Calculus Practice Set 6

Given constants $a, b$, function $f(x) = \begin{cases} (2017+x)^{\frac{1}{b}} & ,x \geq 0 \\ a(1-x)^{\frac{1}{x}} & ,x < 0 \end{cases}$, $f(x)=0$ when $x=0$, find values of $a, b$.

Calculate the limit: $\lim_{x\to +\infty}\left(\sqrt[3]{x^3+x^2+1} - \sqrt[3]{x^3+1}\right)$

Given $f(x) = \arctan x$, find $f^{(2018)}(0)$.

Prove using N as constant: $\lim_{n\to\infty}\frac{2n^4+1}{3n^4+4} = \frac{2}{3}$

Given $f(t) = \begin{cases} \frac{\sin\frac{1}{t}}{t} & ,t \neq 0 \\ 1 & ,t = 0 \end{cases}$, define $F(x) = \int_0^x f(t)\,dt$, find $F'(0)$.

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

Given $y=y(x)$ determined by parametric equations $\begin{cases} x = t - \sin t & ,t \in (0,1) \\ y = 1 - \cos t & ,t \in (0,1) \end{cases}$, find $y'(x), y''(x)$.

Calculate indefinite integral: $\int \arctan\sqrt{\frac{1-x}{1+x}}\,dx$, $x \in (-1,1)$

Find area of region bounded by curve $y=x^3$ and lines $|x|=1$.

Calculate anti-derivative: $\int_1^e \frac{\arctan x}{x^2}\,dx$

Given $y=f(x) > 0$ with continuous derivative passing through origin, the solid of revolution formed by rotating around x-axis has volume at any point equal to the lateral surface area. What is the radius of the sphere with maximum volume?

Given $c < d$ are two real numbers, $f$ is defined on $(c,d)$ with second derivative. For all $x_1,x_2 \in (c,d)$ with $x_1 < x_2$, there exists $f\left(\frac{x_1+x_2}{2}\right) \leq \frac{1}{x_2-x_1}\int_{x_1}^{x_2}f(t)\,dt \leq \frac{f(x_1)+f(x_2)}{2}$.

Prove:

(1) For all $n\in\mathbb{N}$, $\int_0^{\frac{\pi}{2}}(\sin x)^n\,dx = \int_0^{\frac{\pi}{2}}(\cos x)^n\,dx$

(2) For all $n\in\mathbb{N}$, let $I_n = \int_0^{\frac{\pi}{2}}(\sin x)^n\,dx$, prove for all $n\in\mathbb{N}, n\geq 2$, $I_n = \frac{n-1}{n}I_{n-2}$

(3) Prove: For all $n\in\mathbb{N}^+$, $I_n = \frac{(2n-1)!!}{(2n)!!}\cdot\frac{\pi}{2}$, for all $n\in\mathbb{N}$, $I_{2n+1} = \frac{(2n)!!}{(2n+1)!!}$

(4) Prove Wallis formula: $\lim_{n\to\infty}\frac{(n!)^22^{2n}}{(2n)!\sqrt{n}} = \sqrt{\pi}$

Given:

(1) For all $n\in\mathbb{Z}^+$, let $a_n = \frac{n!e^n}{n^{n+1}}$, prove sequence $\{a_n\}$ is strictly increasing and has limit $\alpha$ which exists;

(2) For all $n\in\mathbb{Z}^+$, let $b_n = a_ne^{-\frac{1}{4n}}$, prove sequence $\{b_n\}$ is strictly decreasing. Also $\lim_{n\to\infty}b_n = \alpha$, by which $\alpha > 0$;

(3) Use Wallis formula to prove $a = \sqrt{2\pi}$;

(4) Prove Stirling's formula in its asymptotic form: $n! \sim \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$ as $n\to\infty$.