Calculus 1

$y = x^{2/3}(2-x)$

Find: A. The critical values of x? B. The x coordinate of the local maximum

Find the inflection points for the following function and determine intervals of concave up and concave down on the graph

$f(x) = \ln{(1 - \ln{(x)})}$

Find the inflection points and intervals of concavity for the following function

$y = \frac{x^2 + 1}{x^2}$

A rectangle has a perimeter of 40 feet. Find the width and height of a rectangle with a maximum area.

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)

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

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

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 $\ln{(0.9)}$

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

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.

Find the local linearization of $\ln{(x)}$ at $x = e^2$ and use it to approximate $\ln{(7.4)}$. Also find the error and percentage error.

Find the real zeros of f(x) and find the x intercepts for $f(x) = x^2 -5$

Starting with an initial value $x_1 = 1$, perform 2 iterations of Newton's Method on $f(x) = x^3 - x - 1$ to approximate the root.

Use Newton's Method to approximate a solution to $2\cos{(x)} = 3x$ (Let $x_0 = \frac{\pi}{6}$ and find $x_2$)

Evaluate the following indefinite integral

$\displaystyle\int (\sqrt{x} - 5 \sqrt[3]{x^2}) \ dx$

Evaluate the indefinite integral

$\displaystyle\int (\frac{3}{x^2} - \frac{1}{x}) \ dx$

Evaluate the indefinite integral

$\int \frac{2x^5}{x^6 + 3}^2\ dx$

Evaluate the indefinite integral

$\displaystyle\int (x^4 - \frac{1}{3\sqrt{x}} + \frac{2}{5}x^{-\frac{4}{3}}) \ dx$

Evaluate the following integral

$\displaystyle\int\cos^4(x)\sin(x) \ dx$

Find $\displaystyle\int\sin(x)\sin(\cos{x}) \ dx$