# 009A Sample Final 1

**This is a sample, and is meant to represent the material usually covered in Math 9A for the final. An actual test may or may not be similar. Click on the** ** boxed problem numbers to go to a solution.**

## Problem 1

In each part, compute the limit. If the limit is infinite, be sure to specify positive or negative infinity.

- a)

- b)

- c)

## Problem 2

Consider the following piecewise defined function:

- a) Show that is continuous at

- b) Using the limit definition of the derivative, and computing the limits from both sides, show that is differentiable at

## Problem 3

Find the derivatives of the following functions.

- a)

- b)

## Problem 4

If

compute and find the equation for the tangent line at . You may leave your answers in point-slope form.

## Problem 5

A kite 30 (meters) above the ground moves horizontally at a speed of 6 (m/s). At what rate is the length of the string increasing when 50 (meters) of the string has been let out?

## Problem 6

Consider the following function:

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

- b) Use the Mean Value Theorem to show that has at most one zero.

## Problem 7

A curve is defined implicitly by the equation

- a) Using implicit differentiation, compute .

- b) Find an equation of the tangent line to the curve at the point .

## Problem 8

Let

- a) Find the differential of at .

- b) Use differentials to find an approximate value for .

## Problem 9

- Given the function ,

- a) Find the intervals in which the function increases or decreases.

- b) Find the local maximum and local minimum values.

- c) Find the intervals in which the function concaves upward or concaves downward.

- d) Find the inflection point(s).

- e) Use the above information (a) to (d) to sketch the graph of .

## Problem 10

Consider the following continuous function:

defined on the closed, bounded interval .

- a) Find all the critical points for .

- b) Determine the absolute maximum and absolute minimum values for on the interval .

**Contributions to this page were made by Kayla Murray**