MAT 120 Exam 2 Solutions: Derivatives, Graphs, and Applications

This document presents the complete solutions to the MAT 120 Exam 2, a calculus assignment from Spring 2019. The solutions cover a range of calculus topics, including finding derivatives of various functions using the power rule, chain rule, and product rule, and also includes the differentiation of trigonometric and logarithmic functions. The assignment also involves sketching graphs of piecewise functions and determining differentiability. Furthermore, the solutions encompass finding equations of tangent lines, and the application of related rates problems, such as calculating the rate of change of a balloon's radius and the sliding ladder problem using the Pythagorean Theorem. Each problem is solved with detailed steps, making it a valuable resource for students studying calculus.

Solution 1(a): Given . Differentiate with respect to x we get
Hence,
1(b): Given . Differentiate with respect to x we get
Hence,
1(c): Given . Differentiate with respect to x we get
Hence,
Solution 2(a): Given . Differentiate with respect to x we get
Hence,
2(b): Given . Differentiate with respect to x and use chain rule of
differentiation we get
Hence,
2(c): Given . Differentiate with respect to x and use product rule of
differentiation we get
Hence,
Solution 3(a): Given
The graph of the given piecewise function is shown below

3(b): The function is differentiate for all values of x except . Function is not
differentiatiable at because there is a sharp turning point at .
3(c):
3(d):
The graph of is given below
Solution 4: Given and
(a): Given that . Differentiate with respect to x and use product rule of
differentiation we get,
. Substitute we get
.
Now substitute the values of and we get
Hence,
(b): Given that . Differentiate with respect to x and use quotient rule of
differentiation we get,
Substitute we get
.
Now substitute the values of and we get

Hence,
Solution 5: Given curve is . Differentiate with respect to x and solve
for we get
At ,
The equation of tangent line at point is
Hence, the equation of tangent line is
Solution 6:
(a): From the above table
(b): From the above table,
(c): From the table,
(d): We know that
Solution 7(a): The volume of the spherical balloon whose radius r is by
. Now differentiate with respect to t we get
Given that the spherical balloon is being inflated by a pump at the rate of 2 cubic inches
per second that is the volume of balloon increasing at the rate of that is
. We need to calculate the rate of change at which radius of balloon
increases when radius is inches this means that we need to calculate .
Substitute the value of and value of r in equation (1) we get,

Hence, the radius of balloon increases at the rate of inches per second.
7(b):
Use Pythagorean Theorem in above triangle we get
… (1)
Differentiate both sides with respect to t we get
Since x = 4 feet, use this value in equation we get feet. Given that feet
per second. So from equation (2) we get
.
Hence, the top of the ladder is sliding down (because of the negative sign in the result) at
a rate of .
Solution 8(a): The derivative of with respect to x is the function and is
defined as,
8(b): Given this implies that
Now,
So,

Hence,