Calculus Practice Set 10

Calculate improper integral: $\int_0^{+\infty}e^{-\sqrt{x}}\,dx$

Find limit: $\lim_{x\to+\infty}\left[(x^7+x^6)^{\frac{1}{7}}-(x^7+x)^{\frac{1}{7}}\right]$

Calculate indefinite integral: $\int\frac{1+\sin x}{(1+\cos x)\sin x}\,dx$, $x\in\left(0,\frac{\pi}{2}\right)$

(1) Given $x > 0$, prove: $x - \frac{x^2}{2} < \ln(1+x) < x$

(2) Find: $\lim_{n\to\infty}\left(1+\frac{2}{n^2}\right)\left(1+\frac{2}{n^2}\right)\cdots\left(1+\frac{n}{n^2}\right)$

Find curve $y = \frac{e^x+e^{-x}}{2}$ in region $[0,1]$ arc length.

Given function $f(x) = \ln\cos x$, let $f(x)$ when $x\to 0$ have expansion $f(x) = a_0+a_1x+a_2x^2+a_3x^3+a_4x^4+o(x^4)$, find coefficients $a_0, a_1, a_2, a_3, a_4$.

Find $f(x) = -3x^2+5x^3$ extrema value.

Given parametric equations $\begin{cases} x = e^t\cos t \\ y = e^t\sin t \end{cases}$, find $\frac{dy}{dx}, \frac{d^2y}{dx^2}$, where $t\in\left(0,\frac{\pi}{4}\right)$.

Known sequence $f(x)$ on $[0,+\infty)$ continuous with strictly increasing property, and known $\forall x\in[0,1]$, there exists $g(x)\geq 0$. Prove: There exists $\exists x_0\in[0,1]$ such that $\int_0^1f(x)g(x)\,dx = f(x_0)\int_0^1g(x)\,dx$.

Known sequence $f(x)$ on $[0,+\infty)$ continuous, on $(0,+\infty)$ differentiable with $f(0) < 0$, and for all $\forall x > 0$, $f'(x) > 1$. Prove: There exists $f(x)$ on $[0,+\infty)$ with only one zero.

Let function $f(x)$ on $(0,1)$ with proper definition, and satisfies for all $x_1,x_2\in(0,1)$, $x_1 < x_2$ have $\frac{f(x_2)-f(x_1)}{x_2-x_1} \geq \frac{f(x_3)-f(x_1)}{x_3-x_1} \geq \frac{f(x_3)-f(x_2)}{x_3-x_2}$.

Prove: For all $\forall n\in\mathbb{N}$, let $a_n = (n+2)\left[f\left(\frac{1}{2}\right)-f\left(\frac{1}{2}-\frac{1}{n+2}\right)\right]$ converge. Also, $\lim_{n\to\infty}a_n = \alpha$, where $\alpha$ can be obtained from $\alpha > 0$.

Let $f(x)$ on $[0,1]$ continuous, on $(0,1)$ has derivative with $\forall x\in[0,1]$, there exists $\int_0^1f(t)\,dt \geq \frac{1-x^3}{2}$ hold. Prove: $\int_0^1[f(t)]^2\,dt \geq \frac{5}{12}$