n3+2n2+n) = n2(n2+2n+1) = n2(n+1)2 ∴ ∑ r3 = n2(n+1)2/4 = (n(n+1)/2)2
In other words, the sum of the first n cubes is the square of the sum of the first n natural numbers.<|endoftext|>
| 4.53125 |
250 |
Tech moves fast! Stay ahead of the curve with Techopedia!
Join nearly 200,000 subscribers who receive actionable tech insights from Techopedia.
A remote computer is a computer that a user has no access to physically, but may be able to access it remotely via a network link from another computer. Remote connections are made through the use of a network which connects the computer and the device that is used to access it.
Remote access software is used to be able to control the remote computer just as if the computer is right in front of the user. TeamViewer, VNC and Remote Desktop are just a couple of examples of such software.
The TPC/IP protocol is used for connections, especially ones done over the Internet. Since a computer’s IP address is unique, it is used to specifically identify the computer to be accessed remotely through a known network or the Internet.
Remote computers can be quite handy in certain scenarios. They can be helpful when a user forgot some of his/her files in their home or office computer and wants to access and retrieve them even though it is no longer physically possible. They can also be very useful in business environments, when holding conferences and computer maintenance or assistance.<|endoftext|>
| 3.828125 |
529 |
# growing patterns
Did you know that bacteria, viruses, and every living cell grow in patterns? Mathematicians enjoy investigating growing patterns to understand and describe their growth.
Click on the grade tabs to see different growing pattern activities.
Take a close look at the pattern below:
• What is happening as the pattern grows from Stage 1 to Stage 4?
• What is happening with the blue blocks?
• What is happening with the red blocks?
• Predict what the blocks will look like in Stage 5.
Complete the table below:
Now complete the table for this other pattern:
Now you can draw graph bars of these patterns. Can you complete the ones below?
• How are the bar graphs similar?
• How are they different?
• What makes one graph steeper than the other?
• Is it the red blocks, the blue blocks, or both?
Create a different growing pattern using two colours:
• Build the pattern using cubes
• Complete the table below for your pattern
• Draw a bar graph of your pattern. You may use this Square Grid Handout
• Use words to describe your pattern. What changes? What is constant?
#### Making changes to your pattern
How would you change the pattern to have the following effects?
• Make the bar graph steeper.
• Make the bar graph less steep.
• Make the bar graph taller.
• Make the bar graph shorter.
• Make all the bars the same height.
Watch mathematician Lindi Wahl explaining variables and constants:
#### Keep exploring growing (or shrinking?) patterns!
Here are some more challenges for you:
• Can you make a pattern that is shrinking?
• Can you use 3 colours to build growing patterns?
• Where only 1 colour is growing
• Where 2 colours are growing
#### How bacteria grow
Watch mathematician Lindi Wahl explaining bacteria growing pattern.
#### Using symbols to express patterns
In all the activities so far, you have used blocks, tables, and graphs to express patterns, but the way mathematicians do it is by using symbols.
Look at the example below:
In the following table, we will label the Stage # as “X” and the # of blocks will be “Y”
These are called linear equations because they make a straight line when we graph them.
Can you figure out the expressions for the growing patterns below?
Watch mathematician Lindi Wahl representing patterns through equations:
#### What did you learn?
• What did you learn about constants and variables?
• What surprised you?
• What math insights did you experience?
Share your learning with others, so they can experience math surprise and insight too.<|endoftext|>
| 4.65625 |
420 |
At the North Pole during summer when there is permanent daylight, does the Moon rise and set, and if not, does that mean that during winter with no sunrise the Moon is always up?
The Moon does rise and and set during both summer and winter on the North Pole (or South Pole). The exact movement is complicated, but can be understood the combination of two separate movements:
1) Rotation of the Earth on its axis, which results in movements that change over the course of one day.
2) Orbit of the Moon around the earth, which results in movements that change over the course of one lunar month (about 29 days).
While the Moon does rise during the summer at the North Pole, since the Sun is always up, you generally can't see it, so I'll focus on the movement of the Moon during the winter.
The daily movement from Earth's rotation causes the Moon to circle once around the sky. If you spent the entire day staring at it, you'd have to turn around exactly once. This movement is also the same that the Sun makes during the summer. To give you a better idea of how this looks, here is a video showing how the Sun moves in the sky at the North Pole: Arctic Midnight Sun
The second movement caused by the Moon's orbit around the Earth is analogous to the movement of the Sun over the course of a year only it repeats over the course of a lunar month. Near the new Moon phase, the Moon is near the Sun and therefore never rises during the winter. As the Moon approaches full, it will start to pop up above the horizon. Eventually near the full Moon phase it will be high enough in the sky to stay up all day and circle like the Sun in the video above. The elevation of the circle will rise as the Moon becomes completely full and then start to decrease until it begins to dip below the horizon. Eventually the Moon will stop rising at all as it gets close enough to the new phase. The cycle then repeats.
This page was last updated on July 18, 2015.<|endoftext|>
| 3.703125 |
750 |
Beyond our solar system lie million of stars, nebulae, galaxies, and clusters that are visible from Earth. Often these objects are difficult to see except with a large telescope, but there are also many fascinating deep sky objects that can be seen with small telescopes. Charles Messier, a French astronomer who lived in the late 1700s, created a catalog of 110 objects. His reason for doing this was so that amateur astronomers searching for comets would not mistake bright nebulae, clusters, and galaxies for comets. However, today Messier’s catalog serves as a good guide to the most brilliant objects that can be viewed with a small telescope.
A STAR is, like our sun, an enormous ball of gas that releases energy created through nuclear fusion in the star’s core. All stars are not the same, however. Some are hotter than others, some are more massive, and some are dying while others are being born. The life cycle of a star is complicated, and depends on its mass and other factors. For instance, a very heavy star might explode in a supernova and then become a black hole. A lighter star will eventually run out of fuel, cool down, and become a black dwarf. Stars usually occur in groups called clusters, which are gravitationally bound.
DOUBLE and MULTIPLE STARS are linked by gravity, so that they orbit one another in some way. In this respect, our own sun is somewhat unusual in being single, because about three quarters of all stars are in groups.
OPEN CLUSTERS are groups of dozens to hundreds of stars which are loosely bound by gravitation. Open clusters generally lie in the spiral arms of the Milky Way.
GLOBULAR CLUSTERS are usually spherically shaped, and contain hundreds of thousands of old stars. These clusters lie around our galaxy in a halo.
A NEBULA is a cloud of gas and dust where new stars are being formed. For this reason, nebulae are sometimes called ‘stellar nurseries.’ A nebula is created from the remnants of a star in the last stage of its life. For instance, a supernova explosion will eventually result in the formation of a nebula, which will in turn become a birthplace for new stars. Nebulae are some of the most beautiful objects in the sky, and are often named after their shape, like the North American Nebula, the Rosette Nebula, and the Horsehead Nebula.
A GALAXY is a huge grouping of stars, nebulae, and dust that is held together by gravitational forces. Our galaxy, which has about 100 billion stars in it, is called the Milky Way, because of the bright yet hazy band that we see sweeping across our sky. The nebulae, stars, and clusters that were just discussed are all in our galaxy. From Earth, we can also see through our own galaxy to identify other galaxies in our corner of the universe. There are several different kinds of galaxies: spiral ones have hazy arms that twist around the center (the Milky Way is a spiral galaxy), barred spiral ones have a thick band of stars near the center, and elliptical ones are nearly circular in shape. Galaxies are also grouped together in clusters. Our galactic cluster has about 30 galaxies in it, of which the Andromeda Galaxy is the largest.
REFERENCES FOR THIS PAGE:
Ian Ridpath, Eyewitness Handbooks: Stars and Planets, D.K. Publishing, 1998.
The photographs used on this page are reproduced for educational purposes from the NASA, STScI, and Hubble Space Telescope Picture Galleries.<|endoftext|>
| 4.0625 |
310 |
With the unprecedented proliferation of Internet of Things (IoT), the underpinning technology that makes it possible to communicate and sense, needs to be scaled up. A group of scientists from the University of Virginia and Purdue University have come up with a new method of fabrication in which thin-film and tiny electronic circuits can be peeled from the surface. This method not just enables to do away with the cumbersome manufacturing steps but also the costs associated with them. It also enables an object to unravel its environment and be monitored through the use of Electronic Stickers.
Electronic Stickers Made from Silicon Wafers Can Enable Wireless Communication
Electronic Stickers can bring about wireless communications too in the course of time. In this manner the stickers could be customized so that they can easily be stuck to drones meant to be sent to areas that spell danger. Those could include areas where there is a major gas leak.
The electronic circuits that we see today are built separately on their silicon “wafer,” which forms a rigid and flat substrate. This wafer made from silicon is then capable of withstanding high temperatures and chemical etching. The two are leveraged to remove the circuits from the wafer.
With the help of the new fabrication technique developed by the group of researchers which is known as “transfer printing” cost of manufacturing can come down since it needs just a single wafer for creating numerous thin films holding electronic circuits. The silicon wafer can be peeled off at room temperature without needing high temperature for the same.<|endoftext|>
| 3.921875 |
556 |
# Prove that both the roots of the equation $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$ are real but they are equal only when $a=b=c$.
Given:
Given quadratic equation is $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$.
To do:
We have to prove that both the roots of the equation $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$ are real but they are equal only when $a=b=c$.
Solution:
$(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$
$x^2-ax-bx+ab+x^2-bx-cx+bc+x^2-cx-ax+ac=0$
$3x^2+(-a-b-b-c-c-a)x+(ab+bc+ca)=0$
$3x^2-2(a+b+c)x+(ab+bc+ca)=0$
Comparing the given quadratic equation with the standard form of the quadratic equation $ax^2+bx+c=0$, we get,
$a=3, b=-2(a+b+c)$ and $c=(ab+bc+ca)$.
The discriminant of the standard form of the quadratic equation $ax^2+bx+c=0$ is $D=b^2-4ac$.
$D=[-2(a+b+c)]^2-4(3)(ab+bc+ca)$
$D=4(a^2+b^2+c^2+2ab+2bc+2ca)-12(ab+bc+ca)$
$D=4(a^2+b^2+c^2+2ab+2bc+2ca-3ab-3bc-3ca)$
$D=4(a^2+b^2+c^2-ab-bc-ca)$
$D=2(2a^2+2b^2+2c^2-2ab-2bc-2ca)$
$D=2(a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ca+c^2)$
$D=2[(a-b)^2+(b-c)^2+(c-a)^2]$
$D>0$ or $D=0$ when $a=b=c$
Therefore, the roots of the equation $(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0$ are real but they are equal only when $a=b=c$.
Tutorialspoint
Simply Easy Learning<|endoftext|>
| 4.59375 |
1,317 |
# Degrees of Freedom: What are they?
Hypothesis Testing > Degrees of Freedom
Degrees of freedom are used in hypothesis testing.
## What are Degrees of Freedom?
Degrees of freedom in the left column of the t distribution table.
Degrees of freedom of an estimate is the number of independent pieces of information that went into calculating the estimate. It’s not quite the same as the number of items in the sample. In order to get the df for the estimate, you have to subtract 1 from the number of items. Let’s say you were finding the mean weight loss for a low-carb diet. You could use 4 people, giving 3 degrees of freedom (4 – 1 = 3), or you could use one hundred people with df = 99.
In math terms (where “n” is the number of items in your set):
Degrees of Freedom = n – 1
Why do we subtract 1 from the number of items? Another way to look at degrees of freedom is that they are the number of values that are free to vary in a data set. What does “free to vary” mean? Here’s an example using the mean (average):
Q. Pick a set of numbers that have a mean (average) of 10.
A. Some sets of numbers you might pick: 9, 10, 11 or 8, 10, 12 or 5, 10, 15.
Once you have chosen the first two numbers in the set, the third is fixed. In other words, you can’t choose the third item in the set. The only numbers that are free to vary are the first two. You can pick 9 + 10 or 5 + 15, but once you’ve made that decision you must choose a particular number that will give you the mean you are looking for. So degrees of freedom for a set of three numbers is TWO.
For example: if you wanted to find a confidence interval for a sample, degrees of freedom is n – 1. “N’ can also be the number of classes or categories. See: Critical chi-square value for an example.
## Degrees of Freedom: Two Samples
If you have two samples and want to find a parameter, like the mean, you have two “n”s to consider (sample 1 and sample 2). Degrees of freedom in that case is:
Degrees of Freedom (Two Samples): (N1 + N2) – 2.
## Degrees of Freedom in ANOVA
Degrees of freedom becomes a little more complicated in ANOVA tests. Instead of a simple parameter (like finding a mean), ANOVA tests involve comparing known means in sets of data. For example, in a one-way ANOVA you are comparing two means in two cells. The grand mean (the average of the averages) would be:
Mean 1 + mean 2 = grand mean.
What if you chose mean 1 and you knew the grand mean? You wouldn’t have a choice about Mean2, so your degrees of freedom for a two-group ANOVA is 1.
Two Group ANOVA df1 = n – 1
For a three-group ANOVA, you can vary two means so degrees of freedom is 2.
It’s actually a little more complicated because there are two degrees of freedom in ANOVA: df1 and df2. The explanation above is for df1. Df2 in ANOVA is the total number of observations in all cells – degrees of freedoms lost because the cell means are set.
Two Group ANOVA df2 = n – k
The “k” in that formula is the number of cell means or groups/conditions.
For example, let’s say you had 200 observations and four cell means. Degrees of freedom in this case would be: Df2 = 200 – 4 = 196.
## Why Do Critical Values Decrease While DF Increase?
Thanks to Mohammed Gezmu for this question.
Let’s take a look at the t-score formula in a hypothesis test:
When n increases, the t-score goes up. This is because of the square root in the denominator: as it gets larger, the fraction s/√n gets smaller and the t-score (the result of another fraction) gets bigger. As the degrees of freedom are defined above as n-1, you would think that the t-critical value should get bigger too, but they don’t: they get smaller. This seems counter-intuitive.
However, think about what a t-test is actually for. You’re using the t-test because you don’t know the standard deviation of your population and therefore you don’t know the shape of your graph. It could have short, fat tails. It could have long skinny tails. You just have no idea. The degrees of freedom affect the shape of the graph in the t-distribution; as the df get larger, the area in the tails of the distribution get smaller. As df approaches infinity, the t-distribution will look like a normal distribution. When this happens, you can be certain of your standard deviation (which is 1 on a normal distribution).
Let’s say you took repeated sample weights from four people, drawn from a population with an unknown standard deviation. You measure their weights, calculate the mean difference between the sample pairs and repeat the process over and over. The tiny sample size of 4 will result a t-distribution with fat tails. The fat tails tell you that you’re more likely to have extreme values in your sample. You test your hypothesis at an alpha level of 5%, which cuts off the last 5% of your distribution. The graph below shows the t-distribution with a 5% cut off. This gives a critical value of 2.6. (Note: I’m using a hypothetical t-distribution here as an example–the CV is not exact).
Now look at the normal distribution. We have less chance of extreme values with the normal distribution. Our 5% alpha level cuts off at a CV of 2.
Back to the original question “Why Do Critical Values Decrease While DF Increases?” Here’s the short answer:
Degrees of freedom are related to sample size (n-1). If the df increases, it also stands that the sample size is increasing; the graph of the t-distribution will have skinnier tails, pushing the critical value towards the mean.<|endoftext|>
| 4.625 |
1,664 |
# 3.5 Word problems
Page 1 / 1
This module covers word problems involving simultaneous equations.
Many students approach math with the attitude that “I can do the equations, but I’m just not a ‘word problems’ person.” No offense, but that’s like saying “I’m pretty good at handling a tennis racket, as long as there’s no ball involved.” The only point of handling the tennis racket is to hit the ball. The only point of math equations is to solve problems. So if you find yourself in that category, try this sentence instead: “I’ve never been good at word problems. There must be something about them I don’t understand, so I’ll try to learn it.”
Actually, many of the key problems with word problems were discussed in the very beginning of the “Functions” unit, in the discussion of variable descriptions. So this might be a good time to quickly re-read that section. If you can correctly identify the variables, you’re half-way through the hard part of a word problem. The other half is translating the sentences of the problem into equations that use those variables.
Let’s work through an example, very carefully.
## Simultaneous equation word problem
A roll of dimes and a roll of quarters lie on the table in front of you. There are three more quarters than dimes. But the quarters are worth three times the amount that the dimes are worth. How many of each do you have?
• Identify and label the variables.
• There are actually two different, valid ways to approach this problem. You could make a variable that represents the number of dimes; or you could have a variable that represents the value of the dimes. Either way will lead you to the right answer. However, it is vital to know which one you’re doing! If you get confused half-way through the problem, you will end up with the wrong answer.
Let’s try it this way:
$d$ is the number of dimes
$q$ is the number of quarters
• Translate the sentences in the problem into equations.
• “There are three more quarters than dimes” $\to q=d+3$
• “The quarters are worth three times the amount that the dimes are worth” $\to \text{25}q=3\left(\text{10}d\right)$
• This second equation relies on the fact that if you have $q$ quarters, they are worth a total of $\text{25}q$ cents.
• Solve.
• We can do this by elimination or substitution. Since the first equation is already solved for $q$ , I will substitute that into the second equation and then solve.
$\text{25}\left(d+3\right)=3\left(\text{10}d\right)$ $\text{25}d+\text{75}=\text{30}d$ $\text{75}=5d$ $d=\text{15}$ $q=\text{18}$
So, did it work? The surest check is to go all the way back to the original problem—not the equations, but the words. We have concluded that there are 15 dimes and 18 quarters.
“There are three more quarters than dimes.” $✓$
“The quarters are worth three times the amount that the dimes are worth.” $\to$ Well, the quarters are worth $\text{18}\cdot \text{25}=4\text{.}\text{50}$ . The dimes are worth $\text{15}\cdot \text{10}=1\text{.}\text{50}$ . $✓$
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!<|endoftext|>
| 4.75 |
1,238 |
The Spectrum’s decision-making continuum identifies 11 teaching styles. Each of the style’s decision patterns establish a significantly different developmental opportunity in subject matter and behavior; therefore, the Spectrum styles are referred to as landmark teaching styles.
