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 f(x) = |x|. 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 x=0.

   (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 x=0.

   (a)

   (b)

   (c)

   (d)

7. [3 marks] Evaluate using any method.

   (a) L= -1

   (b) L = 1.75

   (c) L = 3

   (d)

8. [3 marks] Evaluate

   

   (a)
   

   (b)
   

   (c) This limit does not exist

   (d)

9. [3 marks] Which of the following expressions gives the area of the region bounded by the curves y = -x , and between the lines x=1 and x=4?

   (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 graphs of y = 2x and 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.
    
    

    Solution: Rewrite this as Now integrate both sides with respect to x. We see that

    

    Now, let , , on the left and use the Table Method on the right. We get,

    

    or

    

    i.e.,

    

    This is the general solution. [2 marks] (b) Find the particular solution of this differential equation which satisfies y=1 when x=0.

    Solution: We simply set x=0 and y=1 into the general solution and then solve for C. This gives us, or

    

    The particular solution is then given by

    

12. Evaluate the following integrals using any method.

    [4 marks] (a)

    Solution: Write

    

    Integrate by Parts twice or use the "MYCAR" trick! You'll get

    

    where the entry in the Box is given by . See Section 8.3.4 for details. This I is an antiderivative, and so

    

    [4 marks] (b)

    Solution: The degree of the numerator exceeds that of the denominator and so we must divide these expressions. Thus

    

    Next, we use partial fractions on the third integral thus:

    

    where we let , , in the second integral (and didn't have to use partial fractions).

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. Solution: (a) , so only when x=0. This is the only critical point. (b) Use the Sign Decomposition Table of . You see that f is increasing on and decreasing on . (c) . The break-points are . So, the SDT gives that f is concave up on or and concave down on . (d) So y=0 is a horizontal asymptote. There are no vertical asymptotes since f(x) is always finite, for finite x. (e) the graph of f is bell-shaped, dropping down to zero as and its maximum value occurs at x=0.

14. [4 marks] Plutonium 239 (Pu 239) has a half-life of 24,100 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)? Solution: Use the half-life formula, . Given that T = 24, 000, micrograms, we get that N(t) = 1 (what we want) and this forces the equality

    

    Solving this for t using natural logarithms gives us

    

    years.

in Subtotal: 30 marks

Extra Pages for Rough Work: DO NOT UNSTAPLE

Extra Pages for Rough Work: DO NOT UNSTAPLE

Extra Pages for Rough Work: DO NOT UNSTAPLE

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