Calculus Practice Set 11

Find limit: $\lim_{n\to+\infty}\left[\frac{e}{\left(1+\frac{1}{n}\right)^n}\right]^n$

Find limit: $\lim_{x\to 0}\left(\frac{1}{x^2}-\frac{1}{\sin^2 x}\right)$

Given function $f(x) = \begin{cases} \frac{x}{7}+x^2\sin\frac{1}{x} & ,x\neq 0 \\ 0 & ,x=0 \end{cases}$

(1) Find $f'(0)$; (2) Determine whether $\exists\delta > 0$ such that $f(x)$ on $(-\delta,\delta)$ is strictly increasing.

Known first derivative defined by curve formula $y=y(x)$ can be written as $\int_0^{x+y}e^{-t^2}\,dt = \int_0^y u\sin[(u+1)^2]\,du$, find $y(0), y'(0), y''(0)$.

Given function $f(x) = e^{(1+\frac{x}{e})^{\frac{1}{t}}}$, when $x\to 0$ has expansion $f(x) = a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+o(x^4)$, find $a_2, a_3, a_4$.

Calculate improper integral: $J = \int_0^{+\infty}\frac{\arctan e^x}{e^x}\,dx$

Known curve with polar equation $r = 2^a$, $\theta\in[e,\pi]$, find curve length.

Calculate definite integral: $I = \int_0^{\frac{\pi}{4}}\frac{e^{\frac{t}{2}}(\cos t - \sin t)}{\sqrt{\cos t}}\,dt$

Given $f(x)$ on $x=x_0$ continuous, $\lim_{h\to 0}\frac{f(x_0+h)+f(x_0-h)-2f(x_0)}{h^2} = f''(x_0)$.

Given $\alpha$ is a positive integer, $n = 2^{2023}$, prove: For all $0 < x < 1$, $n^\alpha x^n \leq e^{-\alpha n^{\alpha-1}(-\ln x)^{-\alpha}}$.

Known $f(x)$ and $g(x)$ on $[0,1]$ continuous, and for all $x\in[0,1]$ there exists $g(x)\geq 0$. Prove: There exists $\exists x_0\in[0,1]$ such that $\int_0^1f(x)g(x)\,dx = f(x_0)\int_0^1g(x)\,dx$.

Known $f(x)$ on $(-1,2)$ continuous, on $(-1,2)$ except constant is differentiable. And $g(x)$ on $(-1,2)$ is strictly increasing with constant above. Prove: There exists $\exists\xi\in(0,1)$ such that $\int_0^\xi f(x)g(x)\,dx = g(0)\int_0^\xi f(x)\,dx + g(1)\int_\xi^1 f(x)\,dx$.