# Tagged Questions

For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

3answers
123 views

### What is $\dfrac{dr}{d\theta}$?

Suppose we have an equation of a polar curve with usual notation $r=f(\theta).$ I am curious about the geometric meaning of $$\dfrac{dr}{d\theta}=f'(\theta).$$ Also I would like to know the relations ...
1answer
64 views

### how to calculate this limit? without l'hospitalet rule or taylor's or any use of derivatives [closed]

How can i solve this limit? $\displaystyle\lim_{x \to \infty} x^2\ln\left(\dfrac{x^2}{x^2+1}\right)$ I can't use l'hospital rule or taylors or derivatives Just limit modification
1answer
60 views

### Surface area using integration.

I am stuck on the following problem: Find the surface area of the solid formed be revolving $$y=\frac{a}{2}\left(e^\frac{x}{a}+e^\frac{{-x}}{a}\right)$$ about the $x$ axis from $x\in[-a,a]$. Can ...
1answer
75 views

### showing that an inequality holds

I am trying to figure out how to show that for $n\geq 3$, $$(2^n-1)^{\frac{n}{2(n-1)}}\geq (2^{n-1}-1)^{\frac{n-1}{2(n-2)}}+1.$$ I've tried basic algebra and induction, but the inductive hypothesis ...
2answers
35 views

### relation between $\frac{\partial(x,0)}{\partial x}$ and $\left.\frac{\partial(x,t)}{\partial x}\right|_{t=0}$

if $u(x,t)$ differentiable function and i only have $u(x,0)$, then is it right $\frac{\partial(x,0)}{\partial x} = \left.\frac{\partial(x,t)}{\partial x}\right|_{t=0}$ or can i derive $u(x,0)$ to $x$ ...
1answer
95 views

### definite integrals (very interesting question) [closed]

If $u = \displaystyle\int_{0}^{\pi/4}\left(\dfrac{\cos x}{\sin x +\cos x}\right)^2\,dx$ and $v = \displaystyle\int_{0}^{\pi/4}\left(\dfrac{\sin x +\cos x}{\cos x}\right)^2\,dx$ then find $\dfrac{v}{u}$...
1answer
64 views

1answer
26 views

2answers
49 views

1answer
51 views

### The height of right isosceles triangle decreases with the speed proportional to the area of this triangle

The height of right isosceles triangle decreases with the speed proportional to the area of this triangle. At time $t=0$ the area of triangle is $2$, and at time $t=1$ the area of triangle is $\frac12$...
2answers
43 views

### Prove integral inequality of $C^1$ function

Let $f$ be $C^1$ class over $[a,b]$. Asuume that $f(a)=f(b)=0$ prove that: $\displaystyle \sup_{t\in [a,b]} |f(t)| \le \frac{1}{2}\int_a^b |f'(t)| dt$ I tried to show it from C-S ineq. for integrals ...
2answers
65 views

### Find a limit containing an definite integral

Let $f\in C[0,1]$ and $f\ge 0$. Find the limit: $$\lim_{n\to\infty} \left( \int_0^1 (f(x))^n \ dx \right)^{1/n}$$ My thought: We denote $f_n(x) = f(x)^n$. If we could show that $f_n(x)\to f$ ...
1answer
51 views

### Examine the continuity of function $f(x)=\frac{2x^2-4x}{|x+1|+|x-3|-2}$

Using the definition of absolute value for $$|x+1|=\begin{cases} x+1, & x\ge -1\\ -x-1, & x>-1 \end{cases}$$ and $$|x-3|=\begin{cases} x-3, & x\ge 3\\ -x+3, & x>3 \end{cases}$$ ...
1answer
42 views

### Find the Fourier coefficient of $f(x)$

Let $$f(x) = \begin{cases} \sum_{k=0}^\infty \frac{e^{inx}}{1+k^2} &\mbox{if } x \ne 2\pi k \\ 0 & \mbox{if } x = 2\pi k \end{cases}$$ Find the Fourier coefficients of $f(x)$ What I ...
2answers
40 views

### Proving the inequality $2^{-1+\frac{1}{n}}\left(x+1\right)\leq\left(x^{n}+1\right)^{\frac{1}{n}}$ for $n\in\mathbb{N}$ and $x>0$

Prove that for all $n\in\mathbb{N}$ and $x>0$, $$2^{-1+\frac{1}{n}}\left(x+1\right)\leq\left(x^{n}+1\right)^{\frac{1}{n}}$$ The last class was about Taylor polynomial of functions, so I ...
3answers
140 views

### evaluating $\lim_{n \to \infty }(\frac{n!}{n^n})^\frac{2n^4 + 1}{5n^5 + 1}$

I was trying to work out the following limit $$\lim_{n \to \infty }\Big(\frac{n!}{n^n}\Big)^{\Large\frac{2n^4+1}{5n^5+1}}$$ I have some idea about how to do it. Basically I expanded the $n!$ and ...
5answers
813 views

### What is the difference between necessary condition & sufficient condition?

My book says : For having extreme point $a$ of function $f$, the necessary condition is that $f'(a) = 0$. However, it isn't a sufficient condition. Now, what is the difference between necessary ...
4answers
127 views

### Why isn't $0$ an extreme point of $f(x) = x^3$? [closed]

My book writes $0$ isn't a point of extremum for $f(x) = x^3$ inspite of being $f'(0) = 0$. Why is it so?
0answers
68 views

### Can someone explain presheaf to a calculus student?

From reading some stuff online, there is the claim that continuous functions are presheafs, so that the toy functions I play with in class such as $f(x) = x^2$ can also be thought of as presheafs. ...
1answer
23 views