# 009A Sample Final A

**This is a sample final, and is meant to represent the material usually covered in Math 9A. Moreover, it contains enough questions to represent a three hour test. An actual test may or may not be similar. Click on the blue problem numbers to go to a solution.**

## Limits

**Problem 1.** Find the following limits:

(a)

(b)

(c)

(d)

(e)

## Derivatives

**Problem 2.** Find the derivatives of the following functions:

(a)

(b)

(c)

## Continuity and Differentiability

**Problem 3.** (Version I) Consider the following function:

(a) Find a value of which makes continuous at

(b) With your choice of , is differentiable at ? Use the definition of the derivative to motivate your answer.

**Problem 3.** (Version II) Consider the following function:

(a) Find a value of which makes continuous at

(b) With your choice of , is differentiable at ? Use the definition of the derivative to motivate your answer.

## Implicit Differentiation

**Problem 4.** Find an equation for the tangent
line to the function at the point .

## Derivatives and Graphing

**Problem 5.** Consider the function

(a) Find the intervals where the function is increasing and decreasing.

(b) Find the local maxima and minima.

(c) Find the intervals on which is concave upward and concave
downward.

(d) Find all inflection points.

(e) Use the information in the above to sketch the graph of .

## Asymptotes

**Problem 6.** Find the vertical and horizontal asymptotes of the function

## Optimization

**Problem 7.** A farmer wishes to make 4 identical rectangular pens, each with
500 sq. ft. of area. What dimensions for each pen will use the least
amount of total fencing?

## Linear Approximation

**Problem 8.** (a) Find the linear approximation to the function at the point .

(b) Use to estimate the value of .

## Related Rates

**Problem 9.** A bug is crawling along the -axis at a constant speed of .
How fast is the distance between the bug and the point changing
when the bug is at the origin? *(Note that if the distance is decreasing, then you should have a negative answer)*.

## Two Important Theorems

**Problem 10.** Consider the function

(a) Use the Intermediate Value Theorem to show that has at
least one zero.

(b) Use Rolle's Theorem to show that has exactly one zero.