Calculus 1

A cylindrical water tank with a radius of 4 meters is being filled with water at a constant rate of 2 cubic meters per minute. Find the rate at which the water level is rising when the water is 3 meters deep.

Use the function $f(x) = -x^2 + 3x + 10$ to answer the following: a. On the interval [2,6], what is the average rate of change?

b. On the interval (2,6), when does the instantaneous rate of change equal the average rate of change?

Find the derivative of the following function

$y = \sqrt{(x^2 + 1){(x - 1)}^2}$

For the following function draw a rough sketch of the graph for the 3 cases

$f(x) = \frac{(x - a)}{(x - b)(x - c)}$

Case 1: a < b, c

Case 2: b < a < c

Case 3: b, c < a

Find the critical points of the following function

$f(x) = 6x^5 + 33x^4 - 30x^3 + 100$

Use the 2nd Derivative Test to find the inflection points and intervals of concavity for the following function

$f(x) = x^{3}(x - 4)$

Use the second derivative test to find all relative extrema of the following function

$f(x) = \frac{1}{3}x^3 + 2x^2 + 3x$

Find all relative maximum and minimum values for the following function

$f(x) = x^4 - 32x^2 + 256$

An open box is made by cutting and folding a 10x10 inch square of cardboard as shown below. Find the maximum volume the open box can attain by altering the length of cuts, x.

A dudebro has 500 feet of fencing to make a pig pen along a barn as shown below. Find the maximum area.

Use l'Hospital's Rule to find the limit

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

Evaluate the following limit

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

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

Evaluate the following limit

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

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

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

Given a circle with a circumference of 56 inches with an error of $\pm{1.2}$ inches. Find the percent error of the area of the circle.

A 12 by 12 square is produced with an error of $\pm{\frac{1}{64}}$ inches in the length of each side. Find the the percent error in the area of one of these squares.

Use the Newton Raphson method to approximate the real zero close to $x = 1$ until two successive approximations differ by less than 0.005 for the following function

$f(x) = 2x^2 - 3$

Use Newton's Method to approximate the solution to the following equation

$\cos{(x)} = \frac{x}{5}$