Although there are eleven distinct landmark styles, there exists a vast, if not infinite, array of pedagogical variations between styles, called Canopies. Both landmark and canopy styles are comprised of specific sets of decisions that recruit human attributes (characteristics) on different Developmental Channels allowing the objectives of each style or canopy the possibility of achievement. The difference between the two is that canopy decisions and objectives are more alike landmark styles than different. Canopy designs do not indicate significantly different developmental opportunities, because they do not have a unique decision configuration. They are close copies of and similar variations of the landmark style; not uniquely different. Canopies support and reinforce existing landmark decision patterns and objectives. Therefore, canopies are located between the two landmark styles that share their similar decision configuration.
Canopy designs need a distinct designation. They should not be referred to as a landmark style nor should they be labeled with only a name or a letter as we do for the landmark episodes. Canopy designs do not have the same O-T-L-O (decision pattern) as the landmark styles. Therefore, a distinct designation is needed. Canopy designs or variations represent two approaches.
1. A canopy episode can indicate that some of the landmark decisions have been added, deleted or adjusted. The distinct representation for such a canopy is to indicate a carrot symbol ‘^’ above the style name or letter and to indicate the decision change. This carrot symbol distinguishes the canopy teaching episode from a landmark episode and it indicates the styles the canopy is between. This canopy design focuses on changes within the landmark decisions.
2. A canopy episode can indicate a merging of two distinct styles. The distinct representation for such a canopy is to indicate a slash ‘/’ between two style names. This symbol indicates that the essence of two styles are merged in one episode. The first style letter represents the primary objective of the episode and the second style indicates additional parameters on the episode. For example, Canopy H/E ; the primary objective is the discovery of multiple responses to a task and this specific canopy is designed with multiple levels of difficulty.
Both canopy teaching options indicates that a modification has been made to the decision structure of the indicated style(s) influencing the set of corresponding objectives to be highlighted. Canopy designations are critical when researching classroom teaching and learning because landmark and canopy teaching do not have the identical decision expectations and objectives. Assumptions about a landmark style can not be made when teaching with canopies.
Canopy designations indicate: the teaching style and the specific landmark decision(s) that are modified The following examples provide the canopy designation and the verbal behavior for reading the designation.
Read as: Canopy of Style A plus socialization
Meaning: This teaching experience adheres to the landmark decision structure of the Command Style. In addition, the task incorporates socialization decisions.
Read as: Canopy of Convergent Discovery Style and Guided Discovery Style
Meaning: Both Convergent Discovery and Guided Discovery lead the learner to the discovery of an anticipated targeted response. However, these styles are very different in their structure. Guided Discovery is a series of question while Convergent Discovery provides one question and each learner is challenged to design the series of questions that lead to the anticipated response. A canopy G/F is designed to help those learners who get stuck along the way by providing occasional clue questions to clarify the thinking direction.
Read as: canopy of Divergent Discovery Style and Command Style.
Meaning: the intent of this canopy episode is to produce discovered divergent responses in a short period of time. For example: improvisation skits, science robot design competitions, cooking challenges, etc.
The concept underlying the Inclusion Style—the slanted rope concept—can be applied to the content in all teaching styles and canopies. It is possible to design tasks with multiple levels of difficulty in all styles. Implementing canopies with the added concept of inclusion requires clarity of the decisions that will remain or change in the canopy design.
The canopy design variations are infinite. They satisfy a need in learner’s development. The more teachers can diagnose the learning difficulty, the easier it is to design canopy lessons that will help reduce the learners’ content gaps.
The Spectrum is limited to just eleven styles.
Since there are just eleven teaching styles in the Spectrum, there must be times when we are not using Spectrum and using other teaching methods. This misconception leads teachers to say: “I’m not doing Spectrum today rather I’m doing… Cooperative Learning, Sports Education, TGFU, Constructionist activities, fun activities, social responsibility activities, or Character Education, etc.”
This thinking is inconsistent with the very philosophical foundation of the Spectrum, which is: teaching is a chain of decision making and a NON-VERSUS framework. The Spectrum is a universal framework that underlies all teaching methods and approaches. Each of these approaches support a decision configuration for the teacher and the learner; therefore, each have a pedagogical association that is closer to one style than another.
The Spectrum is something we do naturally.
If all teaching methods are comprised of decisions and objectives, then I’m already doing the Spectrum and don’t need to learn it.
This thinking suggests that anything a teacher does is considered quality teaching or Spectrum teaching. There are few professions where its members are “naturally” trained. Spectrum teaching is deliberate teaching. Knowing the learning intent, selecting the corresponding decisions, developing multiple cognitive operations, acknowledging the importance of attributes, and developing a repertoire of alternative teaching approaches is required of all teaching. The Spectrum offers foundational knowledge for all teaching methods. It is a deliberate, non-versus approach to teaching and learning.<|endoftext|>
| 4.0625 |
488 |
# What is the formula for central difference?
## What is the formula for central difference?
f (a) ≈ slope of short broken line = difference in the y-values difference in the x-values = f(x + h) − f(x − h) 2h This is called a central difference approximation to f (a). In practice, the central difference formula is the most accurate.
### What is the first central difference method?
The 1st order central difference (OCD) algorithm approximates the first derivative according to , Plot your results on two graphs over the range , comparing the analytical and numerical values for each of the derivatives.
#### What is the formula for Newton’s forward formula for first derivative?
Newton’s forward differentiation table is as follows. Solution: Equation is f(x)=2×3-4x+1.
What is central finite difference approximation of derivatives?
If a finite difference is divided by b − a, one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems.
Can a central difference be calculated for a first derivative?
For starters, the formula given for the first derivative is the FORWARD difference formula, not a CENTRAL difference. Second: you cannot calculate the central difference for element i, or element n, since central difference formula references element both i+1 and i-1, so your range of i needs to be from i=2:n-1.
## What’s the name of the central difference formula?
This is usually called the forward difference approximation. The reason for the word forward is that we use the two function values of the points x and the next, a step forward, x + h. Similarly, we can approximate derivatives using a point as the central point, i.e. if x is our central point we use x − h and x + h.
### What are the different formulas for numerically approximating derivatives?
There are 3 main difference formulas for numerically approximating derivatives. The central difference formula with step size h is the average of the forward and backwards difference formulas
#### How are finite differences used to solve differential equations?
The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences.<|endoftext|>
| 4.5 |
873 |
# Coordinate Graph: Definition & Examples
Instructor: Laura Pennington
Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.
Learn what a coordinate graph is and about its characteristics. Through definition, story, and example, we will see how to navigate a coordinate graph and how to locate and plot points on these types of graphs.
## A Fly On The Wall
Rene Descartes was a French mathematician. He lived from 1596-1650 and made many contributions to mathematics.
Legend has it that when Descartes was a child, he was often ill and spent a lot of time resting in bed. One time, while in bed, he noticed a fly on the ceiling. As the fly moved around, he noticed that the fly's position was determined by its distance from each of the corners of the walls with the ceiling of the room. After making this realization, he decided to represent the lines in which two of the walls met the ceiling with number lines, and the rest is history!
## Coordinate Graph
Descartes' observation of that fly led him to develop a coordinate graph. A coordinate graph is also sometimes called a coordinate plane, a Cartesian plane, or a Cartesian coordinate system. A coordinate graph consists of two number lines that run perpendicular to each other. These number lines are called axes. The horizontal line is called the x-axis, and the vertical line is called the y-axis. These two axes intersect where they are both equal to zero. This intersection point is called the origin.
## Ordered Pair
Now that we have seen what a coordinate graph looks like, let's consider Descartes' fly observation. Imagine there is a fly somewhere on our graph as shown in the image below.
Notice that if we start at the origin, we have to move 2 units to the left, and then we have to move 3 units up to get to the fly. This position lines up with -2 on the x-axis and 3 on the y-axis. We represent this position using an ordered pair. An ordered pair (x, y) represents a position of a point on a coordinate graph, where x is the number on the x-axis that the point lines up with, and y is the number on the y-axis that the point lines up with. The numbers x and y in the ordered pair (x, y) are called coordinates. The first number in the pair is called the x-coordinate, and the second number in the pair is called the y-coordinate.
In our fly example just described, we said that the fly lines up with -2 on the x-axis and 3 on the y-axis. Therefore the ordered pair representing the position of the fly on our graph is (-2,3).
## Plotting Ordered Pairs
When given an ordered pair (a, b), where a is the x-coordinate and b is the y-coordinate, we can plot it on a coordinate graph. To do so, we follow these steps.
1. Start at the origin, and move a units horizontally along the x-axis. If a is positive, we move a units to the right of the origin, and if a is negative, we move a units to the left of the origin.
2. From the point we just moved to on the x-axis, move vertically b units. If b is positive, we move up b units, and if b is negative, we move down b units.
3. We are now at the position that our ordered pair represents, so we draw a point, and we have plotted our ordered pair.
To unlock this lesson you must be a Study.com Member.
### Register to view this lesson
Are you a student or a teacher?
#### See for yourself why 30 million people use Study.com
##### Become a Study.com member and start learning now.
Back
What teachers are saying about Study.com
### Earning College Credit
Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.<|endoftext|>
| 4.90625 |
1,534 |
If you've ever taken a personality test, it was probably in a lifestyle magazine ("What kind of adventurer are you? Take this quiz to find out!") or maybe at the behest of a friend who's a Meyers-Briggs believer. But these fluffy diversions have a serious, often dark history. In fact, one of the earliest personality tests was developed during World War II to determine who might become an authoritarian and join the Nazi movement.
In 1943, three psychology professors at the University of California at Berkeley were struggling to understand the most horrific European genocide in a generation. As the war raged overseas, Daniel Levinson, Nevitt Sanford, and Else Frenkel-Brunswik decided to use the greatest power at their disposal—scientific rationality—to stop fascism from ever rising again. They did it by inventing a personality test eventually named the F-scale, which they believed could identify potential authoritarians. This wasn't some plot to weed out bad guys. The researchers wanted to understand why some people are seduced by political figures like Adolf Hitler, and they had a very idealistic plan to improve education so that young people would become more skeptical of Hitler's us-or-them politics.
The rise of personality testing
As they cooked up a research plan, the Berkeley group borrowed ideas from a somewhat checkered tradition in psychology that held that personalities could be broken down into discrete character traits. In the late nineteenth century, pseudoscientists like Francis Galton, best known for popularizing the idea of eugenics, believed that human "character" could be measured the same way "the temper of a dog can be tested." This idea gained traction, and the first personality tests were developed by the US Army during World War I so millions of soldiers could be tested for vulnerability to "shell shock," an early term for post-traumatic stress.
If we could test soldiers for shell shock, why not test citizens for anti-Semitism and a tendency to follow dictators? That's what the Berkeley researchers decided to do. Their idea was compelling enough to net them a grant of $500 from the psychology department in 1943. In the year that followed, Sanford, Frenkel-Brunswik, and Levinson created several versions of a personality test they hoped would identify potential authoritarians, their term for people who would follow leaders with fascist or genocidal tendencies. They conducted in-depth personal interviews and administered written personality tests to hundreds of Berkeley students. Some of the answers they got seemed to reflect democratic values, such as when a business student told them that he wanted to hire a diverse workforce and work with people from all over the world. Other answers suggested a less welcoming attitude toward outsiders: a pre-law student claimed he could always "recognize" Jews and called Jewish immigration "a danger" because it meant America would "take on the burdens of people who have been misfits in other countries."
Eventually the Berkeley group's publications caught the eye of prominent social scientist Max Horkheimer, who was on the board of a civil liberties organization called the American Jewish Committee. Founded in 1906 by people who wanted to put a stop to the pogroms killing Jews in Russia, the group was on the lookout for researchers who could explain how everyday prejudices erupted into the Holocaust. Horkheimer secured more funding for the project and introduced the data-minded Berkeley researchers to the rather dystopian political philosopher Theodor Adorno. At that point, the group expanded its scope, bringing qualitative analysis into their quantitative framework. This allowed them to perfect the F-scale test after several false starts.
To create a personality test that actually revealed latent authoritarianism, the researchers had to give up on the idea that there's a strong link between anti-Semitism and authoritarianism. That perspective was too limiting. Though their experiences with the Holocaust suggested a causal connection between hatred of Jews and the rise of fascism, it turned out that people with authoritarian tendencies were more accurately described as ethnocentric. Authoritarians believed their own group was superior and expressed racism against a wide range of other people. Frenkel-Brunswik conducted many of the interviews, and she writes in The Authoritarian Personality that the group adjusted its work accordingly, testing people for prejudice against blacks, Filipinos, and immigrants. It found that the common thread among all the high-scoring authoritarians was a generalized disgust with people who seemed different and therefore "uncanny." When an authoritarian scored low on anti-Semitism, he or she was sure to score high on hatred of another outsider group.
Another discovery was that authoritarians tended to distrust science and strongly disliked the idea of using imagination to solve problems. They preferred to stick to tried-and-true traditional methods of organizing society. Many believed that force was the best way to deal with conflict, partly because war is an inevitable outgrowth of human nature. Another personality trait that emerged had to do with sexuality. Authoritarians were rigidly opposed to homosexuality, occasionally suggesting that homosexuals should be killed or at least jailed. But more generally, they were fascinated by regulating other people's sex lives, often speculating about the "wild sex life" of groups they hated, whether those were artists, "weak" politicians, or racial minorities. Overall, the Berkeley group described authoritarians' outlook as "cynical" because of a tendency to believe the powerful would always rule the weak, and it was best to be on the side of the powerful, which is also likely why authoritarians expressed a desire for politicians who would take charge, set rules, and crush dissent.
As a result of these findings, the F-scale in its final form was intended to measure ethnocentrism, superstition, aggression, cynicism, conservatism, and an inordinate interest in the private sex lives of others as the building blocks for a personality drawn to authoritarian leaders. Now the group began to gather data on a much wider scale. They tested students at the University of Oregon and George Washington University, as well as union members, war vets, the inmates of San Quentin Prison, and patients at a psychiatric clinic.
At this link, we've recreated a version of the original F-scale test that the researchers administered, all questions included. Originally, people had the option to respond to each question using a sliding scale from "strongly agree" to "strongly disagree," but I've simplified this to "agree" and "disagree." As you'll see as you read through the questions, it's somewhat dated—remember, this was created in the late 1940s. To figure out your score, just add up how many times you checked "agree." The higher your score, the closer you are to being authoritarian. You can see why this might make the test easy to game, since the non-authoritarian answer is always "disagree." But the test's structure also created an additional problem. As University of Minnesota political psychologist Christopher Federico pointed out to Ars, there's a widely recognized psychological phenomenon called acquiescence bias, where some people have a predilection for agreeing with anything people say to them. So a high-scoring person might have authoritarian tendencies, but they might just suffer profoundly from acquiescence bias, too.
Though they hardly created the perfect test structure, the researchers were able to gather a fair amount of good data that's still considered relevant by scholars who study authoritarianism today. Harvard political scientist Pippa Norris told Ars that it "set the paradigm in the field of social psychology."<|endoftext|>
| 3.75 |
981 |
# What is the probability of tossing 5 coins?
## What is the probability of tossing 5 coins?
You’re correct that there are 25=32 possible outcomes of tossing 5 coin. That gives us a probability of 532 that exactly one head will face up upon tossing 5 fair coins.
### What is the sample space of tossing 5 coins?
If a coin is tossed once, then the number of possible outcomes will be 2 (either a head or a tail). Since a coin is tossed 5 times in a row and all the events are independent. Therefore the size of the sample space is 25 , where 2 is the number of possible outcomes when tossed once and 5 is the number of trials.
How many outcomes are there if you flip a coin 5 times?
Fair coin is tossed 5 times . Hence total number of outcomes = 2^5 = 32.
What is the probability of getting 5 heads in a toss of 5 coins?
1 in 2 chance
Explanation: When we flip a coin, there is a 1 in 2 chance it will be heads. When we flip 5 coins, each coin has a 1 in 2 chance of being heads. So we have 5 halves.
## What is the probability of getting two heads when 5 coins are tossed simultaneously?
A coin is tossed 2 times, find the probability that at least 5 are heads?…Probability of Getting 2 Heads in 5 Coin Tosses.
for 2 Heads in 5 Coin Flips
Probability P(A) 0.81 0.31
### What is the probability that you get 3 heads if you toss 5 coins?
0.31 is the probability of getting exactly 3 Heads in 5 tosses.
What are the chances of flipping heads 5 times in a row?
So the odds of flipping a coin 5 times and getting 5 heads are 1/2 ^5 (half to the power of 5). Which gives us 1/32 or just over a 3% chance.
What are the odds of a coin landing on heads 6 times in a row?
We find that the percentage odds of correctly calling the outcome of 6 coin tosses exactly 6 times by chance is 1.56%, or rather, the odds are that this exact outcome will occur by chance just once in 64 opportunities.
## What is the probability that a fair coin lands heads 4 times out of 5 flips?
0.19 is the probability of getting 4 Heads in 5 tosses.
### What is the probability that a coin lands heads 4 times in a row?
The probability of getting 4 heads in a row is 1/2 of that, or 1/16.
What is the probability of getting 4 or 5 heads for a coin flipped 5 times?
0.16 is the probability of getting exactly 4 Heads in 5 tosses.
What is the probability you get exactly 2 heads in 4 flips of a coin?
0.38 is the probability of getting exactly 2 Heads in 4 tosses.
## How to calculate the probability of a coin toss?
Solution: When 2 coins are tossed, the possible outcomes can be {HH, TT, HT, TH}. Thus, the total number of possible outcomes = 4. Getting only one head includes {HT, TH} outcomes. So number of desired outcomes = 2. Therefore, probability of getting only one head.
### What is the probability of getting 3 heads in 5 tosses?
Users may refer the below detailed solved example with step by step calculation to learn how to find what is the probability of getting exactly 3 heads, if a coin is tossed five times or 5 coins tossed together. 0.31 is the probability of getting exactly 3 Heads in 5 tosses.
How is the probability of a fair coin determined?
A fair coin has etqual probabilitiers for heads and tails. Therefore the probability can simply be found by counting. There are eight different possibilities for outcomes of three tosses: Head, Head, Head. Head, Head, Tail. Head, Tail, Head. Head, Tail, Tail.
What is the expected number of heads in a coin toss?
If we get a tail immediately (probability 1 2) then the expected number is e + 1. If we get a head then a tail (probability 1 4 ), then the expected number is e + 2. Continue …. If we get 4 heads then a tail, the expected number is e + 5. Finally, if our first 5 tosses are heads, then the expected number is 5.
