Limpet - Testing document 2

Example 1

The following five elementary functions are given:

$f_1(x) = x +1$ ,

$f_2(x) = \frac{1}{x}$ ,

$f_3(x) = 5x$ ,

$f_4(x) = \sin{x}$ ,

$f_5(x) = \sqrt[3]{x}$ .

Choose one function from the options which meets the following condition: $f(x) = f_5 \circ f_4 \circ f_3 \circ f_2 \circ f_1(x)$ .

A. $f(x) = \sqrt[3]{\sin{\frac{5}{x+1}}}$

B. $f(x) = \sqrt[3]{5\sin{\frac{1}{x+1}}}$

C. $f(x) = \sqrt{\sin{\frac{5}{x}}}+1$

Example 2

Figure out the following limits:

$\lim_{x \rightarrow +\infty}\left(\arccos{\frac{1}{x +1}}\right)^3$

$\lim_{x \rightarrow -\infty}\frac{1 +x\arctan{x}}{\sqrt{1 +\cos^2{x}}}$

$\lim_{x \rightarrow 0}\frac{\tan{x}}{3x} +\frac{\arcsin{x}}{x}$

Example 3

Determine the summation $\sum_{n =0}^{\infty}3^{-n}$ and the product $\prod_{n =1}^{\infty}3^n$ . Later verify whether the equation $\lceil(\lfloor x \rfloor +\lfloor y \rfloor)\rceil =\lfloor x \rfloor +\lfloor y \rfloor$ is true for any $x, y \in \mathbb{R}$ .

Example 4

Compute the area of the plane surface $y =|\log_2{x}|$ above the axis $x$ for $x \in \langle 0, 10\rangle$ . Furthermore determine the length of the curve $y =\ln{\cot{x}}$ for $x \in (0, \pi)$ .

Example 5

There are three points on the plane: $\tilde{X} =[1, 2]; \hat{X} = [3, 5]; X\prime = [7, 4]$ . Figure out the measure of the angle $\angle \tilde{X}\hat{X}X\prime$ . Later compute the measure of the angle $\vec{u} = \vec{\tilde{X}\hat{X}}$ from the point $\tilde{X}$ to the point $\hat{X}$ .

Example 6

Determine the definite integral $\int_0^{\ln{2}}xe^{-x}dx$ . Later compute limits $\liminf_{n \rightarrow \infty}a_n$ and $\limsup_{n \rightarrow \infty}a_n$ for $a_n = \frac{2n(-1)^n}{n +1} +\sqrt[n]{2}$ .