Calculus Practice Set 8

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

Calculate: $\lim_{n\to\infty}\left(\frac{n^2+1}{n^2-1} - a\cos\frac{n\pi}{n+1}\right)$

Calculate: $\lim_{x\to 0}\frac{\int_0^x t\sin t\,dt}{x^3}$

Given $f(x) = \begin{cases} \frac{(1+ax)^{\frac{1}{x}}-e}{x} & ,x\neq 0 \\ b & ,x=0 \end{cases}$, determine values of $a,b$ such that $f(x)$ is continuous and differentiable at $x=0$.

Calculate definite integral: $\int_0^1 \frac{\ln(1+x)}{1+x^2}\,dx$

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

Calculate area bounded by curves $y=e^x, y=e^{-x}$ and line $x=1$.

Given parametric curve $\begin{cases} x = a(t-\sin t) \\ y = a(1-\cos t) \end{cases}$, $0\leq t\leq 2\pi, a>0$, find arc length.

Find power series $\sum_{n=1}^{\infty}\frac{x^n}{n}$ sum function, radius and interval of convergence.

Let sequence $\{x_n\}$ satisfy $x_1 = \frac{3}{2}$, $x_{n+1} = \frac{1}{2}\left(x_n+\frac{2}{x_n}\right)$. Prove $\{x_n\}$ converges and find its limit.

Let $f(x)$ be continuous on $[a,b]$ with $f(a) < 0, f(b) > 0$. Prove there exists $\xi\in(a,b)$ such that $f(\xi)=0$.

Let $f(x)$ have continuous second derivative on $[0,1]$ with $f(0)=f(1)=0$ and $\int_0^1 f(x)\,dx=1$. Prove there exists $\xi\in(0,1)$ such that $|f''(\xi)|\geq 8$.

Let $f(x)$ be continuous on $[a,b]$ and differentiable on $(a,b)$. If $f'(x)\neq 0$ for all $x\in(a,b)$, prove $f(x)$ is strictly monotonic on $[a,b]$.

Given sequence defined by $a_1=1$, $a_{n+1}=a_n+\frac{1}{a_n}$. Prove $\lim_{n\to\infty}\frac{a_n}{\sqrt{2n}}=1$.