02/01/2020<|endoftext|>
| 4.78125 |
282 |
A Lectionary is a table of readings from Scripture appointed to be read at public worship. The association of particular texts with specific days began in the 4th century. The Lectionary [1969, revised 1981] developed by the Roman Catholic Church after Vatican II provided for a three-year cycle of Sunday readings. This Roman lectionary provided the basis for lectionary in The Book of Common Prayer 1979 as well as those developed by many other denominations.
The Common Lectionary, published in 1983, was an ecumenical project of several American and Canadian denominations, developed out of a concern for the unity of the Church and a desire for a common experience of Scripture. It was intended as a harmonization of the many different denominational approaches to the three-year lectionary. It has been in trial use in the Episcopal Church and among the member denominations since 1983.
The Revised Common Lectionary, published in 1992, takes into account constructive criticism of the Common Lectionary based on the evaluation of its trial use and like the current prayer-book lectionary is a three-year cycle of Sunday Eucharistic readings in which Matthew, Mark and Luke are read in successive years with some material from John read in each year.
See “The Revised Common Lectionary” The Episcopal Church: The Episcopal Church and the Revised Common Lectionary.<|endoftext|>
| 3.765625 |
4,546 |
In my previous articles Beginners Guide to Statistics in Data Science and The Inferential Statistics Data Scientists Should Know we have talked about almost all the basics(Descriptive and Inferential) of statistics which are commonly used in understanding and working with any data science case study. In this article, lets go a little beyond and talk about some advance concepts which are not part of the buzz.
## Q-Q(quantile-quantile) Plots
Before understanding QQ plots first understand what is a Quantile?
A quantile defines a particular part of a data set, i.e. a quantile determines how many values in a distribution are above or below a certain limit. Special quantiles are the quartile (quarter), the quintile (fifth), and percentiles (hundredth).
An example:
If we divide a distribution into four equal portions, we will speak of four quartiles. The first quartile includes all values that are smaller than a quarter of all values. In a graphical representation, it corresponds to 25% of the total area of distribution. The two lower quartiles comprise 50% of all distribution values. The interquartile range between the first and third quartile equals the range in which 50% of all values lie that are distributed around the mean.
In Statistics, A Q-Q(quantile-quantile) plot is a scatterplot created by plotting two sets of quantiles against one another. If both sets of quantiles came from the same distribution, we should see the points forming a line that’s roughly straight(y=x).
Q-Q plot
For example, the median is a quantile where 50% of the data fall below that point and 50% lie above it. The purpose of Q Q plots is to find out if two sets of data come from the same distribution. A 45-degree angle is plotted on the Q Q plot; if the two data sets come from a common distribution, the points will fall on that reference line.
It’s very important for you to know whether the distribution is normal or not so as to apply various statistical measures on the data and interpret it in much more human-understandable visualization and their Q-Q plot comes into the picture. The most fundamental question answered by the Q-Q plot is if the curve is Normally Distributed or not.
Normally distributed, but why?
The Q-Q plots are used to find the type of distribution for a random variable whether it is a Gaussian Distribution, Uniform Distribution, Exponential Distribution, or even Pareto Distribution, etc.
You can tell the type of distribution using the power of the Q-Q plot just by looking at the plot. In general, we are talking about Normal distributions only because we have a very beautiful concept of the 68–95–99.7 rule which perfectly fits into the normal distribution So we know how much of the data lies in the range of the first standard deviation, second standard deviation and third standard deviation from the mean. So knowing if a distribution is Normal opens up new doors for us to experiment with
Types of Q-Q plots. Source
### Skewed Q-Q plots
Q-Q plots can find skewness(measure of asymmetry) of the distribution.
If the bottom end of the Q-Q plot deviates from the straight line but the upper end is not, then the distribution is Left skewed(Negatively skewed).
Now if upper end of the Q-Q plot deviates from the staright line and the lower is not, then the distribution is Right skewed(Positively skewed).
### Tailed Q-Q plots
Q-Q plots can find Kurtosis(measure of tailedness) of the distribution.
The distribution with the fat tail will have both the ends of the Q-Q plot to deviate from the straight line and its centre follows the line, where as a thin tailed distribution will term Q-Q plot with very less or negligible deviation at the ends thus making it a perfect fit for normal distribution.
### Q-Q plots in Python(Source)
Suppose we have the following dataset of 100 values:
``````import numpy as np
#create dataset with 100 values that follow a normal distribution
np.random.seed(0)
data = np.random.normal(0,1, 1000)
#view first 10 values
data[:10] ``````
``````array([ 1.76405235, 0.40015721, 0.97873798, 2.2408932 , 1.86755799,
-0.97727788, 0.95008842, -0.15135721, -0.10321885, 0.4105985 ])``````
To create a Q-Q plot for this dataset, we can use the qqplot() function from the statsmodels library:
``````import statsmodels.api as sm
import matplotlib.pyplot as plt
#create Q-Q plot with 45-degree line added to plot
fig = sm.qqplot(data, line='45')
plt.show()``````
In a Q-Q plot, the x-axis displays the theoretical quantiles. This means it doesn’t show your actual data, but instead, it represents where your data would be if it were normally distributed.
The y-axis displays your actual data. This means that if the data values fall along a roughly straight line at a 45-degree angle, then the data is normally distributed.
We can see in our Q-Q plot above that the data values tend to closely follow the 45-degree, which means the data is likely normally distributed. This shouldn’t be surprising since we generated the 100 data values by using the numpy.random.normal() function.
Consider instead if we generated a dataset of 100 uniformly distributed values and created a Q-Q plot for that dataset:
``````#create dataset of 100 uniformally distributed values
data = np.random.uniform(0,1, 1000)
#generate Q-Q plot for the dataset
fig = sm.qqplot(data, line='45')
plt.show()``````
The data values clearly do not follow the red 45-degree line, which is an indication that they do not follow a normal distribution.
## Chebyshev’s Inequality
In probability, Chebyshev’s Inequality, also known as “Bienayme-Chebyshev” Inequality guarantees that, for a wide class of probability distributions, only a definite fraction of values will be found within a specific distance from the mean of a distribution.
Source: https://www.thoughtco.com/chebyshevs-inequality-3126547
Chebyshev’s inequality is similar to The Empirical rule(68-95-99.7); however, the latter rule only applies to normal distributions. Chebyshev’s inequality is broader; it can be applied to any distribution so long as the distribution includes a defined variance and mean.
So Chebyshev’s inequality says that at least (1-1/k^2) of data from a sample must fall within K standard deviations from the mean (or equivalently, no more than 1/k^2 of the distribution’s values can be more than k standard deviations away from the mean).
Where K –> Positive real number
If the data is not normally distributed then different amounts of data could be in one standard deviation. Chebyshev’s inequality provides a way to know what fraction of data falls within K standard deviations from the mean for any data distribution.
Credits: https://calcworkshop.com/joint-probability-distribution/chebyshev-inequality/
Chebyshev’s inequality is of great value because it can be applied to any probability distribution in which the mean and variance are provided.
Let us consider an example, Assume 1,000 contestants show up for a job interview, but there are only 70 positions available. In order to select the finest 70 contestants amongst the total contestants, the proprietor gives tests to judge their potential. The mean score on the test is 60, with a standard deviation of 6. If an applicant scores an 84, can they presume that they are getting the job?
The results show that about 63 people scored above a 60, so with 70 positions available, a contestant who scores an 84 can be assured they got the job.
### Chebyshev’s Inequality in Python(Source)
Create a population of 1,000,000 values, I use a gamma distribution(also works with other distributions) with shape = 2 and scale = 2.
``````import numpy as np
import random
import matplotlib.pyplot as plt
#create a population with a gamma distribution
shape, scale = 2., 2. #mean=4, std=2*sqrt(2)
mu = shape*scale #mean and standard deviation
sigma = scale*np.sqrt(shape)
s = np.random.gamma(shape, scale, 1000000)``````
Now sample 10,000 values from the population.
``````#sample 10000 values
rs = random.choices(s, k=10000)``````
Count the sample that has a distance from the expected value larger than k standard deviation and use the count to calculate the probabilities. I want to depict a trend of probabilities when k is increasing, so I use a range of k from 0.1 to 3.
``````#set k
ks = [0.1,0.5,1.0,1.5,2.0,2.5,3.0]
#probability list
probs = [] #for each k
for k in ks:
#start count
c = 0
for i in rs:
# count if far from mean in k standard deviation
if abs(i - mu) > k * sigma :
c += 1
probs.append(c/10000)``````
Plot the results:
``````plot = plt.figure(figsize=(20,10))
#plot each probability
plt.xlabel('K')
plt.ylabel('probability')
plt.plot(ks,probs, marker="o")
plot.show()
#print each probability
print("Probability of a sample far from mean more than k standard deviation:")
for i, prob in enumerate(probs):
print("k:" + str(ks[i]) + ", probability: "
+ str(prob)[0:5] +
" | in theory, probability should less than: "
+ str(1/ks[i]**2)[0:5])``````
From the above plot and result, we can see that as the k increases, the probability is decreasing, and the probability of each k follows the inequality. Moreover, only the case that k is larger than 1 is useful. If k is less than 1, the right side of the inequality is larger than 1 which is not useful because the probability cannot be larger than 1.
## Log-Normal Distribution
In probability theory, a Log-normal distribution also known as Galton’s distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed.
Thus, if the random variable X is log-normally distributed, then Y = ln(X) has a normal distribution. Equivalently, if Y has a normal distribution, then the exponential function of Y i.e, X = exp(Y), has a log-normal distribution.
Skewed distributions with low mean and high variance and all positive values fit under this type of distribution. A random variable that is log-normally distributed takes only positive real values.
The general formula for the probability density function of the lognormal distribution is:
The location and scale parameters are equivalent to the mean and standard deviation of the logarithm of the random variable.
The shape of Lognormal distribution is defined by 3 parameters:
1. σ is the shape parameter, (and is the standard deviation of the log of the distribution)
2. θ or μ is the location parameter (and is the mean of the distribution)
3. m is the scale parameter (and is also the median of the distribution)
The location and scale parameters are equivalent to the mean and standard deviation of the logarithm of the random variable as explained above.
If x = θ, then f(x) = 0. The case where θ = 0 and m = 1 is called the standard lognormal distribution. The case where θ equals zero is called the 2-parameter lognormal distribution.
The following graph illustrates the effect of the location(μ) and scale(σ) parameter on the probability density function of the lognormal distribution:
Source: https://www.sciencedirect.com/topics/mathematics/lognormal-distribution
### Log-Normal Distribution in Python(Source)
Let us consider an example to generate random numbers from a log-normal distribution with μ=1 and σ=0.5 using scipy.stats.lognorm function.
``````import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import lognorm
np.random.seed(42)
data = lognorm.rvs(s=0.5, loc=1, scale=1000, size=1000)
plt.figure(figsize=(10,6))
ax = plt.subplot(111)
plt.title('Generate wrandom numbers from a Log-normal distribution')
ax.hist(data, bins=np.logspace(0,5,200), density=True)
ax.set_xscale("log")
shape,loc,scale = lognorm.fit(data)
x = np.logspace(0, 5, 200)
pdf = lognorm.pdf(x, shape, loc, scale)
ax.plot(x, pdf, 'y')
plt.show()``````
## Power Law distribution
In statistics, a Power Law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another.
For instance, considering the area of a square in terms of the length of its side, if the length is doubled, the area is multiplied by a factor of four.
A power law distribution has the form Y = k Xα,
where:
• X and Y are variables of interest,
• α is the law’s exponent,
• k is a constant.
Source: https://en.wikipedia.org/wiki/Power_law
Power-law distribution is just one of many probability distributions, but it is considered a valuable tool to assess uncertainty issues that normal distribution cannot handle when they occur at a certain probability.
Many processes have been found to follow power laws over substantial ranges of values. From the distribution in incomes, size of meteoroids, earthquake magnitudes, the spectral density of weight matrices in deep neural networks, word usage, number of neighbors in various networks, etc. (Note: The power law here is a continuous distribution. The last two examples are discrete, but on a large scale can be modeled as if continuous).
### Power-law distribution in Python(Source)
Let us plot the Pareto distribution which is one form of a power-law probability distribution. Pareto distribution is sometimes known as the Pareto Principle or ‘80–20’ rule, as the rule states that 80% of society’s wealth is held by 20% of its population. Pareto distribution is not a law of nature, but an observation. It is useful in many real-world problems. It is a skewed heavily tailed distribution.
``````import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import pareto
x_m = 1 #scale
alpha = [1, 2, 3] #list of values of shape parameters
plt.figure(figsize=(10,6))
samples = np.linspace(start=0, stop=5, num=1000)
for a in alpha:
output = np.array([pareto.pdf(x=samples, b=a, loc=0, scale=x_m)])
plt.plot(samples, output.T, label="alpha {0}" .format(a))
plt.xlabel('samples', fontsize=15)
plt.ylabel('PDF', fontsize=15)
plt.title('Probability Density function', fontsize=15)
plt.legend(loc="best")
plt.show()``````
## Box cox transformation
The Box-Cox transformation transforms our data so that it closely resembles a normal distribution.
The one-parameter Box-Cox transformations are defined as In many statistical techniques, we assume that the errors are normally distributed. This assumption allows us to construct confidence intervals and conduct hypothesis tests. By transforming your target variable, we can (hopefully) normalize our errors (if they are not already normal).
Additionally, transforming our variables can improve the predictive power of our models because transformations can cut away white noise.
Original distribution(Left) and near-normal distribution after applying Box cox transformation. Source
At the core of the Box-Cox transformation is an exponent, lambda (λ), which varies from -5 to 5. All values of λ are considered and the optimal value for your data is selected; The “optimal value” is the one that results in the best approximation of a normal distribution curve.
The one-parameter Box-Cox transformations are defined as:
and the two-parameter Box-Cox transformations as:
Moreover, the one-parameter Box-Cox transformation holds for y > 0, i.e. only for positive values and two-parameter Box-Cox transformation for y > -λ, i.e. negative values.
The parameter λ is estimated using the profile likelihood function and using goodness-of-fit tests.
If we talk about some drawbacks of Box-cox transformation, then if interpretation is what you want to do, then Box-cox is not recommended. Because if λ is some non-zero number, then the transformed target variable may be more difficult to interpret than if we simply applied a log transform.
A second stumbling block is that the Box-Cox transformation usually gives the median of the forecast distribution when we revert the transformed data to its original scale. Occasionally, we want the mean and not the median.
### Box-Cox transformation in Python(Source)
SciPy’s stats package provides a function called boxcox for performing box-cox power transformation that takes in original non-normal data as input and returns fitted data along with the lambda value that was used to fit the non-normal distribution to normal distribution.
``````#load necessary packages
import numpy as np
from scipy.stats import boxcox
import seaborn as sns
#make this example reproducible
np.random.seed(0)
#generate dataset
data = np.random.exponential(size=1000)
fig, ax = plt.subplots(1, 2)
#plot the distribution of data values
sns.distplot(data, hist=False, kde=True,
kde_kws = {'shade': True, 'linewidth': 2},
label = "Non-Normal", color ="red", ax = ax[0])
#perform Box-Cox transformation on original data
transformed_data, best_lambda = boxcox(data)
sns.distplot(transformed_data, hist = False, kde = True,
kde_kws = {'shade': True, 'linewidth': 2},
label = "Normal", color ="red", ax = ax[1])
plt.legend(loc = "upper right")
#rescaling the subplots
fig.set_figheight(5)
fig.set_figwidth(10)
#display optimal lambda value
print(f"Lambda value used for Transformation: {best_lambda}")``````
## Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.
In very simple terms, A Poisson distribution can be used to estimate how likely it is that something will happen “X” number of times.
Some examples of Poisson processes are customers calling a help center, radioactive decay in atoms, visitors to a website, photons arriving at a space telescope, and movements in a stock price. Poisson processes are usually associated with time, but they do not have to be.
The Formula for the Poisson Distribution Is:
Where:
• e is Euler’s number (e = 2.71828…)
• k is the number of occurrences
• k! is the factorial of k
• λ is equal to the expected value of k when that is also equal to its variance
Lambda(λ) can be thought of as the expected number of events in the interval. As we change the rate parameter, λ, we change the probability of seeing different numbers of events in one interval. The below graph is the probability mass function of the Poisson distribution showing the probability of a number of events occurring in an interval with different rate parameters.
Probability Mass function for Poisson Distribution with varying rate parameters.Source
The Poisson distribution is also commonly used to model financial count data where the tally is small and is often zero. For one example, in finance, it can be used to model the number of trades that a typical investor will make in a given day, which can be 0 (often), or 1, or 2, etc.
As another example, this model can be used to predict the number of “shocks” to the market that will occur in a given time period, say over a decade.
### Poisson distribution in Python
``````from numpy import random
import matplotlib.pyplot as plt
import seaborn as sns
lam_list = [1, 4, 9] #list of Lambda values
plt.figure(figsize=(10,6))
samples = np.linspace(start=0, stop=5, num=1000)
for lam in lam_list:
sns.distplot(random.poisson(lam=lam, size=10), hist=False, label="lambda {0}".format(lam))
plt.xlabel('Poisson Distribution', fontsize=15)
plt.ylabel('Frequency', fontsize=15)
plt.legend(loc="best")
plt.show()``````
As λ becomes bigger, the graph looks more like a normal distribution.
Feel free to connect me on LinkedIn for any query.
References
Original. Reposted with permission.
Related:<|endoftext|>
| 4.5625 |
1,487 |
# Study tips: Percentages, proportions, ratios & fractions made easy
When I told my family and friends that I was going back to college to study accountancy, they all thought it was a joke. Let’s just say I wasn’t the best at maths.
Percentages were my pet hate and I was still phoning my Dad, years after I’d left home, to check whether I needed to multiply or divide by 100. Then a fantastic tutor told me about the word ‘of’ and how it can be substituted for ‘divide’. In effect, she translated maths into English for me and I’ve never looked back.
Percentages, proportions, ratios and fractions are subtly different, but related enough for the same techniques to be applied and manipulated, as long as we understand the connections.
So here’s how it works.
# Percentages
First we have to know that ‘per cent’ means ‘out of 100’ and that percentages are a way of working out part of a number ie. 66% means 66 out of 100.
We might need to calculate a percentage of a number, say for example 8% of £26,500. The easiest way is to calculate 1% first and then scale that up to 8%.
As £26,500 represents 100% and we want to calculate 1% of it, we change the word ‘of’ to ‘divide’ and the calculation becomes:
£26,500 ÷ 100 = £265.
