Question 1 (10).
Find the derivative of the following functions:
by the product rule and the chain rule.
by the chain rule.
by the quotient rule.
First simplify: this gives
Now it is easy to use the quotient rule:
by the quotient rule. This can be simplfied to
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.
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.
Does not exist, since oscillates between -1 and 1 and does not
approach a single value.
Answer: 0, since for large, and this approaches zero as .
Answer: 4, since
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
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)
. Find .
By the chain rule, and
Question 10 (10).
Find an anti-derivative of the following functions: