Sample Midterm Exam 2 Solutions

Question 1 (10). Find the derivative of the following functions:
a.
Solution:
by the product rule and the chain rule.

b.
Solution:

by the chain rule.

c.
Solution:

by the quotient rule.

d.
Solution:
First simplify: this gives

Now it is easy to use the quotient rule:

by the quotient rule. This can be simplfied to

e.
Solution:

by the product and chain rules.

Question 2 (15). a. State the definition of the derivative of a function at a point .

, if this limit exists.

b. Use the definition of the derivative to compute for .

Taking the limit of this as , we obtain .

Question 3 (10). Find all the vertical and horizontal asymptotes of the graph of
We first notice that there is a simplification,

Then we see that there is a vertical asymptote at . As , we have , and similarly for . So there is one horizontal asymptote, , for both and .

Question 4 (20). For each of the following, either find the limit or state that "no limit exists" and briefly explain why. Show work used to get your answer.

a.
Answer: 2. The limit can be obtained by plugging in, since this does not lead to division by zero or other problems, and the numerator and denominator are continuous.

b. .

Does not exist, since oscillates between -1 and 1 and does not approach a single value.

c. .
Answer: 0, since for large, and this approaches zero as .

d. .

Answer: 4, since when , and

Question 5 (10). For the function

a. Find the equation of the tangent line to the graph of at the point (0,-2).
and at this has value . So the line has slope 2 and goes through the point (0,-2). Using the point-sl ope formula gives the equation
or .

b. Show that at some point.
We know that and . The function is continuous on the interval , so by the Intermediate Value Theorem, there is a point in this interval where

Question 6 (10) a. State the precise definition of what is meant by .
Given an there is a such that whenever then it is true that .

Use the precise definition of the limit to prove that .
To ensure that , or , what can we allow? Simplifying gives , or . So we pick and we satisfy the condition for the limit to equal 4.

Question 7 (5) Give an example of a function which is continuous at but not differentiable at .

The function is an example.

Question 8 (5) Suppose and are functions and Where can you calculate the derivative of ? What is it equal to?
At , the chain rule tells us that

Question 9 (5) Let . Find .
By the chain rule, and

Question 10 (10). Find an anti-derivative of the following functions:
a. .
.

b.
.

c.
.