Math 160

Fall 1999

Exam 1

 

  1. (30 pt) Find the following limits:

    2. a) b) c) d)

     

    e) f)

     

  2. (30 pt) Find the derivatives of the following functions:

    3. a) b) c) d) e)

     

  3. (8 pt) Use the definition of the derivative to find the derivative of the function . You may assume that the function is differentiable.
  4. (20 pt) I fire a cannonball up in the air so that its height at any relevant time is given by the function
.
  1. Find a function that expresses the velocity of the cannonball.
  2. How fast did I shoot the cannonball?
  3. When is the velocity of the cannonball 0?
  4. How high does the cannonball go?
  5. At what time does the cannonball return to the ground?

 

  1. (8pt) Find the horizontal and vertical asymptotes of the function if they exist.
  2. (6 pt) Show that there is a number for which .
  3. (10 pt )Graph the function (greatest integer function…is the greatest integer less than or equal to ). Where is continuous? Where is differentiable? Sketch a graph of .
  4. (8pt) Consider the functions pictured below:

2 g(x) f(x)

1

 

1 2 1 2

 

 

 

Using the pictures show why f(g(x)) is continuous at 1, but not at 0.