Compute the derivative of $f(x)=x^n$.

Show that $$\frac{d}{dx} e^x = e^x$$

Given $\frac{d}{dx} e^x = e^x$, find the value of $\frac{d}{dx} \ln x$.

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

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

Evaluate

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

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.

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

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

Compute $$\int_0^4\frac{dx}{\sqrt{|x-2|}}$$

There is a unique positive real number $a$ such that the tangent line to $y = x^2 + 1$ at $x = a$ goes through the origin. Compute $a$.

Moor has $\$1000$, and he is playing a gambling game. He gets to pick a number k between $0$ and $1$ (inclusive). A fair coin is then flipped. If the coin comes up heads, Moor is given $5000k$ additional dollars. Otherwise, Moor loses $1000k$ dollars. Moor’s happiness is equal to the log of the amount of money that he has after this gambling game. Find the value of k that Moor should select to maximize his expected happiness.

The set of points $(x, y)$ in the plane satisfying $x^{2/5} + |y| = 1$ form a curve enclosing a region. Compute the area of this region.

Estimate the value of $\sqrt[4]{10018}$.

Compute $$\int\frac{1}{ax+b}d{x}$$

Compute $$\int\frac{1}{x^2-a^x}d{x}$$

Compute $$\int\frac{1}{\sqrt{a^2-x^2}}d{x}$$

Compute $$\int\frac{x}{1+x^2}dx$$

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

$\textbf{What Bear}$

Joe leaves his campsite and hikes south for $3$ miles. He then turns east and hikes for $3$ miles. Finally he turns north and hikes for $3$ miles. At this moment he sees a bear inside his tent eating his food! What color is the bear?

$\textbf{Right to Marry}$

Can a man legally marry his widow's sister in the state of California?

$\textbf{Bottle of Bacteria}$

A scientist puts a bacteria in a bottle at exactly noon. Every minute the bacteria divides into two and doubles in size. At exactly $1$ PM the bottle is full. At what time is the bottle half full?

$\textbf{Angle on a Clock}$

The time is $3:15$ now. What is the measurement of the angle between the hour and the minute hands?

A group of $100$ friends stands in a circle. Initially, one person has $2019$ mangos, and no one else has mangos. The friends split the mangos according to the following rules:

• sharing: to share, a friend passes two mangos to the left and one mango to the right.
• eating: the mangos must also be eaten and enjoyed. However, no friend wants to be selfish and eat too many mangos. Every time a person eats a mango, they must also pass another mango to the right.

A person may only share if they have at least three mangos, and they may only eat if they have at least two mangos. The friends continue sharing and eating, until so many mangos have been eaten that no one is able to share or eat anymore. Show that there are exactly eight people stuck with mangos, which can no longer be shared or eaten.

$\textbf{Make Four Liters}$

If you have an infinite supply of water, a $5$-liter bucket, and a $3$-liter bucket, how would you measure exactly $4$ liters of water? The buckets do not have any intermediate scales.