As £265 represents 1% of the whole 100%, we can ‘multiply’ it by 8 to calculate 8%:
£265 x 8 = £2,120
So 8% of £26,500 is £2,120
### Calculating an amount as a percentage of another
We might also need to calculate an amount as a percentage of another, for example £30 as a percentage of £600. We can turn the maths into English here too, as we want to know what 30 is as a percentage of 600. When we substitute ‘divide’ for ‘of’ the calculation becomes:
£30 ÷ £600 = 0.05
However, this is where a connection comes in, as we’ve calculated a decimal, and now need to ‘multiply’ that by 100 to convert it back into a percentage:
0.05 x 100 = 5%
We can check our calculations by re-working it the opposite way, using the first technique, as now we could reasonably expect 5% of £600 to be £30.
But is it?
It’s worth noting that we can convert between percentages and decimals:
• Divide by 100 to change a percentage to a decimal
eg. 15% ÷ 100 = 0.15
• Multiply by 100 to change a decimal to a percentage
eg. 0.46 x 100 = 46%
# Proportions and ratios
Let’s now look at how this relates to proportions and ratios. Firstly we need to know that when we talk about proportions, we simply mean a ‘part’, ‘share’, ‘bit’ or ‘number’ of a whole.
In effect, 80% is a proportion of 100%.
Proportions can be described in general terms; a small proportion of enquiries are generated by referrals.
Or they can be described specifically using ratios, for example, the proportion of enquiries between newspaper advertising, the website and referrals is 10:5:1 respectively.
The word ‘respectively’ means that the order of the numbers in the ratio (10, 5 and 1) relate to the ‘parts’ in the same order. For example, newspaper advertising is 10 as a proportion of the whole, the website is 5 and referrals only make up the small proportion of 1.
Next we need to understand the relationship of each part or proportion to the ‘whole’ and for that we need to calculate what the whole is. We do that by adding all the numbers in the ratio:
10 + 5 + 1 = 16
So in this case 16 represents the ‘whole’.
Finally, we can apply this knowledge and understanding to answer a question such as:
If we had 12,000 enquiries in total, how many were generated by each source?
We can use the same thought process as we did with percentages to find 1/16th of the total enquiries and then scale it up to the number of 16ths needed for each part.
As 12,000 is the total and we want to calculate 1/16th of it, we change the word ‘of’ to ‘divide’ and the calculation becomes:
12,000 ÷ 16 = 750
As 750 represents 1/16th of the total, we can ‘multiply’ it by 10 to calculate the proportion generated by newspaper advertising:
750 x 10 = 7500
Again we can check our answer, in this case by calculating the other two ‘parts’ and ensuring that when all three ‘parts’ are added together the answer is the total.
Newspaper Advertising 750 x 10 7500 Website 750 x 5 3750 Referrals 750 x 1 750 Total 1200
# Fractions
By using the technique to calculate 16ths , we have in effect used fractions to help us calculate the proportions in the correct ratios. This is because, like percentages, proportions and ratios, a fraction is another way of expressing ‘a part of a whole’.
10/16th’s is a fraction which means that the ‘whole’ amount has been split into 16 and we are looking at 10 of those 16 bits!
Instead of using a ratio, we could have been told that the website is the source of 5/16 of total enquiries. Even if we only had this information and knew nothing about the newspaper advertising or referrals, we still could have performed the calculation above.
Alternatively, we could read the fraction as 5 divided by 16, which would be:
5 ÷ 16 = 0.3125
As we now have a decimal we could turn it into a percentage if needed, but in this case we are trying to calculate the actual number of enquiries that 0.3125 is as a proportion of the 12,000 that were made in total.
Therefore, we can just multiply the decimal by the total:
0.3125 x 12,000 = 3,750
This brings us full circle back to the first calculation we started with for percentages, and that’s because all of these basic maths concepts and calculations are related.
It’s also why the techniques we’ve looked at can be applied to them all. The trick is to understand the connections and turn the maths into English or a fraction into a percentage.
Read more on the AAT Foundation Certificate in Accounting;
Browse the full range of AAT study support resources here
## Free Excel webinar
Learn how to present effectively in Excel from expert Deborah Ashby. To view the recorded webinar please register your details below
Gill Myers is a self-employed accounts consultant. She has taught AAT qualifications since 2005 and written numerous articles and e-learning resources.<|endoftext|>
| 4.75 |
900 |
Today is the World Day of Social Justice, where the ongoing need to tackle poverty, inequality and exclusion is recognized. Global inequalities regarding access to clean water supplies is something that doesn’t always receive the same coverage as other inequalities but they still persist. Universal access to clean water and sanitation is number 6 of the UN Sustainable Development Goals to be achieved by 2030. In this article, we take a brief look at the challenges to overcome.
Global water inequalities – the current picture
Although billions of people have gained access to clean and safe drinking water over the past 25 years, there are still a lot of inequalities that remain. The UN aims to ensure available and sustainably managed water and sanitation for all by 2030, but current data and predictions show that there is still a long way to go…
- According to a 2017 WHO/Unicef joint report, around 844 million people worldwide still lack access to basic clean drinking water;
- The same report found that 2.3 billion people lack access to even basic sanitation;
- The problem of water inequality is linked to general poverty and social inequality, with problems being more drastic in least developed regions. Over half of those, without access to safe drinking water, live in sub-Saharan Africa;
- With water scarcity predicted to get worse over the next 20-25 years, there is a danger that these inequalities could worsen despite current efforts.
Inequalities within countries – the urban-rural and rich-poor divide
In addition to inequalities between countries, there also exists big disparities within countries when it comes to accessing clean water and sanitation services. Two of the biggest gaps occur between richest and poorest sections within countries and between urban and rural populations. This highlights how improvements over the last couple of decades have been concentrated largely within the wealthiest, urbanised regions of developing nations.
- 8 out of 10 people without access to clean water live in rural communities;
- In Angola, there has been high overall improvement in water access but a 40% gap between urban and rural areas and a 65% gap between richest and poorest sections;
- In Bangladesh, there has been great overall improvement in sanitation but problems such as open defecation still persist in the most deprived areas;
Impact on gender inequalities
Inequalities regarding access to clean drinking water and basic sanitation also cut across other social inequalities. This is most clearly seen when it comes to gender inequalities, with the burden of sourcing and collecting safe water largely falling on women and girls. Households in many countries in sub-Saharan Africa as well as countries such as India and Pakistan cannot source water within 30 minutes of home.
- Families without a nearby water supply can spend up to five hours a day collecting water;
- Females in the household bear the responsibility for water collection in 8 out of 10 households in poorer regions;
- Households without clean water or sanitation are also more prone to health problems and diseases, with females being largely responsible for caring for sick family members;
- This has education implications, as young girls often have to miss school to perform water collection and caring chores.
Other inequalities that exist
Looking beyond the basic inequalities regarding accessing clean water supplies and basic sanitation, the WHO/Unicef joint report identified a number of other issues highlighting water inequalities:
- Many countries that are able to access clean water supplies can’t do so every day. For example, water shortages in South Africa in recent years have meant that some parts of the country have been without piped water supply for 5-6 days at a time;
- Some countries have water supplies categorised as “safe” but are still highly risky. For example, e-coli is still detected in over 50% of the drinking water in Ghana and over 80% in Nepal;
- The costs of water supply vary greatly across the world. In Tanzania, access to improved water supply has increased but expenditure on safe drinking water accounts for 5% of overall spending for 10% of the population.
These statistics show that there is a long way to go to get close to universal equitable access to clean water over the next decade, even though things have been moving in the right direction. Progress will undoubtedly depend not only on the success in tackling other sustainability goals such as poverty and overall inequality, but also in our attempts to manage existing global water resources and avoid the water shortage crisis that many experts are now warning about.<|endoftext|>
| 4.0625 |
414 |
At the top of your handout, respond to the question “What do you know about Pocahontas?”. What evidence do we have to support our beliefs? Does the evidence come from primary or secondary sources? Did Pocahontas save John Smith’s life? Do you believe the movie? Is this what actually happened between Pocahontas and John Smith? ….We should probably look at some primary sources…. We are reading with a purpose… -Sources -Power Verbs What is the author saying? In the left margin summarize each paragraph into one sentence. (10 words or less) Primary sources – written by John Smith about the events he experienced. 1. True Relations 2. General History Why would Smith add on to his earlier story? Why might Smith lie or exaggerate and invent new information? Why wouldn’t Smith lie about the story? Secondary sources – written by historians about the events. Paul Lewis J.A. Leo Lemay Which historian interpretation do you find more convincing? WHY? What evidence did both historians use to support their argument? Could there be a third interpretation/explaination? What did the movie get right and what did it get wrong? Why would Disney choose to make the movie that way? Our Pocahontas activity was just like the lunchroom fight activity and Snapshot Autobiography. In all cases, we needed to ask, “How do we know what we know?” What’s the evidence? In history, most of the evidence is written in documents. When we read those documents, we have to ask certain questions: 1. Who wrote the document? 2. Why did they write it? 3. What else was going on at the time? 4. Do other sources agree or disagree with this account? Right now we are developing the skills to conduct our own high level historical analysis. Soon you will answer similar questions in essay form using the evidence from primary source documents. Complete your Colonial America map! Binder Check on Wednesday!<|endoftext|>
| 3.890625 |
1,470 |
# How can the use of mental
This question absorbed all his mental powers. He had a mental list of what they should take when they left. Though he harbored no regrets in declining her invitation to sex, he knew he could and should have handled so obviously unstable a person in such a mental state far better than he did. She awoke in a mental institution.
Multiplication Facts for 0 through 10[ edit ] Patterns for 0, 1 and 10[ edit ] Chances are you already know the pattern for multiplying by 0, 1 and Anything times 1, is itself still, and anything times 10 has a zero added to the end, so 29x10 is Patterns for 2, 4, and 8[ edit ] Multiplying by 2 is simply doubling a number, and so the pattern for doubling with addition is the same.
Multiplying by 4 is doubling twice. If you multiply by a quadratic a number that can continuously divide in half until reaching 1, without involving fractionsthen this method will always work.
Patterns for 9[ edit ] Multiplying by 9 has a special pattern. When multiplying a single digit by 9, the answer will always start with a How can the use of mental one less than the number, and then the other number will add to 9.
This may sound complex, but lets look at an example. If we want to do 9x6, simply have your first digit being one less than 6, so we know the answer will start with a 5, then next digit must add up to 9, so the number that when added to 5 will equal 9, is 4, so the answer of 9x6 is Patterns for 5[ edit ] Any number multiplied by 5 will end in either a 5 or a 0.
One way of finding the answer is to multiply by 10 first, and then divide your answer in half.
Another is to count by fives. Multiplying Larger Numbers[ edit ] When multiplying larger numbers it is very important to pick the correct sums to do. If you multiply by straight off it can be very difficult, but it is actually a very easy sum if approached in the right way.
Rounding[ edit ] One of the first things to do is to look if the numbers are near anything easy to work out. In this example there is, very conveniently, the numberwhich is next to There is, however, an easy way of multiplying by which can also apply to other numbers.
You multiply by then divide by 4. An even more effective way in some circumstances is to know a simple rule for a set of circumstances. There are a large number of rules which can be found, some of which are explained below. Factoring[ edit ] If you recognize that one or both numbers are easily divisible, this is one way to make the problem much easier.
For example, 72 x 39 may seem daunting, but if taken as 8 x 9 x 3 x 13, it becomes much easier. First, rearrange the numbers in the hardest to multiply order. Then multiply them one at a time. When the number A is a two digit or one digit number, the result would be A - 1 following by - A.
People who get appropriate care can recover from mental illness and addiction and lead full, rewarding lives. See Resources for Stress and Mental Health for campus and community resources. *Adapted from the National Mental Health Association/National Council for Community Behavioral Healthcare. In particular, can drug use cause a person to develop a mental illness? To understand the answer to this question, it may be important to know how dual diagnoses work, what they are, common symptoms of drug abuse and mental illness, and what treatments may be best for a person affected by both disorders. I used mental math to round the numbers in the chart. I then added the rounded numbers together to find out how much Marcus spent. + + + = points.
For example, when we multiply 65 by 99, we get Similarly, to multiply a number A byyou can multiply A by and then subtract A from the result. When the number A is a three digit, two digit or one digit number, the result would be A - 1 following by - A.
For example, when we multiply bywe get This same idea can be used for multiplication by any large number consisting only 9s. Then stick the first answer at the start of the second to get the answer If the result from the multiplication of the unit digits is less than 10, simply add a zero in front of the number i.
Squaring a Number That Ends with 5[ edit ] This is a special case of the previous method. Discard the 5, and multiply the remaining number by itself plus one. Then tack on a 25 which as in the previous section, is 5x5.
When squaring two digit numbers that are only 1 higher from a number ending in zero, you can also use the basic algebraic formulas: For example, when squaring For example, take the square of 46, using the "5" rule above you know that 45 squared is Doing this with an adjacent known square that is below is a bit more challenging depending on how you feel about doing subtraction in your head.News You Can Use is a weekly mental health and wellness e-communication that covers a diverse selection of topics..
Each week you'll receive a short message that contains three links to connect you, a client, a family member or friend to an interesting article, resources or mental health information. Check out our interactive infographic to see progress toward the Mental Health and Mental Disorders objectives and other Healthy People topic areas.
Mental health is a state of successful performance of mental function, resulting in productive activities, fulfilling relationships with other people. The Connection Between Mental Illness and Substance Abuse Home The Most Common Co-Occurring Disorders The Connection Between Mental Illness and Substance Abuse The National Bureau of Economic Research (NBER) reports that there is a “definite connection between mental illness and the use of addictive substances” and that mental .
When two disorders or illnesses occur in the same person, simultaneously or sequentially, they are described as comorbid. 1 Comorbidity also implies that the illnesses interact, affecting the course and prognosis of both.
1,2 This research report provides information on the state of the science in the comorbidity of substance use disorders with mental illness and physical health conditions. In particular, can drug use cause a person to develop a mental illness?
To understand the answer to this question, it may be important to know how dual diagnoses work, what they are, common symptoms of drug abuse and mental illness, and what treatments may be best for a person affected by both disorders.
## Search form
Mental Sentence Examples. This question absorbed all his mental powers. He had a mental list of what they should take when they left. Final causes, vital and mental forces, the soul itself can, if they act at all, only act through the .
Mental Math - Wikibooks, open books for an open world<|endoftext|>
| 4.5 |
581 |
Midterm 2 Preparation
# Midterm 2: Version A
Find the solution set of the system graphically.
1. $\left\{ \begin{array}{rrrrr} x&+&2y&=&-5 \\ x&-&y&=&-2 \end{array}\right.$
For problems 2–4, find the solution set of each system by any convenient method.
1. $\left\{ \begin{array}{rrrrr} 4x&-&3y&=&13 \\ 5x&-&2y&=&4 \end{array}\right.$
2. $\left\{ \begin{array}{rrrrr} x&-&2y&=&-5 \\ 2x&+&y&=&5 \end{array}\right.$
3. $\left\{ \begin{array}{rrrrrrr} x&+&y&+&2z&=&0 \\ 2x&&&+&z&=&1 \\ &&3y&+&4z&=&0 \end{array}\right.$
Reduce the following expressions in questions 5–7.
1. $28 - \{5x - \left[6x - 3(5 - 2x)\right]^0 \} + 5x^2$
2. $4a^2 (a - 3)^2$
3. $(x^2 + 2x + 3)^2$
Divide using long division.
1. $(2x^3 - 7x^2 + 15) \div (x - 2)$
For problems 9–12, factor each expression completely.
1. $2ab + 3ac - 4b - 6c$
2. $a^2 - 2ab - 15b^2$
3. $x^3 + x^2 - 9x - 9$
4. $x^3 - 64y^3$
Solve the following word problems.
1. The sum of a brother’s and sister’s ages is 35. Ten years ago, the brother was twice his sister’s age. How old are they now?
2. Kyra gave her brother Mark a logic question to solve: If she has 20 coins in her pocket worth $\2.75$, and if the coins are only dimes and quarters, how many of each kind of coin does she have?
3. A 50 kg blend of two different grades of tea is sold for $\191.25.$ If grade A sells for $\3.95$ per kg and grade B sells for $\3.70$ per kg, how many kg of each grade were used?
Midterm 2: Version A Answer Key<|endoftext|>
| 4.5 |
409 |
## Chapter 7 Triangles
Chapter 7.TrianglesDefinition:- a closed figure formed by three intersecting lines is called a triangle. (‘Tri’ means ‘three’). A triangle has three sides, three angles and three vertices. For example, in triangle ABC, denoted as ABC ; AB, BC, CA are the three sides, <A, <B, <C are the three angles and A, B, C are … Read more
## Chapter 5. INTRODUCTION TO EUCLID’S GEOMETRY
1. The word ‘geometry’ comes form the Greek words ‘geo’, meaning the ‘earth’, and ‘metrein’, meaning ‘to measure’. Geometry appears to have originated from the need for measuring land. 2. Theorem 5.1 : Two distinct lines cannot have more than one point in common. Proof : Here we are given two lines l and m. … Read more
## Chapter 3.Coordinate Geometry
Important Points to Remember 1. What are the coordinates of the origin O? It has zero distance from both the axes so that its abscissa and ordinate are both zero. Therefore, the coordinates of the origin are (0, 0). 2. the axes (plural of the word‘axis’) divide the plane into four parts. These four parts … Read more
## 1.Number System
Class IX(Chapter 1.Numbers System) 1.Natural Number (N) : All number s used in counting are known as natural numbers such as 1,2,3,….., is called natural number.Denoted by N.N={1,2,3,……,n}. 2.Whole Number(W): All natural numbers including zero , is called Whole numbers.Denoted by W. W={0,1,2,3,…..n}. 3.Integers(I):- All number with +ve and –ve sign including 0, is known … Read more<|endoftext|>
| 4.40625 |
429 |
Skip to 0 minutes and 7 secondsPrimary school science is the first formal opportunity to develop children's curiosity about the world in which they live.
Skip to 0 minutes and 19 secondsIt's the starting point for building their science capital and a lifelong appreciation of the importance of science. Through hands-on exploration, children are able to ask questions, test their ideas, and in turn, develop a greater understanding of the world in which they live. Creating learning opportunities that will develop a sense of excitement and curiosity is fundamental in supporting children's learning in science. [Teacher] It's really important for the children to be hands on, actively engaged... [Teacher] A lot of the lesson was being able to explore how it works and how the sound does come out of it. Through practical enquiry, children can develop the scientific knowledge required to understand the uses and implications of science for today and for the future.
