query: $y'(x)=y^2-y$ and $y(0)=\frac 12$. Show that $y(x) \in (0,1)$ $\forall x$
positive_passages.docid: 155864
positive_passages.text: Prove or disprove: if $y'=y-y^2$ and $y(0)=a$ where $00$
negative_passages.docid: 105999
negative_passages.text: An application of derivatives problem- to show that $f(x+2)-f(x)>2$ for $f(x)=x\cos(1/x)$
negative_passages.docid: 19828
negative_passages.text: Show that $\frac{x^3}{x^2+y^2}$ is not differentiable at $(0,0)$, even though all directional derivatives exist
negative_passages.docid: 129394
negative_passages.text: understanding $\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}=-1$
negative_passages.docid: 69172
negative_passages.text: $y'=$ ${-x+\sqrt{(x^2+4y)}}\over 2$ , $y(2)=-1$
negative_passages.docid: 33523
negative_passages.text: If $f(0)=0$ and $f'$ is increasing, then $\frac{f(x)}{x}$ is increasing.
negative_passages.docid: 71127
negative_passages.text: How do I find $f(0)$, $f'(0)$, and $f'(x)$ given $f(x+y)=f(x)+f(y)+x^2y+xy^2$ and $\lim_{x\to0}\frac{f(x)}{x}=1$?
negative_passages.docid: 107727
negative_passages.text: $m\{x\in [0,1]:f'(x)=0\}>0$
negative_passages.docid: 128764
negative_passages.text: How does $y[x] = a x^r$ when $y'[x] = \frac{(r y[x])}{x}$ and $y[1] = a$
negative_passages.docid: 88374
negative_passages.text: How to separate variables in DiffEq of $y'=x-y$
negative_passages.docid: 30204
negative_passages.text: Prove that $f'(0)$ exists and $f'(0) = b/(a - 1)$
negative_passages.docid: 129847
negative_passages.text: Show that $\forall (x,y)$ in the first quadrant: $\frac {x^2+y^2}{4}\leq e^{x+y-2}$
negative_passages.docid: 81437
negative_passages.text: Find the value of $y(1)$ of the ODE $y'+y=|x|$.
negative_passages.docid: 92054
negative_passages.text: $X$ and $Y$ i.i.d., $X+Y$ and $X-Y$ independent, $\mathbb{E}(X)=0 $ and $\mathbb{E}(X^2)=1$. Show $X \sim N(0,1)$
negative_passages.docid: 103706
negative_passages.text: If $\{f(0)\}^{2} + \{f'(0)\}^{2} = 4$ then there is a $c$ with $f(c) + f''(c) = 0$
negative_passages.docid: 78319
negative_passages.text: Show that $x \mapsto \frac{f(x)}{x}$ is strictly increasing on (0,1) given that f '(x) is strictly increasing on (0,1) and that f(0)=0
negative_passages.docid: 116946
negative_passages.text: Ordinary differential equation $y'(t)=\sin(f(t,y))$