Calculus Practice Set 4

Let $y = \frac{x^3}{3}\arccos x - \frac{1}{9}(x^3+2)\sqrt{1-x^2}+\ln 5$, find $dy$.

Given $y=y(x)$ determined by $\begin{cases} x(t) = \sqrt{1+t^2} \\ y(t) = \int_1^t \frac{3^u}{\sqrt{1+u}}\,du \end{cases}$, find $\frac{dy}{dx}, \frac{d^2y}{dx^2}$.

Given $y=y(x)$ determined by $y^3+xy+x^2y^2+x^2y-3=0$, find tangent line of curve $y=y(x)$ at point (1,1).

Calculate: $\int_0^1 x^3\sqrt{2x-x^2}\,dx$

Calculate the improper integral: $\int_0^{+\infty}\frac{\arctan x}{(\sqrt{1-x^2})^3}\,dx$

Given $f(x) = \int_1^x \frac{\ln(1+t)}{t}\,dt$, calculate $\int_1^2 \frac{f(x)}{\sqrt{x}}\,dx$.

Find: $\lim_{x\to 0}\frac{2\sin x + x^2\cos\frac{1}{x}}{(x^3+1)\ln(1+x^2)}$

Find: $\lim_{x\to 0}\frac{e^{\sin x}-e^x}{\sqrt{1+x^3}-1}$

Given sequence $a_n = \frac{\sin\frac{\pi}{n+1}}{n+1} + \frac{\sin\frac{2\pi}{n+1}}{n+\frac{1}{2}} + \cdots + \frac{\sin\pi}{n+\frac{1}{n}}$, $n=1,2,\ldots$, find $\lim_{n\to\infty}a_n$.

Given $f(x) = \int_x^e \sqrt{1+t^2}\,dt + \int_1^x \sqrt{1+t^2}\,dt$, find all zeros of $f(x)$.

Find curve $f(x) = x\arctan x - \ln\sqrt{1+x^2}$ with its derivative's sign change intervals.

Given sequences $\{a_n\}, \{b_n\}$ satisfy $0<a_n<\frac{\pi}{2}$, $0<b_n<\frac{\pi}{2}$, $\cos a_n - a_n = \cos b_n$, and $\sum_{n=1}^{\infty}b_n$ converges.

(1) Prove $\lim_{n\to\infty}a_n=0$;

(2) Prove series $\sum_{n=1}^{\infty}\frac{a_n}{b_n}$ converges.

Given curve $y=x^2$ in region $A(a,a^2)$ where $a>0$, let $D$ be the region bounded by curve and tangent at another point, rotated around x-axis.

(1) Find the volume of solid;

(2) When curve and tangent cut plane at equal areas, find the volume of solid.

Given continuous function $f(x)$ on [-1,1] with 2 derivatives, $f(1)=-1$, prove:

(1) There exists $\xi\in(0,1)$ such that $f'(\xi)=-1$;

(2) There exists $\eta\in(-1,1)$ such that $\eta f''(\eta)-f'(\eta)=-1$.