Skip to 1 minute and 6 seconds[Teacher] Some of the children on the table go: 'no, no that needs to happen' before any teacher prompting because they started to connect the dots themselves, and that's what we want.
Skip to 1 minute and 12 secondsTeaching Primary Science: Getting Started is for those new to teaching primary science or for teachers who would like to develop their skills further. We will explore a range of techniques to deliver primary science in an engaging and practical way. You'll be able to identify the different types of enquiry and learn how to embed them in science teaching. You will also be able to trial activities and think about how they impact on learning and engagement in science. As a participant on this course, you will be able to develop your confidence in carrying out practical science for 5 to 11 year olds. You'll share ideas on how to plan for effective practical work in primary science.
Skip to 1 minute and 53 secondsAnd you'll collaborate with peers, educators, and mentors to share and critique ideas from teaching practical science. Join us online to get started with primary science through engaging, hands-on, practical activities that will inspire our scientists of the future.<|endoftext|>
| 4.09375 |
2,495 |
Ovulation induction is an artificial reproductive technique (ART) that uses hormonal therapy to initiate the development and release of an egg for fertilisation. It is often used to stimulate the maturation of multiple eggs in a single menstrual cycle, thereby increasing the chances of fertilisation.
What is ovulation?
Every woman is born with a lifetimes supply of egg cells in her ovaries. Each egg cell is contained in a small sac called a follicle. Follicle stimulating hormone (FSH) and luteinising hormones (LH) are two hormones produced by the brain to stimulate and regulate the monthly release of an egg. At the onset of puberty, one egg from an ovary is released every month until menopause (cessation of periods). Each egg released is picked up by the adjacent fallopian tube. The egg can fertilise with a sperm and lead to pregnancy or in the absence of sperm, it will dissolve and be flushed out with the menstrual flow. Each stage of ovulation is governed by many hormones secreted by the hypothalamus, pituitary gland, adrenal gland, thyroid gland and ovaries.
This process of ovulation and its timing within the menstrual cycle plays a key role in determining fertility and achieving pregnancy. A normal menstrual cycle is between 28-32 days, measured from the first day of period to the first day of the next menstrual flow. Some cycles may be shorter or longer. Ovulation usually occurs on the 11th to 21st day, starting from the first day of the last menstrual period (LMP). This is considered a fertile period when couples wishing to conceive can have intercourse and increase their chances of pregnancy.
What are the ovulation disorders?
Many conditions that affect ovulation can lead to infertility. These may include:
- Polycystic ovarian syndrome (PCOS): formation of multiple cysts where the egg does not develop or does not get released as it should; It is the most common cause of infertility.
- Hypothalamic dysfunction: disruption in the release of hormones by the hypothalamus
- Premature ovarian insufficiency: egg production ceases prematurely at an early age, much before menopause
What is the percentage of population affected by ovulation disorders?
Being the most common ovulation disorder that causes infertility, PCOS affects 5-10% women of childbearing age.
How does ovulation disorder affect you?
Ovulation disorders disrupt or prevent the ovulation process and can thereby lead to infertility. It is a cause of infertility when associated with irregular or no periods (amenorrhea). It is rarely a cause of infertility when associated with regular menstrual periods without premenstrual symptoms, such as mood swings, breast tenderness or lower abdominal swelling.
Which part of the body is affected?
Eggs are contained in two small oval-shaped ovaries present on either side of the womb in the pelvic region. At the time of ovulation, one mature egg is released from the ovary and picked up by finger-like structures at the ends of the fallopian tube. The tube then transports the egg to the womb.
What are the causes of ovulation disorder?
Ovulation disorders occur when one part of the ovulation process malfunctions. This can happen when:
- The hypothalamus stops secreting gonadotropin-releasing hormone, which signals the pituitary gland to release LH and FSH, the two hormones that trigger ovulation.
- The pituitary gland produces low levels of LH and FSH.
- The pituitary gland produces high levels of prolactin, which results in low levels of hormones that stimulate ovulation. This may be caused because of a pituitary gland tumour.
- The ovaries release low levels of oestrogen hormone.
- The adrenal glands release high levels of male hormones (such as testosterone).
- The thyroid glands release very high or very low levels of thyroid hormones, which regulates the pituitary gland and ovaries.
Ovulation problems may occur as a result of a disorder such as
- Polycystic ovary syndrome (PCOS)
- Early menopause
Who is at risk of ovulation disorder?
You may be at a risk for ovulation problems if you have the following:
- Psychological stress
- Are on certain drugs (such as estrogens, progestins and antidepressants)
- Excessive exercise
- Weight loss
What are the signs and symptoms of ovulation disorder?
The signs and symptoms of ovulation problems depend on the related disorder and may include:
- Absent or irregular periods
- Unusual spotting, light or heavy periods
- Pain during periods and intercourse
- Lack of premenstrual symptoms such as bloating or breast tenderness
- Pelvic pain
- Acne, excessive growth of facial hair
What are the related disorders that occur with ovulation induction?
Ovulation induction is associated with PCOS, diabetes and infertility.
How is the ovulation disorder diagnosed?
When you visit your doctor with problems in your menstrual cycle or inability to attain a pregnancy despite trying without contraception for a year, your doctor will review your menstrual history and perform a thorough physical examination. You may be asked to record your daily body temperature as an increase in your body temperature indicates ovulation. This can also be performed using a home ovulation predictor kit. Other tests may include ultrasound or blood tests.
What are the consequences of not treating ovulation disorder?
Left untreated, you may not be able to have biological offspring.
What are the treatment options for ovulation problems?
Ovulation problems can be treated with lifestyle changes such as diet and maintaining a healthy weight. Other treatments may include:
- Fertility drugs
- In vitro fertilisation
- Surgery to open blocked fallopian tubes and remove endometrial tissue
What are the conditions considered for ovulation induction?
You will be considered for ovulation induction if you suffer from:
- Anovulation: absence of ovulation (ovaries do not release an egg)
- Oligo-ovulation: irregular ovulation
- Luteal phase deficiency (LPD): insufficient production of progesterone
- Unexplained infertility
- During IVF, to trigger ovulation for the treatment of male factor infertility
- During IVF, to increase the number of eggs released
What are the prerequisites for ovulation induction?
The required pre-requisites for ovulation induction are:
- Healthy fallopian tubes that are open and not blocked.
- Absence of moderate/severe endometriosis, especially involving the tubes and/or the ovaries, and pelvic inflammatory disease
- Fertile sperm test
- Adequate ovarian reserve
How do I prepare for ovulation induction?
Before ovulation induction, your doctor will suggest:
- Semen analysis to test for sperm count and number and presence of sperm antibody to exclude male infertility
- Hysterosalpingogram (HSG) or laparoscopy, to test the viability of fallopian tubes
How is ovulation induction performed?
Ovulation induction involves the following steps:
- Ovulation stimulation: This involves the administration of certain drugs to initiate the release of egg(s) from your ovary. The following are commonly used drugs for ovulation induction:
- Clomiphene citrate: oral medication that is administered on the 3rd to 5th day after the onset of your periods. This drug works by blocking oestrogen receptors, making your body believe that the levels of oestrogen are low. The body thus initiates the production FSH and induces ovulation. This method of treatment requires monitoring.
- Human menopausal gonadotropin (hMG): injectable medication that includes FSH and LH that is administered early in the menstrual cycle and continued for 8 to 14 days until the maturation of one or more follicles.
- Follicle stimulating hormone (FSH): daily injectable medication administered for 5 to 12 days until the maturation of one or more follicles.
- Synthetic gonadotropin releasing hormone (FSH/LH inhibitor): causes an initial surge of LH and FSH followed by the suppression of these hormones. It is used in preparation for ovulation induction cycles as it improves the hormonal control, enhances egg production and prevents spontaneous ovulation.
- Cycle monitoring: Throughout your treatment, your doctor will monitor how you respond to treatment through blood tests. This helps your doctor change the treatment if needed. Vaginal ultrasound may be ordered to check the number and size of the follicles. You will not be restricted from intercourse during this period.
- Ovulation: Once the follicles mature, your doctor will inject human chorionic gonadotropin (hCG) to trigger the release of the egg from the follicle, which happens within 36 hours. Two things can happen during this time:
- You will be advised on the appropriate time to have intercourse.
- Your doctor will perform an intrauterine insemination as part of the IVF procedure, where sperm will be directly inserted into your uterus for fertilisation.
- Luteal phase: Approximately 10 days after your hCG injection, you are required to confirm ovulation through a blood test. If your menstrual flow does not start, you are required to perform a pregnancy test after 16 days of your injection.
What can I expect after ovulation induction?
After confirming positive for pregnancy, your doctor will continue to monitor the progress of your pregnancy with weekly hCG tests for about 8 weeks of pregnancy. Then, an ultrasound is performed to determine the presence of a pregnancy sac with a foetus and the presence of a foetal heartbeat.
What are the advantages of ovulation induction?
The advantages of ovulation induction are:
- Ovulation induction therapy stimulates the development and release of an egg for fertilisation in women who have problems with ovulation.
- When performed as part of an IVF cycle, ovulation induction helps mature multiple eggs in a single cycle, thereby increasing the chances of pregnancy.
- It also enhances the quality and quantity of the ovulation.
What are the outcomes of ovulation induction?
Ovulation induction helps release a healthy egg for fertilisation or multiple eggs for IVF treatment.
What are the potential complications of ovulation induction?
Some of the potential risks of ovulation induction are
- Multiple gestation
- Ovarian hyperstimulation
- Increased time commitment
What is the downtime of ovulation induction?
Ovulation induction is started on the third day of your menstrual cycle for about 7 to 10 days. This may continue for a few cycles until you achieve pregnancy. However, this does not require you to be off work. You can go about your normal routine.
What is the cost of ovulation induction?
Any costs involved will be discussed with you prior to your surgery.
How can ovulation disorders be prevented?
Not all forms of ovulation disorders can be prevented. However, maintaining a normal weight can lower your risk of ovulation disorders.
What are the lifestyle recommendations to manage ovulation disorder?
Eating healthy and engaging in moderate exercise are recommended for managing ovulation disorders.
What is the current research regarding ovulation induction?
Extensive research is being done to find better treatment outcomes of ovulation induction. Some of the recent studies are listed below:
- Horowitz E, Levran D, Weissman A. Extension of the clomiphene citrate stair-step protocol to gonadotropin treatment in women with clomiphene resistant polycystic ovarian syndrome. Gynecol Endocrinol. 2017 Apr 28:1-4.
- Haller L, Severac F, Rongieres C, et al. Intra-uterine insemination at either 24 or 48hours after ovulation induction: Pregnancy and birth rates. [French] Gynecol Obstet Fertil Senol. 2017 Apr;45(4):210-214.<|endoftext|>
| 3.734375 |
994 |
Functions are built-in formulas that you can use to make your worksheet computations easier. For example, the SUM function adds all cells within a given range. For instance, if you wanted to add cells B8, B9, B10, B11, B12, B13, and B14, you wouldn’t need to enter cell references and operators by typing or pointing; you would just insert the name of the function and the range of cells you want to add. In other words, instead of entering a long string, such as
you would enter
where B8:B14 is the reference for the range of cells to include in the sum.
Using the SUM Function
The SUM function is one of the most frequently used functions. It totals the numeric value of all cells in the ranges it references. Cells in the referenced range that contain text or error values are ignored. You can enter the range references by typing or pointing.
To use the SUM function:
- Select the cell where you want the SUM function result to appear.
- Type =
- Type SUM(
- Type or select the appropriate range reference.
- Type )
- Click the Enter button.
- Press Enter
In the following exercise, you will using the SUM function to total data in column D.
|1.||If necessary, select cell D12|
|4.||Using the mouse, select the range D6:D11||As you select the range, the range reference appears in the formula bar and in cell D12. The range is surrounded by a moving dashed border.|
|5.||Type )||The moving dashed border disappears.|
|6.||Press Enter||The result is displayed in D12.|
|7.||Select cell B12|
|9.||Click the Enter button||The result is displayed in B12.|
|10.||Save and close January|
Using the AutoSum Button
The most efficient way to sum a contiguous range of cells is by clicking the AutoSum button on the Standard toolbar. AutoSum automatically enters the SUM function and inserts the cell references that Excel assumes you want to add, which is usually the column above or the row to the left of the selected cell. If the selected cells are not correct, you can edit them by typing or pointing, or you can start again.
The AutoSum button is placed on the EDIT section of the HOME ribbon.
To use the AutoSum button:
- Select the cell in which you want the sum to appear. The chosen cell will need to have numeric data cells adjacent and above it (it will also work with other ranges that it can guess you want to SUM, as stated above).
- Click the AutoSum button on the Home ribbon or Formulas tab.
- Verify or select the cells to be totaled. Press Enter
- Click the Enter button.
Using the Formula Palette
In Excel you can enter functions using the formula palette. The formula palette is activated by typing the equals (=) sign in a cell. Depending on the function you select from the drop-down list, the expanded formula palette will contain different features. In addition to allowing you to view the cell and range references pertaining to the function, the expanded formula palette also lets you preview the result and explains what the entry boxes mean.
All built-in functions in Excel consist of a function name, such as SUM, and a set of arguments. Arguments appear in parentheses after the function name and consist of cell or range references, text, values, names, labels, and other functions.
To use the formula palette:
- Select the cell in which you want the formula to appear.
- On the Formula bar, click the Edit Formula button.
- In the Function Box, click the drop‑down arrow, and then select a function.
- In the expanded formula palette, examine the argument boxes.
- If necessary, type or select different arguments.
- Choose OK.
In the following exercise, you will use the formula palette to make a calculation.
|1.||Select cell B12|
|2.||On the Formula bar, click the Edit Formula button||The formula palette appears.|
|3.||In the Function box, click the drop‑down arrow, and then select the SUM function||The expanded formula palette appears, specific to the SUM function.|
|4.||In the Number 1 argument box, make sure B6:B11 appears|
|5.||Choose OK||The result and formula appear.|
|6.||Use the same steps in cell B23 to calculate the sum of the Boxes of Fresh Products sold|
Learn more about Microsoft Excel on our regular training sessions in Glasgow and Edinburgh.<|endoftext|>
| 4 |
1,583 |
Resources for the Late Eighteenth Century United States
The aim of this page is to contextualize how people of various classes, races, and genders experienced politics and society in the late eighteenth century. Rather than providing a comprehensive list, we have selected some helpful resources to better understand social and political life during time period. For general resources on the military, economics, religion, and material culture, please see the bottom of this page.
Politics and political beliefs continued to be a topic of debate after the end of the Revolutionary War as statesmen sought to establish a federal government. The Articles of Confederation, drafted in 1777, established the first federal government in the United States of America, and by 1781 all 13 states had ratified it. By 1787, it was clear that a stronger federal government was required, and the Constitutional Convention began drafting what would become the Constitution. During this time, men and women continued to debate what the role of government should be, who is considered a citizen in the United States, what role recognized citizens should play in all levels of government, and what rights and freedoms those citizens should enjoy.
Resources on politics:
- From Colonial Williamsburg Foundation: Continuing Revolutions
- From the National Museum of American History: A Letter from George Washington, November 30, 1785
- From the Digital Public Library of America: Declaration of the Rights of Man and of the Citizen and Creating the US Constitution
- From the New England Historical Society: The 1795 Trial of Samuel Livermore and John Jay
- From the American Yawp: A New Nation and The Early Republic
Americans in different social classes frequently held dissimilar thoughts and opinions on government in the newly created United States. Many members of the middling and lower classes thought that the Articles of Confederation was beneficial primarily to those in the upper classes, and in some cases, such as Shays’ Rebellion from 1786-1787, men took up arms against the new government to revolt against a weak federal power. While today it is common to think that all Americans supported both the Articles of Confederation and the Constitution, it was a highly contested document, and even some in the upper classes did not support the newly created federal government.
Resources on social classes:
- From History Matters:
- William Manning, “A Laborer,” Explains Shays Rebellion in Massachusetts: “In as Plain a Manner as I am Capable";
- “All Men Are Born Free and Equal”: Massachusetts Yeomen Oppose the “Aristocratickal” Constitution, January, 1788; and
- “The Sentiments of a Labourer”: William Manning Inquires in the Key of Liberty, 1798
- From the Digital Public Library of America: Shay’s Rebellion
Abigail Adams famously wrote to John Adams to “remember the ladies” while he was at the Constitutional Convention crafting the new federal government. White women held a secondary position in early American society, and were to be subordinate to their fathers and husbands. However, many women debated political ideas and engaged in overt political acts throughout the eighteenth and nineteenth century. Everyday acts, such as spinning bees and purchasing food and goods for the home, became politicized in the wake of the boycott of British goods prior to the Revolution. Up until the early decades of the nineteenth century, women in some states, such as New Jersey, were legally able to vote in elections, but then their suffrage rights were rolled back by white men who wanted to restrict the franchise. White women were granted the right to vote in 1920; black women could not vote until the passage of the Voting Rights Act of 1965. Despite men’s expectations that women should not be involved in politics, women nevertheless held and exercised political opinions.
Resources on women and gender:
- From History Matters:
- From the New England Historical Society: How the Daughters of Liberty Fought for Independence
- From the Massachusetts Historical: Letter from Paul Revere to William Eustis, 20 February 1804
African American men and women held a precarious position in early American society. While there were some free blacks throughout the United States who enjoyed limited rights, whites enslaved the majority of black men and women and forced them to work in typically awful conditions with no pay, and ensured that they did not have any significant social, political, or economic rights in the newly created United States of America. Many black men and women had been forced by whites into slavery in Africa, had to endure a grueling and frequently fatal Atlantic crossing in slave ships, and forced to work on farms and plantations throughout the colonies and later the United States. With new technologies like the invention of the cotton gin in 1793, the demand for enslaved labor increased exponentially. In addition, the transformation of the Upper South from farming tobacco to farming mixed crops led to the expansion of the domestic slave trade, and whites bought and sold enslaved men, women, and children, who were forced to work on rice, indigo, and sugar plantations and farms in the Lower and Deep South. Southern whites who did not own large plantations most often owned several black men, women, and children to work their smaller farms or plots of land. Even after emancipation, as they did during their enslavement, black men and women continue to fight for their political, social, and economic rights.
