PART 1: Multiple-Choice Questions
Please circle only one answer.

1. [3 marks] Let . Evaluate . In other words, find the second derivative of f at x=0.

   (a)

   (b)

   (c)

   (d)

2. [3 marks] Let . Calculate

   

   (a) L=0

   (b) L=-1

   (c) L=1

   (d) This limit does not exist

3. [3 marks] Let . Evaluate . In other words, find the derivative of f at .

   (a)

   (b)

   (c)

   (d)

4. [3 marks] Let . Evaluate . In other words, find the derivative of f at x.

   (a)

   (b)

   (c)

   (d)

5. [3 marks] Let . Evaluate . In other words, find the derivative of f at x.

   (a)

   (b)

   (c)

   (d)

6. [3 marks] Let . Calculate . In other words, find the derivative of f at .

   (a)

   (b)

   (c)

   (d)

7. [3 marks] Evaluate using any method.

   (a)

   (b) L = 0

   (c) This limit does not exist.

   (d)

8. [3 marks] Evaluate

   

   (a) This limit does not exist

   (b)
   

   (c)
   

   (d)
   

9. [3 marks] Which of the following expressions gives the area of the region bounded by the curves y = 1 and between the lines x=0, and x=1?

   (a)
   

   (b)
   

   (c)
   

   (d)
   

10. [3 marks] Which of the following expressions gives the volume of the solid of revolution obtained when the region bounded by the curves y = 1 and between the lines x=0, and x=1 is revolved about the y-axis?

    (a)
    

    (b)
    

    (c)
    

    (d)
    

    Subtotal : 30 marks

    PART 2
    Please show all work here.

11. [6 marks] (a) Find the general solution of the differential equation

    

    using the method of separation of variables and any method of integration.

    [2 marks] (b) Find the particular solution of this differential equation which satisfies y=0 when x=0.

12. Evaluate the following integrals using any method.

    [4 marks] (a)

    [4 marks] (b)

13. [10 marks] Sketch the graph of the function f defined by by providing the following information:

    [2 marks] (a) Find the critical points of f,

    [2 marks] (b) Find the intervals where the graph of f is increasing and decreasing,

    [2 marks] (c) Find the intervals where the graph of f is concave up and concave down,

    [2 marks] (d) Find all asymptotes,

    [2 marks] (e) Sketch the graph of f.

14. [4 marks] Plutonium 240 (Pu 240) has a half-life of 6500 years. This radionuclide is an extremely toxic carcinogen and occurs as a by-product of nuclear activity. How long would it take for a 1 gram sample of Pu 239 to decay to 1 microgram ( grams)?

Subtotal: 30 marks

ANSWERS:

1. d
2. a
3. c
4. d
5. b
6. a
7. b
8. d
9. c
10. b
11. This is a separation of variables problem. Use the Table Method on the right and a substitution on the left. The general solution is given by

    

    The particular solution such that y=0 when x=0 is given by solving for C above. This gives .

12. This is a three-row problem. The answer is

    

13. Use long division first. This gives

    

    Using Partial Fractions on the right (and the Cover-Up method) we find,

    

    The final answer is

    

14. 129,564 years. See Example 460.

• About this document ...