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.

.