Resources on African American history and slavery:
- From History Matters:
- “Having Tasted the Sweets of Freedom”: Cato Petitions the Pennsylvania Legislature to Remain Free;
- “Is It Not Enough that We are Torn from Our Country and Friends?”: Olaudah Equiano Describes the Horrors of the Middle Passage, 1780s; and
- White Slaveowners Fear that the Haitian Revolution Has Arrived in Charleston, South Carolina, 1797
- From the Digital Public Library of America: Cotton gin and the expansion of slavery
- From the New England Historical Society:
- From the American Yawp: The Cotton Revolution
Indigenous peoples in the late eighteenth century had already endured centuries of violence at the hands of white Europeans who were actively working to strip them of their rights. White settlers had occupied and taken land that belonged to Indigenous men and women. Native Americans attempted to work with white settlers and later the United States government to negotiate their rights to their land and sought to secure political rights in the newly established federal government. Early political leaders in the United States met with Indigenous leaders to create compromises and treaties, but whites frequently manipulated these documents to solidify and legalize their supremacy over Indigenous communities. Native Americans continued to fight for political, social, and economic legitimacy in the eyes of whites and sought to retain control of their lands, a fight that still continues today.
Resources on Indigenous people
- From History Matters:
- From the New England Historical Society: Joseph Orono, The Blue Eyed Indian Who Helped the Revolutionary Cause
- Military - from History Matters: “Laying Close Siege to the Enemy”: Joseph Plumb Martin at the Battle of Yorktown, 1781
- Economics - from Mount Vernon: Ten Facts About the American Economy in the 18th Century
- Economics - from the American Yawp: The Market Revolution
- Religion - from the Library of Congress: Religion and the Founding of the American Republic
- Religion - from the American Yawp: Religion and Reform
- Material culture - from Colonial Williamsburg: Historic Threads: Three Centuries of Clothin
- Other resources - from the National Museum of American History: Peopling the Expanding Nation, 1776-1900
- Other resources - from the New York Public Library: Historical Maps of North America<|endoftext|>
| 3.875 |
940 |
# Thread: Finding the Constants in partial fractions
1. ## Finding the Constants in partial fractions
Find constants A, B, and C such that:
1/(x^3 - x) = A/(x-1) + B/(x) + C/(x+1)
Thanks
2. Originally Posted by Mr_Green
Find constants A, B, and C such that:
1/(x^3 - x) = A/(x-1) + B/(x) + C/(x+1)
Thanks
Note: 1/(x^3 -x)=1/x(x^2 - 1) = 1/x(x+1)(x-1) = A/(x-1) + B/x + C/(x+1)
The Lcm of A/(x-1) + B/x + C/(x+1) is x(x+1)(x-1), so we can rewrite A/(x-1) + B/x + C/(x+1) as [A(x^2 + x) + B(x^2 -1) + C(x^2 -x)]/x(x+1)(x-1)
Expanding the numerator we get:
(Ax^2 + Ax + Bx^2 - B + Cx^2 - Cx)/x(x+1)(x-1)
Grouping like powers of x we get:
[(A+B+C)x^2 + (A-C)x - B]/x(x+1)(x-1)
So now we have 1/x(x+1)(x-1) = [(A+B+C)x^2 + (A-C)x - B]/x(x+1)(x-1)
Since the denominators are equal, the numerators must be equal to maintain the equation.
=> 1 = (A+B+C)x^2 + (A-C)x - B
Now we equate like powers of x (Note we can think of 1 as 0x^2 + 0x +1)
Then A+B+C=0-------------(1)
A-C=0----------------(2)
B = -1 ----------------(3)
=> A+C=1 -------------------Plugged in the value of B
A-C=0 --------------------rewrite equation 2
Adding these equations eliminates C, so we end up with:
2A=1 => A=1/2.
But A+C = 1, => C=1/2.
Thus the constants are A=1/2, B=-1, C=1/2
So 1/(x^3 - x) = (1/2)/(x-1) - 1/(x) + (1/2)/(x+1), and you can check this.
There is another major method used to come up with the 3 equations, but this is the mainstream one I think.
3. Hello, Mr_Green!
This is a problem in Partial Fractions.
Find constants A, B, and C such that:
. . 1/(x³ - x) = A/(x - 1) + B/(x) + C/(x + 1)
. . . . . . . . . . . . . .1 . . . . . . . .A . . . .B . . . . C
We have: . ----------------- . = . --- + ------ + -------
. . . . . . . . x(x - 1)(x + 1) . . . . x . . .x - 1 . . x + 1
Multiply through by x(x - 1)(x + 1):
. . 1 .= .A(x - 1)(x + 1) + Bx(x + 1) + Cx(x - 1)
Let x = 0: . 1 .= .A(-1)(1) + B(0)(1) + C(0)(-1) . . A = -1
Let x = 1: . 1 .= .A(0)(2) + B(1)(2) + C(1)(0) . . B = ½
Let x = -1: .1 .= .A(-2)(0) + B(-1)(0) + C(-1)(-2) . . C = ½
4. Yup, that's a much nicer way to do it, gets to the point quickly, good for exams. (I think you switched the values for A and B though).<|endoftext|>
| 4.46875 |
399 |
Current bird species distributions are correlated with climate. Fossil record reveals that past periods of climatic instability have produced dramatic shifts in species’ geographic ranges.
Adaptation to climate change is less likely when climate change occurs rapidly.
The current rate of climate change is unusually rapid. Short-term responses may include shifts in distribution and changes in the timing of migration and breeding.
Phenological shifts in prey bases have been shown to affect bird populations: failure to adapt to a change in the timing of insect emergence has been cited as a cause of population collapse among some European bird species.
Some predicted seasonal changes in Vermont that may affect breeding birds include: warmer winters, more overwintering of insects, multiple melt events in the winter, earlier arrival of spring, earlier bloom dates of many plant species, earlier last spring frost, earlier ice-out of lakes and ponds, hotter summers, more heavy rain events, greater frequency of 1-2 month droughts, increased warm-weather insects, and warmer fall temperatures.
Changing distributions are likely to be observed within the next 25 years.
- Montane forests of spruce and fir are apt to be most sensitive to the changing climate.
- Not all birds will be adversely affected. Under one scenario, 38 percent of bird species in Vermont were predicted to occupy smaller ranges in the future, whereas 44 percent were predicted to expand their range. (see reference)
- Some of Vermont’s iconic species such as Common Loon and Hermit Thrush, and several Neotropical migrants like Wood Thrush, Veery, and Bobolink, are projected to decline significantly under all climate-change scenarios
- Species that have expanded into Vermont from the south (e.g., Tufted Titmouse, Northern Cardinal) and species compatible with suburban habitats (e.g., Canada Goose) are expected to benefit.
Webinar: Regional Impacts of Climate Change on Forests and Bird Communities<|endoftext|>
| 3.921875 |
748 |
August 24, 2016 report
Cuttlefish found to have number sense and state-dependent valuation
Human beings very clearly have number sense—from a very early age, they are able to make judgments about objects in the world around them based on how many of them there are. But few other animals have this ability. In this new effort, the research pair wondered if cuttlefish might because prior studies have shown them to have one of the more complex invertebrate brains. To find out, they set up a series of experiments all based around a type of food: shrimp.
The experiments consisted of offering young cuttlefish choices for a meal—a dead shrimp or a live one, a large shrimp or a small one, or two different quantities of shrimp—and then noting how the cuttlefish responded.
The researchers found that the cuttlefish, quite naturally, preferred the live shrimp over the dead ones, but their decision making was more nuanced than that. They preferred a large shrimp over two smaller ones when they were hungry, for example, but chose the small ones when they were not. Perhaps most interesting were the results found when the cuttlefish were given quantity options, e.g., one versus two shrimp, two versus three, three versus four and four versus five—they almost always chose the larger quantity option.
The researchers suggest these findings indicate that cuttlefish have number sense (they could see and comprehend that there were more or fewer shrimp in different groups) and that their choice of prey could depend on how hungry they were or the quality of the available prey—a form of state-dependent valuation. Such behavior also suggests that cuttlefish are capable of using both external (environmental conditions) and internal (their own preferences) information when making prey choices. Indeed, the researchers report that it appeared the cuttlefish at times took a moment to think over their choices before deciding which they preferred.
Identifying the amount of prey available is an important part of an animal's foraging behaviour. The risk-sensitive foraging theory predicts that an organism's foraging decisions with regard to food rewards depending upon its satiation level. However, the precise interaction between optimal risk-tolerance and satiation level remains unclear. In this study, we examined, firstly, whether cuttlefish, with one of the most highly evolved nervous system among the invertebrates, have number sense, and secondly, whether their valuation of food reward is satiation state dependent. When food such as live shrimps is present, without training, cuttlefish turn toward the prey and initiate seizure behaviour. Using this visual attack behaviour as a measure, cuttlefish showed a preference for a larger quantity when faced with two-alternative forced choice tasks (1 versus 2, 2 versus 3, 3 versus 4 and 4 versus 5). However, cuttlefish preferred the small quantity when the choice was between one live and two dead shrimps. More importantly, when the choice was between one large live shrimp and two small live shrimps (a prey size and quantity trade-off), the cuttlefish chose the large single shrimp when they felt hunger, but chose the two smaller prey when they were satiated. These results demonstrate that cuttlefish are capable of number discrimination and that their choice of prey number depends on the quality of the prey and on their appetite state. The findings also suggest that cuttlefish integrate both internal and external information when making a foraging decision and that the cost of obtaining food is inversely correlated with their satiation level, a phenomenon similar to the observation that metabolic state alters economic decision making under risk among humans.
© 2016 Phys.org<|endoftext|>
| 3.828125 |
670 |
ML Aggarwal Class 7 Solutions Chapter 2 Fractions and Decimals Check Your Progress for ICSE Understanding Mathematics acts as the best resource during your learning and helps you score well in your exams.
## ML Aggarwal Class 7 Solutions for ICSE Maths Chapter 2 Fractions and Decimals Check Your Progress
Question 1.
What fraction is 270 gram of 3 kilograms?
Solution:
Question 2.
Simplify the following:
Solution:
Question 3.
A shirt was marked at ₹ 540. It was sold at $$\frac { 3 }{ 4 }$$ of the marked price. What was the sale price?
Solution:
Question 4.
In a class of 56 students, $$\frac { 1 }{ 4 }$$ are in blue house and $$\frac { 3 }{ 14 }$$ are in yellow house. Out of the remaining, $$\frac { 1 }{ 3 }$$ are in the greenhouse and the rest are in the red house. Find the number of students in each house.
Solution:
Question 5.
Rohit bought a motorcycle for ₹ 36000. He paid $$\frac { 1 }{ 6 }$$ of the price in cash and the rest in 12 equal monthly installments. Find the amount he had to pay every month.
Solution:
Question 6.
Mr Mukerjee gave $$\frac { 5 }{ 14 }$$ of his money to his son, $$\frac { 2 }{ 3 }$$ of the remainder to his daughter and the rest to his wife. If his wife got ₹ 36000, what was that total amount?
Solution:
Question 7.
Solution:
Question 8.
Convert the following numbers to fractions (in simplest form):
(i) 0.025
(ii) 0.876
(iii) 4.3125
Solution:
Question 9.
Write the following fractions as decimals:
(i) 1$$\frac { 3 }{ 8 }$$
(ii) $$\frac { 47 }{ 125 }$$
(iii) 2$$\frac { 9 }{ 40 }$$
Solution:
Question 10.
By how much does the sum of 17.443 and 29.657 exceed the sum of 13.687 and 18.548?
Solution:
Question 11.
Simplify the following:
(i) 4.27 × 0.036
(ii) 0.09 × 1.04
(iii) 1.32 ÷ 0.8
(iv) 0.7038 ÷ 0.34
Solution:
Question 12.
If one kg of rice costs ₹ 52.70, then find the cost of 12.5 kg rice.
Solution:
Question 13.
A piece of cloth is 24.5 m long. How many pieces, each of length 1.75 m, can be cut from it?
Solution:
Question 14.
The product of two decimal numbers is 1.599 and one of them ¡s 0.65, find the other.
Solution:
Question 15.
Simplify the following:
Solution:<|endoftext|>
| 4.5625 |
2,070 |
Can you write a Python program to check a sequence of numbers is an arithmetic progression or not?
To refresh your memory, a sequence is a set of things (usually numbers) that are in order. In an Arithmetic Sequence the difference between one term and the next is a constant. In other words, we just add the same value each time.
For example, the sequence 5, 7, 9, 11, 13, 15 ... is an arithmetic progression with common difference of 2.
We can write an Arithmetic Sequence as a rule:
xn = a + d(n−1)
How would you write it using Python? Try it yourself. If you cannot figure it out, see code on the next page.
Image from Pixabay
The Fibonacci sequence was first observed by the Italian mathematician Leonardo Fibonacci in 1202. He was investigating how fast rabbits could breed under ideal circumstances. He made the following assumptions:
Fibonacci asked how many pairs of rabbits would be produced in one year.
Can you create the numbers yourself? Remember to count the 'pairs' of rabbits and not the individual ones. Try it.
Were you able to come up with the Fibonacci numbers? If not, here is how you would do it.
The pattern comes out to be 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.
Fibonacci numbers are of interest to biologists and physicists because they are frequently observed in various natural objects and phenomena. For example, the branching patterns in trees and leaves are based on Fibonacci numbers.
On many plants, the number of petals is a Fibonacci number: buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.
How can we create a rule (algorithm) for the fibonacci series (sequence)?
First, the terms are numbered from 0 onwards like this:
n = 0 1 2 3 4 5 6 7 8 9 10 ...
xn =0 1 1 2 3 5 8 13 21 34 55 ...
What rule can we create here? Well, if you look, x3 = x2 + x 1 (2 = 1 + 1) and x4 = x2 + x3 (3 = 1 + 2), etc.
So we can write the rule (algorithm) as: xn = x(n-1) + x(n-2).
Example: term 7 is calculated as:
x7= x(7-1) + x(7-2)
= x6 + x5
= 13 + 8
Let's write programs in Python to calculate the Fibonacci numbers.
1. With looping:
a,b = 1,1
for i in range(n-1):
a,b = b,a+b
1. With recursion:
if n==1 or n==2:
N.B: No not copy and paste the python code as identation is important.
The greatest common divisor (GCD) or the highest common factor (HCF) of two numbers is the largest positive integer that perfectly divides the two given numbers. Solving this problem for a specific set of numbers is easy. For example, find the GCD of 12 and 18. The The divisors of 12 are 1, 2, 3, 4, 6, 12 and for 18 are 1, 2, 3, 6, 9, 18. The common factors are 1, 2, 3, and 6. So the greatest common factor is 6.
How would you find the GCD for any number? Here the problem is more challenging. Here is one solution. Let's take two integers a and b passed to a function which returns the GCD. In the function, we first determine the smaller of the two number since the GCD (HCF) can only be less than or equal to the smallest number. For example, the GCD of 12 and 14 can only be less than 12 and not greater. We then use a for loop to go from 1 to that number.
In each iteration, we check if our number perfectly divides both the input numbers. If so, we store the number as the GCD. At the completion of the loop we end up with the largest number that perfectly divides both the numbers.
Below is the algorithm in python.
def computeGCD(a, b):
if a < b:
smaller = a
smaller = b
for i in range(1, smaller+1):
if (a % i == 0) & (b % i == 0):
gcd = i
N.B: Do not cut and paste the above code. Make sure the indentation is correct.
The above method is easy to understand and implement but not efficient. A much more efficient method to find the GCD (HCF) is the Euclidean algorithm.
The Euclidean algorithm is based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number. That is a mouthful! Let's make it simple by taking an example. 21 is the GCD of 252 and 105 (as 252 = 21 × 12 and 105 = 21 × 5), and the same number 21 is also the GCD of 105 and 147 (252 − 105). Since this replacement reduces the larger of the two numbers, repeating this process gives successively smaller pairs of numbers until the two numbers become equal. When that occurs, they are the GCD of the original two numbers.
A more efficient version of the algorithm shortcuts these steps, instead we divide the greater by smaller and take the remainder. Now, divide the smaller by this remainder. Repeat until the remainder is 0.
For example, if we want to find the H.C.F. of 54 and 24, we divide 54 by 24. The remainder is 6. Now, we divide 24 by 6 and the remainder is 0. Hence, 6 is the required GCD.
Python code for Euclidean Algorithm
def euclidAlgo(a, b):
a, b = b, a % b
Python code for Euclidean Algorithm using recursion:
def euclidAlgo(a, b):
if (b == 0):
return euclidAlgo(b, a % b)
Sources: Wikipedia; https://www.programiz.com/python-programming/examples/hcf
Free Image from Pixabay
Finding Prime Numbers
Prime numbers are very important, yet many students do not see the value of learning them. Primes have several applications, most importantly in information technology, such as public-key cryptography, which relies on the difficulty of factoring large numbers into their prime factors. One key challenge is to find prime numbers. Interestingly, Prime numbers and their properties were first studied extensively by the ancient Greek mathematicians. Euclid, for example, proved that there are infinitely many prime numbers.
Just to refresh our memory, a number greater than 1 is called a prime number, if it has only two factors, namely 1 and the number itself.
Proof by Contradiction
One of the first known proofs is the method of contradiction. It is used to calculate prime factors of large numbers. Calculating prime factors of small numbers is easy. For example, the factors of 17 is 1 and 17, so it is a prime number. What about large numbers? Let's look at the proof by contradiction method.
If a number n is not a prime, it can be factored into two factors a and b, such that n = a*b. For example, let's say a * b = 100, for various pairs of a and b.
If a = b, then they are equal, we have a*a = 100, or a^2 = 100, or a = 10, the square root of 100. If one of the numbers is less than 10, then the other has to be greater to make it to 100. For example, take 4 x 25 = 100. 4 is less than 10, the other number has to be greater than 10. In other words, if a * b, if one of them goes down, the other number has to get bigger to compensate so the product stays at 100. Put mathematically, the numbers revolve around the square root of their product.
Let's test if 101 is prime number. You could start dividing 101 by 2, 3, 5, 7, etc, but that is very tedious. A better way is to take the square root of 101, which is roughly equal to 10.049875621. So you only need to try the integers up through 10, including 10. 8, 9, and 10 are not themselves prime, so you only have to test up through 7, which is prime.
