Math 165

Spring 2002

Exam 1

1. (30 pt) Evaluate the following limits:

a) b) c)

d) e)

2. (10 pt) Use the definition of the derivative to find the derivative of

3. (24 pt) Find the derivatives of the following functions:

a) b) c)

d)

4. (16 pt) Consider the function

a) Find all asymptotes of the function.

b) Find the derivative of the function.

c) Find where the derivative is positive and where the derivative is negative.

d) Use this information to sketch the graph of the function.

5. (10 pt) Let be a line.

a) Find the inverse function,

b) Use the result of part a) to explain why if is a one to one, differentiable function, and the slope of the tangent line to at the point is then the slope of the tangent line to at is

6. (15 pt) You are standing on the ground and you throw a ball up in the air with initial velocity its height above the ground () is given by (Here is in seconds and is in feet).

a) At what time does the ball reach its highest point?

b) What is its highest point?

c) How fast must you throw the ball (that is, what is ) so that the ball will go 100 feet in the air?

7. (5 pt) Let be a function that is differentiable everywhere. Suppose, in addition, that is an even function (that is, ). Show that