Which one of the numbers below is larger?

$$\int_0^{\pi} e^{\sin^2x}dx\qquad\text{and}\qquad \frac{3\pi}{2}$$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a periodic continuous function of period $T > 0$, that is $f(x+T)=f(x)$ holds for any $x\in\mathbb{R}$. Show that

$$\lim_{x\to\infty}\frac{1}{x}\int_0^xf(t)dt=\frac{1}{T}\int_0^Tf(t)dt$$

Prove the absolute convergence testing rule using the comparison testing rule. That is, if a series $\{|a_n|\}$ converges, then the series $\{a_n\}$ must be convergent.

For what pairs $(a, b)$ of positive real numbers does the the following improper integral converge?

$$\int_b^{\infty}\left(\sqrt{\sqrt{x+a}-\sqrt{x}}-\sqrt{\sqrt{x}-\sqrt{x-b}}\right)dx$$

Show that the function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ given by

$$f(x,y)=x^4+6x^2y^2 + y^4 -\frac{9}{4}x-\frac{7}{4}$$

achieves its minimal value, and determine all the points $(x, y)\in\mathbb{R}^2$ at which it is achieved.

For $n=1, 2,\dots$, let $x_n=\displaystyle\sum_{k=n+1}^{9n}\frac{k}{9n^2 + k^2}$. Find the value of $\displaystyle\lim_{n\to\infty}x_n$.

Let $s\in\mathbb{R}$. Prove that

$$\sum_{n\ge 1}(n^{\frac{1}{n^s}}-1)$$

converges if and only if $s > 1$.

A right circular cone $\mathbb{C}$ has altitude $40$ and a circular base of radius $30$ inches. A sphere $\mathbb{S}$ is inscribed in $\mathbb{C}$. Compute the volume of the region inside $\mathbb{C}$ which is above $\mathbb{S}$.

Let $$S_n=\sum_{k=1}^{2n}\frac{1}{n+k}=\frac{1}{n+1}+\frac{1}{n+2}+\cdots + \frac{1}{3n}$$

Does $\displaystyle\lim_{n\to\infty}S_n$ exist? If so, find its value. If not, prove the claim.

Show that $1-\cos{x} < x^2$ holds for all $x > 0$.

Determine whether the following series converge? $$\sum_{n=1}^{\infty}\left(1-\cos{\frac{\pi}{n}}\right)$$

Note that the point $(2, 1)$ is always on the curve $x^4 + ky^4 = 16+k$ regardless of the value of $k$. If for a particular non-zero value of $k$, $y'(2)=y''(2)$ along this curve. Find this $k$.

According to Newton’s law of cooling, the rate at which a cup of coffee cools is proportional to the difference between its temperature and that of the room it is in. A certain cup of coffee cools from $164^{\circ}$ to $140^{\circ}$ (all temperatures Fahrenheit) in five minutes, and then from $140^{\circ}$ to $122^{\circ}$ in the next five minutes. What is the temperature of the room?

It is well-known that the solution to the Fibonacci sequence is

$$F_n=\frac{1}{\sqrt{5}}\left(\left(\frac{1+\sqrt{5}}{2}\right)^n-\left(\frac{1-\sqrt{5}}{2}\right)^n\right)$$

Show that

$$\lim_{n\to\infty}\frac{F_{n+1}}{F_n}=\frac{1+\sqrt{5}}{2}$$

Compute $$\lim_{n\to\infty}\left(\sqrt{n+1}-\sqrt{n}\right)$$

Show that $$\lim_{n\to\infty}\int_0^1 x^n(1-x)^n dx = 0$$

Without explicitly evaluating the integral, show that

$$\lim_{n\to\infty}\int_1^2\ln^n{x}dx =0\quad\text{and}\quad\lim_{n\to\infty}\int_2^3\ln^n{x}dx = \infty$$

Compute $$\int_0^{\frac{\pi}{4}}(\cos{x} - 2\sin{x}\sin(2x))dx$$

Let $f_0(x)=(\sqrt{e})^x$ , and recursively define $f_{n+1}(x) = f'_n(x)$ for integers $n\ge 0$. Compute $$\sum_{k=0}^{\infty}f_k(1)$$

Consider the parabola $y=ax^2 + 2019x + 2019$. There exists exactly one circle which is centered on the $x$-axis and is tangent to the parabola at exactly two points. It turns out that one of these tangent points is $(0, 2019)$. Find $a$.

What is the smallest natural number $n$ for which the following limit exists?

$$\lim_{x\to 0}\frac{\sin^nx}{\cos^2x(1-\cos{x})^3}$$

Turn the graph of $y=\frac{1}{x}$ by $45^{\circ}$ counter-clockwise and consider the bowl-like top part of the curve (the part above $y=0$). We let a $2D$ fluid accumulate in this $2D$ bowl until the maximum depth of the fluid is $\frac{2\sqrt{2}}{3}$. What’s the area of the fluid used?

Compute

$$\lim_{x\to 0}\frac{\frac{x^2}{2}+1-\sqrt{1+x^2}}{(\cos{x}-e^{x^2})\sin(x^2)}$$

Calculate $$\lim_{n\to\infty}\frac{1}{n^2}\sum_{k=1}^{n}\left(k\sin\frac{k\pi}{n}\right)$$