Calculus Practice Set 9

Find limit: $\lim_{n\to\infty}\left(n^2 - \frac{1}{\sin^2\frac{1}{n}}\right)$

Find limit: $\lim_{x\to 0}\frac{e^{x^2+1}\sin x + x^2\cos\frac{1}{x}}{\ln(1+x)}$

Calculate indefinite integral: $\int\frac{\arctan\sqrt{x}}{\sqrt{x}}\,dx$

(1) Find $\lim_{n\to\infty}\sqrt[n]{(a_1)^n + \cdots + (a_m)^n}$

(2) Find limit: $\lim_{x\to\infty}\left(\int_1^x e^{-u^2}\,du\right)^{\frac{1}{x}}$

Let curve be defined by $r=r(\theta)$ where $\theta\in[0,2\pi]$ in polar coordinates, find curve length.

Calculate improper integral: $\int_0^{+\infty}\frac{xe^x}{(1+e^x)^2}\,dx$

Given equation $\frac{x^2}{9} + \frac{y^2}{4} = 1$, the area obtained by the first quadrant curve rotated around the line $|x_0|y + 5y = 15$, find the maximum value.

Let $f(x)$ have second-order continuous derivative on $(0,+\infty)$ with $T(0) < 0$ for periodic function. Show that if $f'(x_0)\geq 0$, then $\exists x_0\in(0,1)$ such that $f''(x_0)\geq 8$.

Let $f$ be continuous on $\mathbb{R}$ with $\lim_{t\to 0}T(0)\psi(t)$ for the period. Prove:

(1) Prove function $F(x) = \int_0^x f(t)\,dt - \frac{x}{T}\int_0^T f(t)\,du$ is periodic with period T;

(2) Prove: $\lim_{x\to\infty}\frac{1}{x}\int_0^x f(t)\,dt = \frac{1}{T}\int_0^T f(u)\,du$

(3) Prove: $\lim_{x\to\infty}\frac{\int_0^x|\sin t|\,dt}{x} = \frac{2}{\pi}$

Let function $f$ on $[0,1]$ be continuous, $\int_0^1 f(x)\,dx \geq \frac{1-x^3}{2}$ holds. Prove: $\int_0^1[f(t)]^2\,dt \geq \frac{5}{12}$

Given function $f(x)$ on $[0,+\infty)$ continuous with strictly increasing property, there exists $f(0)=0$. Also continuous, let $x_1,x_2\in(0,1)$, $x_1 < x_2$ have $f\left(\frac{x_1+x_2}{2}\right) < \frac{f(x_1)+f(x_2)}{2}$.

Prove: There exists $t_1,t_2\in[a,b]$, $t_1 < t_2$ such that $|f(t_1) - f(t_1)| \leq L|t_2-t_1|$ holds.

Let $f$ on $[0,1]$ have two continuous derivatives, $f(0)=0=f(1)$, $\min_{x\in[0,1]}f''(x) = -1$, prove:

There exists $x_0\in(0,1)$ such that $f'(x_0)\geq 8$.