Because if there's a pair of factors with one of the numbers bigger than 10, the other of the pair has to be less than 10. If the smaller one doesn't exist, there is no matching larger factor of 101.
Let's now build an algorithm using this method to test any number for primality.
Algorithm in Python
if (num < 2):
for i in range(2, int(math.sqrt(num)) + 1):
if num % i == 0:
N.B: Do not just copy the code because you have to be careful with indentation in python.
Try the above algorithm and let us know if you found it useful or have alternative solutions.
Write something about yourself. No need to be fancy, just an overview.<|endoftext|>
| 3.984375 |
917 |
## How do you find the equation of a parabola given the vertex and a point?
Vertex form of a quadratic equation is y=a(x-h)2+k, where (h,k) is the vertex of the parabola.The vertex of a parabola is the point at the top or bottom of the parabola.’h’ is -6, the first coordinate in the vertex.’k’ is -4, the second coordinate in the vertex.’x’ is -2, the first coordinate in the other point.
## How do you find the equation of a parabola with the vertex?
Insert the vertex coordinates into the equation y= a(x-h)^2 + k, where h is the x-value and k is the y-value. The value of a comes from the original equation. y = 3(x+1)^2+5 This is the vertex form of the parabola’s equation.
## How do you find the equation of a parabola?
This straight line outside the parabola is called the directrix. For parabolas that open either up or down, the standard form equation is (x – h)^2 = 4p(y – k). For parabolas that open sideways, the standard form equation is (y – k)^2 = 4p(x – h). The vertex or tip of our parabola is given by the point (h, k).
## How do you find the equation of a parabola given two points?
Using the vertex form of a parabola f(x) = a(x – h)2 + k where (h,k) is the vertex of the parabola.The axis of symmetry is x = 0 so h also equals 0.a = 1.Substituting the a value into the first equation of the linear system:k = 3.f(1) = 4 = (1 – 0)2 + 3 = 1 + 3.f(2) = 7 = (2 – 0)2 + 3 = 4 + 3.
## What is the formula for Vertex?
(The vertex formula is derived from the completing-the-square process, just as is the Quadratic Formula. For a given quadratic y = ax2 + bx + c, the vertex (h, k) is found by computing h = b/2a, and then evaluating y at h to find k.
## How do you find the vertex of an equation?
Lesson SummaryGet the equation in the form y = ax2 + bx + c.Calculate -b / 2a. This is the x-coordinate of the vertex.To find the y-coordinate of the vertex, simply plug the value of -b / 2a into the equation for x and solve for y. This is the y-coordinate of the vertex.
You might be interested: Nernst equation ph
## What is Vertex form of an equation?
The vertex form of a quadratic is given by. y = a(x – h)2 + k, where (h, k) is the vertex. The “a” in the vertex form is the same “a” as. in y = ax2 + bx + c (that is, both a’s have exactly the same value). The sign on “a” tells you whether the quadratic opens up or opens down.
## What is the vertex of a graph?
The vertex of a parabola is the point where the parabola crosses its axis of symmetry. If the coefficient of the x2 term is negative, the vertex will be the highest point on the graph, the point at the top of the “ U ”-shape.
### Releated
#### Convert to an exponential equation
How do you convert a logarithmic equation to exponential form? How To: Given an equation in logarithmic form logb(x)=y l o g b ( x ) = y , convert it to exponential form. Examine the equation y=logbx y = l o g b x and identify b, y, and x. Rewrite logbx=y l o […]
#### H2o2 decomposition equation
What does h2o2 decompose into? Hydrogen peroxide can easily break down, or decompose, into water and oxygen by breaking up into two very reactive parts – either 2OHs or an H and HO2: If there are no other molecules to react with, the parts will form water and oxygen gas as these are more stable […]<|endoftext|>
| 4.75 |
5,789 |
# Logarithm
Logarithms to various bases: red is to base e, green is to base 10, and purple is to base 1.7. Each tick on the axes is one unit. Logarithms of all bases pass through the point (1, 0), because any number raised to the power 0 is 1, and through the points (b, 1) for base b, because any number raised to the power 1 is itself. The curves approach the y axis but do not reach it, due to the singularity of a logarithm at x = 0.
In mathematics, the logarithm (or log) of a number x in base b is the power (n) to which the base b must be raised to obtain the number x. For example, the logarithm of 1000 to the base 10 is the number 3, because 10 raised to the power of 3 is 1000. Or, the logarithm of 81 to the base 3 is 4, because 3 raised to the power of 4 is 81.
In general terms, if x = bn, then the logarithm of x in base b is usually written as
${\displaystyle \log _{b}(x)=n.\,}$
(The value b must be neither 0 nor the root of 1.)
A useful way of remembering this concept is by asking: "b to what power (n) equals x?" When x and b are restricted to positive real numbers, the logarithm is a unique real number.
Using one of the examples noted above, 3 raised to the power of 4 is usually written as
${\displaystyle 3^{4}=3\times 3\times 3\times 3=81\,}$
In logarithmic terms, one would write this as
${\displaystyle \log _{3}(81)=4\,}$
In words, the base-3 logarithm of 81 is 4; or the log base-3 of 81 is 4.
The most widely used bases for logarithms are 10, the mathematical constant e (approximately equal to 2.71828), and 2. The term common logarithm is used when the base is 10; the term natural logarithm is used when the base is e.
The method of logarithms simplifies certain calculations and is used in expressing various quantities in science. For example, before the advent of calculators and computers, the method of logarithms was very useful for the advance of astronomy, and for navigation and surveying. Number sequences written on logarithmic scales continue to be used by scientists in various disciplines. Examples of logarithmic scales include the pH scale, to measure acidity (or basicity) in chemistry; the Richter scale, to measure earthquake intensity; and the scale expressing the apparent magnitude of stars, to indicate their brightness.
The inverse of the logarithmic function is called the antilogarithm function. It is written as antilogb(n), and it means the same as ${\displaystyle b^{n}}$.
## History
The method of logarithms was first publicly propounded in 1614, in a book titled Mirifici Logarithmorum Canonis Descriptio, by John Napier,[1] Baron of Merchiston in Scotland. (Joost Bürgi, independently discovered logarithms, but he did not publish his discovery until four years after Napier.)
This method contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, it was used constantly in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved method of prosthaphaeresis, which relied on trigonometric identities as a quick method of computing products. Besides their usefulness in computation, logarithms also fill an important place in higher theoretical mathematics.
At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers." Later, he formed the word logarithm to mean a number that indicates a ratio: λόγος (logos) meaning proportion, and ἀριθμός (arithmos) meaning number. Napier chose that because the difference of two logarithms determines the ratio of the numbers for which they stand, so that an arithmetic series of logarithms corresponds to a geometric series of numbers. The term "antilogarithm" was introduced in the late seventeenth century and, while never used extensively in mathematics, persisted in collections of tables until they fell into disuse.
Napier did not use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to base 1/e. For interpolation purposes and ease of calculation, it is useful to make the ratio r in the geometric series close to 1. Napier chose r = 1 - 10−7 = 0.999999 (Bürgi chose r = 1 + 10−4 = 1.0001). Napier's original logarithms did not have log 1 = 0 but rather log 107 = 0. Thus if N is a number and L is its logarithm as calculated by Napier, N = 107(1 − 10−7)L. Since (1 − 10−7)107 is approximately 1/e, this makes L/107 approximately equal to log1/e N/107.[2]
### Tables of logarithms
Part of a twentieth-century table of common logarithms in the reference book Abramowitz and Stegun.
Prior to the advent of computers and calculators, using logarithms meant using tables of logarithms, which had to be created manually. Base-10 logarithms are useful in computations when electronic means are not available.
In 1617, Henry Briggs published the first installment of his own table of common logarithms, containing the logarithms of all integers below 1000 to eight decimal places. This he followed, in 1624, with his Arithmetica Logarithmica, containing the logarithms of all integers from 1 to 20,000 and from 90,000 to 100,000 to fourteen places of decimals, together with a learned introduction, in which the theory and use of logarithms were fully developed.
The interval from 20,000 to 90,000 was filled by Adriaan Vlacq, a Dutch mathematician; but in his table, which appeared in 1628, the logarithms were given to only ten places of decimals. Vlacq's table was later found to contain 603 errors, but "this cannot be regarded as a great number, when it is considered that the table was the result of an original calculation, and that more than 2,100,000 printed figures are liable to error."[3] An edition of Vlacq's work, containing many corrections, was issued at Leipzig in 1794, under the title Thesaurus Logarithmorum Completus by Jurij Vega.
François Callet's seven-place table (Paris, 1795), instead of stopping at 100,000, gave the eight-place logarithms of the numbers between 100,000 and 108,000, in order to diminish the errors of interpolation, which were greatest in the early part of the table; and this addition was generally included in seven-place tables. The only important published extension of Vlacq's table was made by Mr. Sang 1871, whose table contained the seven-place logarithms of all numbers below 200,000.
Briggs and Vlacq also published original tables of the logarithms of the trigonometric functions.
Besides the tables mentioned above, a great collection, called Tables du Cadastre, was constructed under the direction of Gaspard de Prony, by an original computation, under the auspices of the French republican government of the 1700s. This work, which contained the logarithms of all numbers up to 100,000 to nineteen places, and of the numbers between 100,000 and 200,000 to twenty-four places, exists only in manuscript, "in seventeen enormous folios," at the Observatory of Paris. It was begun in 1792; and "the whole of the calculations, which to secure greater accuracy were performed in duplicate, and the two manuscripts subsequently collated with care, were completed in the short space of two years."[4] Cubic interpolation could be used to find the logarithm of any number to a similar accuracy.
## The logarithm as a function
The function logb(x) depends on both b and x, but the term logarithm function (or logarithmic function) in standard usage refers to a function of the form logb(x) in which the base b is fixed and so the only argument is x. Thus there is one logarithm function for each value of the base b (which must be positive and must differ from 1). Viewed in this way, the base-b logarithm function is the inverse function of the exponential function bx. The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.
### Graphical interpretation
The natural logarithm of a is the area under the curve y = 1/x between the x values 1 and a.
### Irrationality
For integers b and x > 1, the number logb(x) is irrational (that is, not a quotient of two integers) if either b or x has a prime factor which the other does not. In certain cases this fact can be proved very quickly: for example, if log23 were rational, we would have log23 = n/m for some positive integers n and m, thus implying 2n = 3m. But this last identity is impossible, since 2n is even and 3m is odd. Much stronger results are known. See Lindemann–Weierstrass theorem.
## Integer and non-integer exponents
If n is a positive integer, bn signifies the product of n factors equal to b:
${\displaystyle \underbrace {b\times b\times \cdots \times b} _{n}.}$
However, if b is a positive real number not equal to 1, this definition can be extended to any real number n in a field (see exponentiation). Similarly, the logarithm function can be defined for any positive real number. For each positive base b not equal to 1, there is one logarithm function and one exponential function, which are inverses of each other.
Logarithms can reduce multiplication operations to addition, division to subtraction, exponentiation to multiplication, and roots to division. Therefore, logarithms are useful for making lengthy numerical operations easier to perform and, before the advent of electronic computers, they were widely used for this purpose in fields such as astronomy, engineering, navigation, and cartography. They have important mathematical properties and are still widely used today.
## Bases
The most widely used bases for logarithms are 10, the mathematical constant e ≈ 2.71828… and 2. When "log" is written without a base (b missing from logb), the intent can usually be determined from context:
• Natural logarithm (loge, ln, log, or Ln) in mathematical analysis
• Common logarithm (log10 or simply log) in engineering and when logarithm tables are used to simplify hand calculations
• Binary logarithm (log2) in information theory and musical intervals
• Indefinite logarithm when the base is irrelevant, for example, in complexity theory when describing the asymptotic behavior of algorithms in big O notation.
To avoid confusion, it is best to specify the base if there is any chance of misinterpretation.
### Other notations
The notation "ln(x)" invariably means loge(x), that is, the natural logarithm of x, but the implied base for "log(x)" varies by discipline:
• Mathematicians generally understand both "ln(x)" and "log(x)" to mean loge(x) and write "log10(x)" when the base-10 logarithm of x is intended.
• Many engineers, biologists, astronomers, and some others write only "ln(x)" or "loge(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log10(x) or, sometimes in the context of computing, log2(x).
• On most calculators, the LOG button is log10(x) and LN is loge(x).
• In most commonly used computer programming languages, including C, C++, Java, Fortran, Ruby, and BASIC, the "log" function returns the natural logarithm. The base-10 function, if it is available, is generally "log10."
• Some people use Log(x) (capital L) to mean log10(x), and use log(x) with a lowercase l to mean loge(x).
• The notation Log(x) is also used by mathematicians to denote the principal branch of the (natural) logarithm function.
• A notation frequently used in some European countries is the notation blog(x) instead of logb(x).
This chaos, historically, originates from the fact that the natural logarithm has nice mathematical properties (such as its derivative being 1/x, and having a simple definition), while the base 10 logarithms, or decimal logarithms, were more convenient for speeding calculations (back when they were used for that purpose). Thus, natural logarithms were only extensively used in fields like calculus while decimal logarithms were widely used elsewhere.
As recently as 1984, Paul Halmos in his "automathography" I Want to Be a Mathematician heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. (The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at Berkeley.) As of 2005, many mathematicians have adopted the "ln" notation, but most use "log."
In computer science, the base 2 logarithm is sometimes written as lg(x) to avoid confusion. This usage was suggested by Edward Reingold and popularized by Donald Knuth. However, in Russian literature, the notation lg(x) is generally used for the base 10 logarithm, so even this usage is not without its perils.[5] In German, lg(x) also denotes the base 10 logarithm, while sometimes ld(x) or lb(x) is used for the base 2 logarithm.[2]
### Change of base
While there are several useful identities, the most important for calculator use lets one find logarithms with bases other than those built into the calculator (usually loge and log10). To find a logarithm with base b, using any other base k:
${\displaystyle \log _{b}(x)={\frac {\log _{k}(x)}{\log _{k}(b)}}.}$
Moreover, this result implies that all logarithm functions (whatever the base) are similar to each other. So to calculate the log with base 2 of the number 16 with your calculator:
${\displaystyle \log _{2}(16)={\frac {\log(16)}{\log(2)}}.}$
## Uses of logarithms
Logarithms are useful in solving equations in which exponents are unknown. They have simple derivatives, so they are often used in the solution of integrals. The logarithm is one of three closely related functions. In the equation bn = x, b can be determined with radicals, n with logarithms, and x with exponentials. See logarithmic identities for several rules governing the logarithm functions. For a discussion of some additional aspects of logarithms see additional logarithm topics.
### Science and engineering
Various quantities in science are expressed as logarithms of other quantities.
• The negative of the base-10 logarithm is used in chemistry, where it expresses the concentration of hydronium ions (H3O+, the form H+ takes in water), in the measure known as pH. The concentration of hydronium ions in neutral water is 10−7 mol/L at 25 °C, hence a pH of 7.
• The bel (symbol B) is a unit of measure that is the base-10 logarithm of ratios, such as power levels and voltage levels. It is mostly used in telecommunication, electronics, and acoustics. It is used, in part, because the ear responds logarithmically to acoustic power. The Bel is named after telecommunications pioneer Alexander Graham Bell. The decibel (dB), equal to 0.1 bel, is more commonly used. The neper is a similar unit which uses the natural logarithm of a ratio.
• The Richter scale measures earthquake intensity on a base-10 logarithmic scale.
• In spectrometry and optics, the absorbance unit used to measure optical density is equivalent to −1 B.
• In astronomy, the apparent magnitude measures the brightness of stars logarithmically, since the eye also responds logarithmically to brightness.
• In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation.
• In computer science, logarithms often appear in bounds for computational complexity. For example, to sort N items using comparison can require time proportional to N log N.
### Exponential functions
The natural exponential function exp(x), also written ${\displaystyle e^{x}}$ is defined as the inverse of the natural logarithm. It is positive for every real argument x.
The operation of "raising b to a power p" for positive arguments ${\displaystyle b}$ and all real exponents ${\displaystyle p}$ is defined by
${\displaystyle b^{p}=\exp({p\ln b}).\,}$
The antilogarithm function is another name for the inverse of the logarithmic function. It is written antilogb(n) and means the same as ${\displaystyle b^{n}}$.
### Easier computations
Logarithms switch the focus from normal numbers to exponents. As long as the same base is used, this makes certain operations easier:
Operation with numbers Operation with exponents Logarithmic identity
${\displaystyle \!\,ab}$ ${\displaystyle \!\,A+B}$ ${\displaystyle \!\,\log(ab)=\log(a)+\log(b)}$
${\displaystyle \!\,a/b}$ ${\displaystyle \!\,A-B}$ ${\displaystyle \!\,\log(a/b)=\log(a)-\log(b)}$
${\displaystyle \!\,a^{b}}$ ${\displaystyle \!\,Ab}$ ${\displaystyle \!\,\log(a^{b})=b\log(a)}$
${\displaystyle \!\,{\sqrt[{b}]{a}}}$ ${\displaystyle \!\,A/b}$ ${\displaystyle \!\,\log({\sqrt[{b}]{a}})={\frac {\log(a)}{b}}}$
These relations made such operations on two numbers much faster and the proper use of logarithms was an essential skill before multiplying calculators became available.
The ${\displaystyle \log(ab)=\log(a)+\log(b)}$ equation is fundamental (it implies effectively the other three relations in a field) because it describes an isomorphism between the additive group and the multiplicative group of the field.
To multiply two numbers, one found the logarithms of both numbers on a table of common logarithms, added them, and then looked up the result in the table to find the product. This is faster than multiplying them by hand, provided that more than two decimal figures are needed in the result. The table needed to get an accuracy of seven decimals could be fit in a big book, and the table for nine decimals occupied a few shelves.
The discovery of logarithms just before Newton's era had an impact in the scientific world which can be compared with the invention of the computer in the twentieth century, because many calculations which were too laborious became feasible.
When the chronometer was invented in the eighteenth century, logarithms allowed all calculations needed for astronomical navigation to be reduced to just additions, speeding the process by one or two orders of magnitude. A table of logarithms with five decimals, plus logarithms of trigonometric functions, was enough for most astronomical navigation calculations, and those tables fit in a small book.
To compute powers or roots of a number, the common logarithm of that number was looked up and multiplied or divided by the radix. Interpolation could be used for still higher precision. Slide rules used logarithms to perform the same operations more rapidly, but with much less precision than using tables. Other tools for performing multiplications before the invention of the calculator include Napier's bones and mechanical calculators: see history of computing hardware.
