MATH 221  Final Exam                                                                       Name_____________

Instructions:  Answer all 12 questions on Part I.  You may not use a calculator on this part.  Each question is worth five points.

 

Find the indicated limits:

1.                                                                          1.____________

 

 

 

2.                                                                           2.____________

 

 

 

3.                                                                           3.____________

 

 

4.  =                                                                                4.____________

 

 

 

Find  for each of the following:

 

5.                                                                            5.____________

 

 

 

6.                                                                        6.____________

 

 

 

7.                                                                             7.____________

 

 

 

8. Find the slope of the tangent line to  at x = -2.                  8.____________

Evaluate each of the following integrals:

9.                                                                              9.____________

 

 

 

 

 

 

10.                                                                                  10.____________

 

 

 

 

 

 

11.  =                                                                               11.____________

 

 

 

 

 

 

12.                                                                                           12.____________

 

MATH 221  Final Exam Part II                                                Name________________

Instructions:  Answer any five of the six questions on part II.  You may use your calculator on this part.  Show all your work so that partial credit may be given.

Each question is on a separate page. Each question is worth 18 points.

 

1.      Find the derivative of  using the limit definition of the derivative.

 

2.      A cube is growing at a rate of 100 cubic inches per minute. How fast are the side of the cube and the surface area of the cube growing when the volume is 1000 cubic inches? 


3.      Find and classify the critical points of each of the following functions.

 

 

a)     

 

 

 

 

 

 

 

 

 

 

       b) 

 

 

 

 

 

 

 

 

 

       c)  


4.      The derivative, , of a function, f(x), is graphed below.  Answer the following questions based on this graph.

 

 

a)      On what intervals is f(x) increasing?

 

 

 

b)      What are the critical points of f(x)?

 

 

 

 

c)      On what intervals is f(x) concave downwards?

 

 

 

 

 

d)      Where are the points of inflection of f(x)?

 

 

 

 

 

e)      Where does f(x) have a local minimum point (or points)?

 

5.      Let  and let R be the region bounded by y = 0, x = ½, x = 2,  and the graph of f(x).  Write the integral necessary to find each of the following quantities and evaluate the integrals.

 

a)      Find the area of R.

b)      Find the volume of the solid that results when R is rotated about the x-axis.

c)      Find the volume of the solid that results when R is rotated about the line x = -5.

d)      Find the length of the graph of f(x) from x = ½ to x = 2.

 

6.      a)  Solve the following inequality and express your solution in interval notation.

      

 

 

 

 

 

 

 

 

 

 

 

 

 

b)      Give examples (by formula) of functions f(x) and g(x) such that

i)                    f(x) has a removable discontinuity at x = 2 and f(2) = 0

ii)                   g(x) has a jump discontinuity at x = 1 and g(1) = 4.