Limits

$$\text{lim}_{x \rightarrow \infty} \frac{\sqrt{4x^3}+5x^2}{\sqrt{2x^4+7x+1}}$$

Derivatives

Let $f(x) = 12 \sqrt[3]{x} + \frac{2x}{x^6} - x^5 e^{3x}$. Find the equation for the tanglent line to $f(x)$ at $x = 1$.

Compute the derivative of $y = (\sin x)^{\sqrt{x}}$ for $x>0$.

You can't do too many derivatives.

$$\begin{array}{lcl} f(x) = \frac{x^2+bx+c}{a} & & y(t) = \frac{3}{\sqrt{t}+2} \\ g(t) = \sqrt[3]{\tan(5t)} & & y = \ln \sqrt{5+x^2} \\ f(t) = e^{1/t} & & \end{array}$$

Optimization Problem

Find two numbers that sum to 100 and have the largest possible product.

Parametric Equations

Find all the points on the given curve $$x = t^2 + t + 3, \hspace{5mm} y = t^3 -2 $$ where the tangent line has slope 1.

Integrals

$$\int \frac{1}{x \ln x} dx$$

There are several ways to solve at least one of these. $$\begin{array}{lcl} \int \frac{\tan^3 x}{\cos^3 x} dx & & \int \frac{x^5+1}{x^3-3x^2-10x} dx \\ \int \sqrt{\frac{1-x}{1+x}}dx & & \int \frac{xe^x}{\sqrt{1+e^x}} dx \end{array}$$

Volume: Method of Discs

Use the method of discs to find the volume between $x^2+4y^2 = 4$, rotated about $x = 2$.