### Calculus
The derivative of the natural logarithm function is
${\displaystyle {\frac {d}{dx}}\ln(x)={\frac {1}{x}}.}$ (A proof is shown below.)
By applying the change-of-base rule, the derivative for other bases is
${\displaystyle {\frac {d}{dx}}\log _{b}(x)={\frac {d}{dx}}{\frac {\ln(x)}{\ln(b)}}={\frac {1}{x\ln(b)}}={\frac {\log _{b}(e)}{x}}.}$
The antiderivative of the logarithm is
${\displaystyle \int \log _{b}(x)\,dx=x\log _{b}(x)-{\frac {x}{\ln(b)}}+C=x\log _{b}\left({\frac {x}{e}}\right)+C.}$
See also: table of limits of logarithmic functions, list of integrals of logarithmic functions.
#### Proof of the derivative
The derivative of the natural logarithm function is easily found via the inverse function rule. Since the inverse of the logarithm function is the exponential function, we have ${\displaystyle \ln '(x)={\frac {1}{\exp '(\ln(x))}}}$. Since the derivative of the exponential function is itself, the right side of the equation simplifies to ${\displaystyle {\frac {1}{\exp(\ln(x))}}={\frac {1}{x}}}$, the exponential canceling out the logarithm.
## Computers
When considering computers, the usual case is that the argument and result of the ${\displaystyle \ln(x)}$ function is some form of floating point data type. Note that most computer languages uses ${\displaystyle \log(x)}$ for this function while the ${\displaystyle \log _{10}(x)}$ is typically denoted log10(x).
As the argument is floating point, it can be useful to consider the following:
A floating point value x is represented by a mantissa m and exponent n to form
${\displaystyle x=m2^{n}.\,}$
Therefore
${\displaystyle \ln(x)=\ln(m)+n\ln(2).\,}$
Thus, instead of computing ${\displaystyle \ln(x)}$ we compute ${\displaystyle \ln(m)}$ for some m such that ${\displaystyle 1\leq m<2}$. Having ${\displaystyle m}$ in this range means that the value ${\displaystyle u={\frac {m-1}{m+1}}}$ is always in the range ${\displaystyle 0\leq u<{\frac {1}{3}}}$. Some machines uses the mantissa in the range ${\displaystyle 0.5\leq m<1}$ and in that case the value for u will be in the range ${\displaystyle -{\frac {1}{3}} In either case, the series is even easier to compute.
## Generalizations
The ordinary logarithm of positive reals generalizes to negative and complex arguments, though it is a multivalued function that needs a branch cut terminating at the branch point at 0 to make an ordinary function or principal branch. The logarithm (to base e) of a complex number z is the complex number ln(|z|) + i arg(z), where |z| is the modulus of z, arg(z) is the argument, and i is the imaginary unit.
The discrete logarithm is a related notion in the theory of finite groups. It involves solving the equation bn = x, where b and x are elements of the group, and n is an integer specifying a power in the group operation. For some finite groups, it is believed that the discrete logarithm is very hard to calculate, whereas discrete exponentials are quite easy. This asymmetry has applications in public key cryptography.
The logarithm of a matrix is the inverse of the matrix exponential.
A double logarithm, ${\displaystyle \ln(\ln(x))}$, is the inverse function of the double exponential function. A super-logarithm or hyper-logarithm is the inverse function of the super-exponential function. The super-logarithm of x grows even more slowly than the double logarithm for large x.
For each positive b not equal to 1, the function logb (x) is an isomorphism from the group of positive real numbers under multiplication to the group of (all) real numbers under addition. They are the only such isomorphisms that are continuous. The logarithm function can be extended to a Haar measure in the topological group of positive real numbers under multiplication.
## Notes
1. James Mills Peirce, The Elements of Logarithms with an Explanation of the Three and Four Place Tables of Logarithmic and Trigonometric Functions (1873).
2. Math Forum, Logarithms: History and Use Retrieved November 20, 2018.
3. Great Britain Institute of Actuaries, Journal of the Institute of Actuaries and Assurance Magazine, 1873, Vol. 17 (Forgotten Books, 2018, 978-0366971244).
4. Charles Knight, English Cyclopaedia, Biography, Vol. IV., article "Prony."
5. MathWorld, Common Logarithm. Retrieved November 20, 2018.
## ReferencesISBN links support NWE through referral fees
• Great Britain Institute of Actuaries, Journal of the Institute of Actuaries and Assurance Magazine, 1873, Vol. 17. Forgotten Books, 2018. 978-0366971244
• Knight, Charles. The English Cyclopaedia, Vol. 4. Forgotten Books, 2012.
• Peirce, James Mills. The Elements of Logarithms with an Explanation of the Three and Four Place Tables of Logarithmic and Trigonometric Functions. Andesite Press, 2015. ISBN 978-1297495465
• Przeworska-Rolewicz, D. Logarithms and Antilogarithms: An Algebraic Analysis Approach with an appendix by Zbigniew Binderman (Mathematics and Its Applications). New York, NY: Springer, 1998. ISBN 0792349741.
• REA. Math Made Nice & Easy #2: Percentages, Exponents, Radicals, Logarithms and Algebra Basics (Math Made Nice & Easy). Piscataway, NJ: Research & Education Association, 1999. ISBN 0878912010.
• Ryffel, Henry, Robert Green, Holbrook Horton, and Edward Messal. Mathematics at Work. New York, NY: Industrial Press, Inc., 1999. ISBN 0831130830.<|endoftext|>
| 4.75 |
257 |
Anti-Slave Trade Patrols
America withdrew from the transatlantic slave trade in 1808. With The Treaty of Ghent, ending the War of 1812, both the United States and Great Britain agreed to work towards ending the slave trade. The U.S. Navy's role in the struggle against slavery began in 1820 when warships deployed off West Africa to catch American slave ships. Enforcement of the slave trade ban was sporadic until the Navy deployed a permanent African Squadron in 1842. This deployment was due to the Webster-Ashburton Treaty, between the United States and Great Britain signed that August to suppress the slave trade. Despite the vigilance of American, as well as British and French, warships in African waters, the overseas slave trade increased in the 1850s, owing to the high demand for slaves in Latin America. The U.S. Navy's participation lasted until the start of the U.S. Civil War, April 1861.
Interesting artifacts in the Anti-Slavery Patrol exhibit include:
- Model of USS Shark (Schooner)
- Painting, oil on canvas, of Captain Elisha Peck (Captain of Sloop-of-War Portsmouth)
Click on the model below to review a photographic history of USS Shark.<|endoftext|>
| 3.75 |
420 |
As KB students progress in their counting skills we’ve introduced this open ended task that is challenging and allows all students to represent their mathematical thinking in various ways.
Students started by looking at the pictures below and commenting on what they “See, Think, and Wonder”. Then we set off to solve one of their toughest wonderings, “How many EGGS are in the tower”.
I.P. gave the class the idea that they should figure out how many boxes are in the tower first then they can count how many eggs! Each student set off to solve this question in ways that made sense for them.
Then students shared their thinking with the class. We came to the conclusion that mathematicians in our class used numbers, words, counting strategies, addition, symbols, and made models to help them figure out the question.
Students have now been working for 5 days to figure out this same question: How Many eggs are in the tower?
Each day students find more information that gets them closer to the answer. Sometimes they start over and use different strategies to help them figure it out. Some choose to work in groups on one day and then by themselves on another day.
No matter how they choose to work our focus is on HOW the mathematicians are thinking and HOW they are showing their thinking and getting it out of their brain. Each day ends with students sharing how they tried to solve the problem. Although we may not get the answer, we are all using our skills, learning new skills from friends and teachers, and are building our stamina as mathematicians.
Today’s GO WORD is “struggle”. KB students have discussed the importance of struggling to sovle a problem, but continuing to try as a way to build connections between the synapses in our brains. Finding the answer to a problem is not always the most important part and this challenging tasks serves as our reminder. We might not find the number answer to this task, but we are growing our brains every single time we try.<|endoftext|>
| 4.21875 |
499 |
Courses
Courses for Kids
Free study material
Offline Centres
More
Store
# The breaking stress of a wire depends uponA. The length of the wireB. The radius of the wireC. The material of the wireD. The shape of the cross-section
Last updated date: 09th Sep 2024
Total views: 428.7k
Views today: 9.28k
Verified
428.7k+ views
Hint: We know that the equation for the relation of stress and strain is $\dfrac{{Stress}}{{Strain}} = \gamma$ . This means that the stress on an object depends on $\gamma$ . Now $\gamma$ as we already know depends upon the material that makes up the wire.
Before starting the actual solution, it would be good to discuss stress, stress, and their relationship.
Stress – Stress is the force applied per unit area on a material.
Strain – Strain is the change in the dimensions of material when it is under stress.
The relation of stress-strain is given below
$\dfrac{{Stress}}{{Strain}} = \gamma$
Here, $\gamma =$ The proportionality of linear expansion
The stress-strain graph is shown below
In this graph, you can see the point E which is called the breaking point where the object can longer bear the stress and just breaks into two. At this point even if we reduce the stress the strain still increases.
By the stress-strain equation, i.e. $\dfrac{{Stress}}{{Strain}} = \gamma$ , we can see that the strain of a body does not depend on the length of the wire, the shape of the cross-section of the wire, and the radius of the wire.
But stress does depend on $\gamma$ . Now $\gamma$ depends on the material of the wire, so the breaking stress also depends on the material of the wire.
In simple words, it means that two objects made of different material will have different breaking stress (the limit of strain the body can bear)
So, the correct answer is “Option C”.
Note:
The concept of breaking stress that we discussed in the solution above is of great importance to us. Every object that we use whether it be lifts, bridges, mobiles phones, etc. are designed in such a manner that the stress on the object does not exceed the safe limit in normal day to day usage.<|endoftext|>
| 4.5 |
795 |
## How to Find Eulerian Paths in Graph Theory
Edges: Degrees 1:
Output: `Press calculate`
# How to Find Eulerian Paths in Graph Theory
Graph theory is a fascinating field of mathematics that finds applications in computer science, engineering, social sciences, and many other domains. One of its intriguing problems is that of finding Eulerian paths, named after the brilliant mathematician Leonhard Euler. An Eulerian path is a trail in a graph that visits every edge exactly once. But how do you determine whether such a path exists for a given graph? Let’s dive into the details and uncover the mystery behind Eulerian paths!
## Understanding Eulerian Paths
To comprehend Eulerian paths, it's important to grasp some basic concepts of graph theory. A graph comprises vertices (nodes) and edges (connections between nodes). Eulerian paths are special because they traverse every edge precisely once.
• Eulerian Path: A trail that visits every edge of the graph exactly once.
• Eulerian Circuit: A cycle that visits every edge of the graph exactly once and returns to the starting vertex.
• Degree of a Vertex: The number of edges connected to the vertex.
## Conditions for Eulerian Paths
Discovering whether a graph possesses an Eulerian path or circuit is subject to specific conditions:
• Eulerian Circuit: All vertices must have an even degree.
• Eulerian Path: Exactly zero or two vertices should have an odd degree.
If these conditions are met, the graph has an Eulerian path or circuit; otherwise, it does not.
## Finding Eulerian Paths
### 1. Identify Vertex Degrees
The first step is to assess the degrees of all vertices. Count the number of edges connected to each vertex.
### 2. Check the Conditions
• If every vertex has an even degree, the graph contains an Eulerian circuit and thus an Eulerian path.
• If exactly two vertices have an odd degree, the graph has an Eulerian path starting at one odd-degree vertex and ending at the other.
• If the graph does not meet these criteria, it lacks an Eulerian path.
VertexDegree
A2
B3
C2
D3
In this example, vertices B and D have odd degrees, fulfilling the condition for an Eulerian path.
## Real-Life Example of Eulerian Paths
Imagine you're planning a drone delivery route and need to traverse every street in your delivery area. By representing streets as edges and intersections as vertices, you can apply Eulerian path concepts to find an optimal route. If there are exactly two intersections with an odd number of streets, you have an Eulerian path. If all intersections are even, your route is an Eulerian circuit.
## FAQs
### What is an Eulerian Path?
An Eulerian path is a trail in a graph that visits every edge exactly once.
### What conditions are needed for an Eulerian path?
At most, two vertices should have an odd degree for an Eulerian path to exist.
### Can a graph have both an Eulerian path and circuit?
Yes, a graph with an Eulerian circuit (all even-degrees vertices) inherently contains an Eulerian path.
### Is there an Eulerian path in a disconnected graph?
No, a disconnected graph cannot contain an Eulerian path.
### What is a real-life application of Eulerian paths?
Eulerian paths can optimize routes for delivery systems, garbage collection routes, and network data traversal.
## Summary
Eulerian paths in graph theory open up a world of efficient problem-solving. By understanding the conditions that define these paths and applying them to various scenarios, from transportation to network analysis, one can greatly enhance operational efficiency. Leonhard Euler's discovery continues to influence modern algorithms and solutions today. Whether you're a student or a professional, mastering Eulerian paths equips you with a powerful tool to solve complex issues with elegance and precision.
Tags: Math, Graph Theory, Algorithms<|endoftext|>
| 4.53125 |
537 |
Glaciers in Antarctica are remote and difficult to review, which suggests a lot of regarding their behavior is unknown. These data gaps mean scientists cannot be certain however they’re going to answer global climate change.
However, with new technology researchers are able to study them in bigger detail than ever before, even searching through kilometers-thick glaciers to look at processes occurring at their bases. For the study, scientists used ice-penetrating measuring device mounted on planes to see through the ice mass and determine what’s happening at its base.
Scientists have connected an enormous 130-km-long channel on the surface of an Antarctic floating shelf ice to the landscape 2 kilometers below the ice sheet upstream. The channel is thought to be a point of instability on the ice shelf. If the surface ice melts, water will preferentially run down these features, carving out a deeper channel and creating further weaknesses.
According to scientists, the study will help pinpoint potential regions of Antarctica that may be of enhanced risk of change because of ice shelf channeling.
Project lead Professor Martin Siegert, from the Grantham Institute – Climate Change and the Environment at Imperial, said, “It also reminds us that we can’t ignore subsurface processes – even if they are below two kilometers of ice in some of the most remote places on Earth.”
“Surface melting on Antarctic ice shelves has been noticed in the last few years, and further warming of atmospheric conditions would lead to increased levels – adding to the need to restrain global warming to 1.5C above pre-industrial levels.”
The region that scientists studied, is known as one of the poorest known parts of Antarctica, near the edge of the ‘grounded’ ice sheet, resting on land rather than water.
The grounded ice sheet feeds ice into the ocean, then contributes to water level modification. Holding the ice sheet back are the floating ice shelves, which offer a ‘back force’ to scale back the speed of the grounded ice. Weakness within the floating ice will so result in an accelerated flow of the grounded ice, and water level rise.
The team found that when the base of the glacier encountered a large solitary hill at the same point it starts to float, a gap emerged under the ice downstream of the hill. This gap was crammed by water from around the base of the ice mass, that sliced a gouge upwards into the ice. This gouge was 800 meters high in some places and LED to the intensive channel seen on the surface of the ice.
The study is published in the journal Nature Communications.<|endoftext|>
| 4.1875 |
335 |
3.6. Positive and Negative Lookahead
Naturally, a regular expression is matched against a string in a linear fashion (with backtracking as necessary). Therefore there is the concept of the "current location" in the stringrather like a file pointer or a cursor.
The term lookahead refers to a construct that matches a part of the string ahead of the current location. It is a zero-width assertion because even when a match succeeds, no part of the string is consumed (that is, the current location does not change).
In this next example, the string "New World" will be matched if it is followed by "Symphony" or "Dictionary"; however, the third word is not part of the match:
s1 = "New World Dictionary" s2 = "New World Symphony" s3 = "New World Order" reg = /New World(?= Dictionary| Symphony)/ m1 = reg.match(s1) m.to_a # "New World" m2 = reg.match(s2) m.to_a # "New World" m3 = reg.match(s3) # nil
Here is an example of negative lookahead:
reg2 = /New World(?! Symphony)/ m1 = reg.match(s1) m.to_a # "New World" m2 = reg.match(s2) m.to_a # nil m3 = reg.match(s3) # "New World"
In this example, "New World" is matched only if it is not followed by "Symphony."<|endoftext|>
| 3.984375 |
539 |
In the rainforests along the border of Myanmar and China lives a monkey with a nose so upturned the rain is said to cause the primate to sneeze uncontrollably. It’s called the Burmese snub-nosed monkey (Rhinopithecus strykeri) and it was just discovered in 2010.
Scientists reporting in the International Journal of Primatology note that for first time, they’ve had success using camera traps to capture images of these elusive monkeys. Direct observation on eight days lead to 222 pictures (three of which are seen above) from 30 camera traps. The new pictures, according to the BBC, reveal that these monkeys are most similar to the black-and-white snub-nosed monkey and that they live in clusters, groups containing one male and multiple females.
In March of 2014, Fauna & Flora International (FFI) got the first video of these animals, already in trouble due to hunting and logging. FFI said there are likely just 300 of these individuals left. Their strange anatomical feature doesn’t help.
When the primates were first uncovered five years ago, scientists noticed that during the rainy season, they were less hidden because they were sneezing — loudly — all the time. “Normally they’re pretty quiet,” Frank Momberg, FFI’s Asia-Pacific development director, told National Geographic. Given their nose problems, these monkeys supposedly spend much of any downpour bent with their heads between their legs, according to locals and reported by Discovery, though it hasn’t yet been scientifically proven.
Other seasons hinder these monkeys, too. They spend the summer (i.e., dry season) at higher altitudes rich in food, moving to lower altitudes when the snows come, putting the primates closer to villages — and potentially more at risk.
The International Union for Conservation of Nature and Natural Resources already lists the Burmese snub-nosed monkey as critically endangered, noting an “urgent need” to put conservation measures in place, a feeling shared by FFI. “We are committed to taking immediate conservation action to safeguard the survival of this important new species,” Mark Rose, FFI CEO, said back in 2010.
Allergy-sufferers and anyone who’s had a cold can relate to how frustrating uncontrollable sneezing can be. For these monkeys, the difference between a loud and soft ah-choo could save their lives.
MORE FROM WEATHER.COM: Summer-Loving Animals Beat the Heat (PHOTOS)<|endoftext|>
| 3.703125 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.