Find the derivative of function $y=\sin{x}$.

Find the derivative of $\arcsin{x}$.

Let $f(x)$ be an odd function which is differentiable over $(-\infty, +\infty)$. Show that $f'(x)$ is even.

Compute the limit of the power series below as a rational function in $x$:

$$1\cdot 2 + (2\cdot 3)x + (3\cdot 4)x^2 + (4\cdot 5)x^3 + (5\cdot 6)x^4+\cdots,\qquad (|x| < 1)$$

Compute $$1-\frac{1\times 2}{2}+\frac{2\times 3}{2^2}-\frac{3\times 4}{2^3}+\frac{4\times 5}{2^4}-\cdots$$

Construct one polynomial $f(x)$ with real coefficients and with all of the following properties:

• it is an even function
• $f(2)=f(-2)=0$
• $f(x) > 0$ when $-2 < x < 2$, and
• the maximum of $f(x)$ is achieved at $x=\pm 1$.

Find the coordinates of the center of mass of the $\frac{1}{4}$ disc defined by

$$\{(x, y) | x\ge 0, y\ge 0, x^2 + y^2 \le 1\}$$

assuming the density is uniform.

Consider the ellipse $x^2+\frac{y^2}{4}=1$. What is the area of the smallest diamond shape with two vertices on the $x$-axis and two vertices on the $y$-axis that contains this ellipse?

Compute $$I=\int \frac{x\cos{x}-\sin{x}}{x^2 + \sin^2{x}} dx$$

Find the maximum and minimal values of the function

$$f(x)=(x^2-4)^8 -128\sqrt{4-x^2}$$

over its domain.

Find all quadratic polynomials $p(x)=ax^2 + bx + c$ such that graphs of $p(x)$ and $p'(x)$ are tangent to each other at point $(2, 1)$.

Determine if the following infinite series is convergent or divergent:

$$\sum_{n=2}^{\infty}\frac{1}{(\ln n)^{\ln \ln n}}$$

Show that $\ln x < \sqrt{x}$ holds for all positive $x$.

Evaluate $$\int_{0}^{\pi}\frac{x\sin{x}}{1+\cos^2 x}dx$$

Let $f(x)=\int_1^x\frac{\ln{x}}{1+x}dx$ for $x > 0$. Find $f(x)+f(\frac{1}{x})$.

Compute $$\lim_{x\to 0}\frac{\int_0^x\sin(xt)^2dt}{x^5}$$

Compute

$$\int_0^{\infty}\frac{x^2}{1+x^4}dx$$

Evaluate $\displaystyle\lim_{n\to\infty}S_n$ where

$$S_n = 1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}-\cdots + (-1)^{n-1}\frac{1}{n}$$

Determine the values of $\alpha$ and $\beta$ such that 

$$\lim_{n\to\infty}\frac{n^{\alpha}}{n^{\beta}-(n-1)^{\beta}}=2020$$

Evaluate

$$\int_0^1 x\arcsin{x}d{x}$$

Evaluate

$$\int_0^1 \sqrt{1-x^2} d{x}$$

Compute

$$I= \iiint \limits_S \frac{dx dy dz}{(1+x+y+z)^2}$$

where $S=\{x\ge 0, y\ge 0, z\ge 0, x+y+z\le 1\}$.

Compute $$\int \ln{x} dx$$