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MATH 105 PRACTICE EXAM 1

Please work out each of the given problems.  Credit will be based on the steps that you show towards the final answer.  Show your work.

  Key

PROBLEM 1  Please answer the following true or false.  If false, explain why or provide a counter example.  If true, explain why.

 

A)     (7 Points)  If f(x)  and g(x)  are differentiable functions with

           f '(5)  =  10    and    g'(5)  =  4

then if

                           f(x)
     h(x)   =           -  3g(x)
                      2 

        then

            h'(5) = -7

  Solution

B)     (7 Points)  Let f(x)  and g(x)  be continuous functions
     f(1)  >  g(1)       and      f(2) < g(2)
then if
     h(x) = f(x) - g(x)
h(x)  has a root for some value of x between 1 and 2.

  Solution

C)     (7 Points)  Let f(x)  and g(x)  be continuous functions such that

    and    

            Then h(x) has a vertical asymptote at x = 2.

  Solution

PROBLEM 2 Find the following limits if they exist:

A) (8 Points)

  Solution

B) (8 Points) 

  Solution

C)    (8 Points) 

  Solution

 

PROBLEM 3

 

 

A)   (8 Points)  Find the following limits if they exist

                   i)     ii)      iii)      iv)      v)

  Solution

B)    (8 Points)  At which values is f(x) not continuous?

  Solution

C)   (8 Points)  At which values is f(x) not differentiable?

  Solution

 

PROBLEM 4   (20 Points)  Below is the function y = f(x).  Sketch a graph of the derivative y = f (x).

 

  Solution

 

PROBLEM 5   Find f ' (x) for the following

A)    (10 Points) 

  Solution

B)     (11 Points) 

  Solution

 

PROBLEM 6 Let

A)    (10 Points)  Use the limit definition of the derivative to find f (x).

  Solution

B)     (10 Points)  Prove using the e-d  definition of the limit that

  Solution

 

PROBLEM 7 (20 Points)

The position of a robin flying through the wind is given by

           s(t)  =  -5t + tcost

Find its acceleration when t is 2 seconds.

  Solution

 

Extra Credit:  Write down one thing that your instructor can do to make the class better and one thing that you want to remain the same in the class.

(Any constructive remark will be worth full credit.)

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