Calculus Practice Set 7

For all $n\in\mathbb{Z}^+$, let $a_n = \left(1+\frac{1}{n}\right)^n$, $b_n = \left(1+\frac{1}{n}\right)^{n+1}$. We already know sequence $\{a_n\}^{\infty}_{n=1}$ is strictly increasing, sequence $\{b_n\}^{\infty}_{n=1}$ is strictly decreasing. Prove using fixed $\varepsilon > 0$ definition that: For all $n\in\mathbb{Z}^+$, there exists $\frac{1}{n+1} < \ln(1+\frac{1}{n}) < \frac{1}{n}$.

Find limit: $\lim_{x\to 0}\frac{(3+2x)^x - 3^x}{\arcsin(x^2)}$

Given $f(x) = \frac{1}{1+\frac{1}{x^2}}$, find $f^{(2019)}(0)$.

Given function $f(x) = \begin{cases} e^{\frac{x}{x-1}} & x\neq 0 \\ 0 & x=0 \end{cases}$, let $f(x) = B_0 + \frac{B_1}{1!}x + \frac{B_2}{2!}x^2 + \cdots + \frac{B_n}{n!} + o(x^n)$, $x\to 0$ (mid-point formula holds), find coefficients $B_0, B_1, B_2$ values.

Given $f(x) = x^{\frac{1}{x}}$, $x > 0$, find $f(x)$ maximum value on $(0,+\infty)$.

Given curve $y=y(x)$ determined by $x=0$ when all points satisfy certain derivative condition, and determines $\lim_{x\to 0}\left(1+x+\frac{y(x)}{x}\right)^{\frac{1}{x}} = e^{2019}$, find $y(0), y'(0), y''(0)$.

Given curve $y=y(x)$ satisfies $y''+y^2+3xy=1$, find $dy|_{x=1}$.

Calculate indefinite integral: $\int\frac{x\ln x}{(1+x^2)^2}\,dx$

Given anti-derivative: $\int_0^{\frac{\pi}{2}}\frac{\sin x}{x}\,dx = \frac{\pi}{2}$, calculate anti-derivative: $\int_0^{+\infty}\frac{(\sin x)^2}{x^2}\,dx$

Find arc length of parametric curve: $\begin{cases} x = t - \sin t & ,t\in[0,1] \\ y = 1 - \cos t & ,t\in[0,1] \end{cases}$

Let $\{a_n\}$ be a real sequence, $a$ is a real number, define $N$ as positive integer. Prove that: the limit by $N$ definition $\sum_{i=1}^{\infty}a_i$ converges and equals $T-a(2)$ if and only if the summation $\sum_{i=1}^{\infty}a_i$ does not converge and equals $T-a$.

Let $f$ be defined on region $(0,1)$, continuous, and satisfies for all in mid-point formula form $0 < x_1 < x_2 < x_3$, there exists $\frac{f(x_2)-f(x_1)}{x_2-x_1} \leq \frac{f(x_3)-f(x_1)}{x_3-x_1} \leq \frac{f(x_3)-f(x_2)}{x_3-x_2}$. For any $x_0\in(0,1)$, prove at least one of the limits $f'(x_0^-), f'(x_0^+)$ exists.

Let $f$ on $[0,+\infty)$ have continuous derivative with strictly increasing property, let $f(0)=0$. Also $\alpha > 0, b > 0$ are two real constants, prove the following inequality holds: $\int_0^a f(x)\,dx + \int_0^b f^{-1}(y)\,dy \geq ab$.

For all $n\in\mathbb{Z}^+$, let $a_n = \sum_{i=1}^{n}\frac{\sqrt{i}}{n\sqrt{n+\frac{1}{i}}}$, prove: $\lim_{n\to\infty}a_n = \frac{2}{3}$