Calculus Practice Set 3

Let $f(x) = (x-1)(x^2-2)(x^3-3)(e^{100}-100)$, find $f'(1)$.

Given parametric equations $\begin{cases} x = t^3+3t-1 \\ y = e^t \end{cases}$, find the concave/convex intervals of curve $y=y(x)$ (expressed in parameter t), and identify inflection points (in (x,y) form).

Given curve $y=y(x)$ determined by $x^4 = \int_0^y e^{-t^2}\,dt$, find the radius of curvature at $x=0$.

Find: $\lim_{x\to 0}\left(\frac{1}{x^2} - \frac{\cos^2 x}{\sin^4 x}\right)$

Let $f(x) = \lim_{n\to\infty}\frac{x^{2n}+1+(a-1)x^n+1}{x^{2n}-ax^n+1}$ be continuous on $(0,+\infty)$, find the value of constant $a$.

Find all asymptotes of curve $y = \frac{x^2}{x^2+1-e^{-x}}$.

Calculate: $\int_1^e (x-1)^3\sqrt{4-x^2}\,dx$

Calculate the improper integral: $\int_1^{+\infty}\frac{\arctan x}{x^4}\,dx$

Let constant $a>0$, $a_n = \int_0^a\sqrt{a+x^n}\,dx$. Discuss whether series $\sum_{n=1}^{\infty}(-1)^n a_n$ converges, converges absolutely, or diverges. Prove your answer.

Let $f(x) = (1+\sin 2x)^x$ for $x\neq 0$, and $f(x)$ is continuous at $x=0$. Find $f(0)$ and the tangent line equation of curve $y=f(x)$ at $x=0$.

Cycloid L has parametric equations $\begin{cases} x = a(t-\sin t) \\ y = a(1-\cos t) \end{cases}$ where $0<t<2\pi, a>0$. Find the volume when region D enclosed by curve L and x-axis is rotated around line $y=-2a$.

Find the radius of convergence, interval of convergence, and sum function of power series $\sum_{n=0}^{\infty}\frac{\ln^2 n + 4n - 3}{2n+1}x^n$.

(1) For $0<x<+\infty$, prove there exists $\eta\in(0,1)$ such that $\sqrt{x+1}-\sqrt{x} = \frac{1}{2\sqrt{x+\eta}}$.

(2) Based on the result above, find $\lim_{n\to\infty}$ of the expression involving n, and when $0<x<+\infty$, determine the range of function $\eta=\eta(x)$.

Prove:

(1) is positive

(2) For all $\alpha\in\left(0,\frac{\pi}{2}\right]$, $\int_0^{2\pi}\frac{\sin x}{x}\,dx > \sin\alpha\ln\frac{\pi^2-\alpha^2}{2\pi-\alpha}$