Calculus 1

Find two non-negative numbers, x and y, whose sum is 9 with a maximum product of x and the square of y, $p(x,y) = xy^2$

Find the maximum value of $f(x) = xe^{-x}$ on $(-\infty, \infty)$

A company makes wrenches. It costs them an initial \$10,000 to set up and an additional \$5.00 for each wrench. A market study indicates that if the price of a wrench is \$5.00 then the company will sell 200,000 wrenches. However, for every \$10 that the company raises the price per wrench, only half as many people will buy the wrench. What price should they charge per wrench to maximize the companies profits.

You have been asked to design a 1 liter can in the shape of a right circular cylinder. What dimensions use the least amount of material for the can? (minimize surface area)

A man has 1000 feet of fencing material and he wants to enclose three adjacent pens for his three dogs as shown below. What dimensions should be used to maximize the total enclosed area? What is the final maximum total area?

Use l'Hospital's Rule to find the following limit

$\lim_{x\rightarrow 1} \frac{x^a - 1}{x^b - 1}$

Compute the following limit

$\lim_{x\rightarrow 0} \frac{\sin{(5x)}}{x}$

Use l'Hospital's Rule to find the limit

$\lim_{x\rightarrow 0} \frac{x^2 - 6x + 2}{x + 1}$

Evaluate the following limit

$\lim_{x\rightarrow \infty} \frac{\ln{(1 + e^{3x})}}{2x + 5}$

Explain why the following limit can not be found using l'Hospital's Rule then find the limit using a different method.

$\lim_{x\rightarrow \infty} \frac{x + \cos{(x)}}{x}$

Evalute $\lim_{x\rightarrow 0} \frac{x + \cos{(2x)} - e^x}{x}$

Evaluate the following limit

$\lim_{\theta\rightarrow 0} \frac{\tan{(\theta)} - \theta}{\theta - \sin{(\theta)}}$

Evaluate the following limit

$\lim_{x\rightarrow 0} \frac{\sin{(3x)}}{\sin{(4x)}}$

Let $f(x) = \ln{(x)}$ Find the linearization of $f$ at $1$ and use it to evaluate $\ln{(0.9)}$

Find the linearization of $f(x) = {(1 + x)}^{P}$ at $0$, and approximate $f(\sqrt{0.99})$

Suppose that a spherical container has a radius of $1 \pm 0.001 m$. Approximate the corresponding possible error in the calculated volume.

Use Newton's method for approximating roots of functions to approximate $\sqrt{0.99}$

Approximate $\sqrt[4]{75}$ using the Newton Raphson method

Use linear approximations to estimate $\sqrt{8}$. Also find the error and percentage error.

Find the linearization of $\csc{x}$ at $x = \frac{\pi}{4}$ and use it to approximate $\csc{1}$. Also find the error and percentage error.