Math 102 Section 106
Midterm exam 1
Friday, 3 October 2008
Name (▯rst and last)SOLUTIONS
1. Ensure that your full name and student number appear on this page.
2. No calculators, books, notes, or electronic devices of any kind are permitted.
3. Unless otherwise indicated, show all your work. Answers not supported by calculations
or reasoning may not receive credit. Messy work will not be graded. The value of each
question is shown in the right margin.
4. Exposing your test paper, copying from another student’s paper, or sharing information
about this test constitutes academic dishonesty. Such behaviour may jeopardize your
grade on this test, in this course, and your standing at this university.
5. Five minutes before the end of the test period you will be given a verbal notice. After
that time, you must remain seated until all test papers have been collected.
6. When the test period is over, you will be instructed to put away writing implements.
Put away all pens and pencils at this point. Continuing to write past this instruction
will be considered as cheating.
7. Please remain seated and pass your test paper down the row to the nearest indicated
aisle. Once all the test papers have been collected, you are free to leave.
1 1. Use the rules of di▯erentiation to ▯nd the second derivative of the function (3)
R(c) = :
k n c
Solution: To ▯nd the derivative, we make use of the quotient rule,
0 K(k n c) ▯ Kc(1)
R (c) = 2
We di▯erentiate once more to ▯nd the second derivative. Since the numerator of the
▯rst derivative is constant, we do not need the quotient rule.
R (c) = ▯2Kk (k +nc)n
Page 2 of ?? 0
2. The graph of the derivative, f (x), of a certain function f(x) is shown below.
(a) Where is the original function, f(x), increasing? (3)
(b) Where does the original function, f(x), have an in
ection point? (4)
(a) The sign of the derivative tells us where the function is increasing or decreasing.
In particular, when f (x) > 0, then f(x) is increasing. This appears to be from
about x = 65 to about x = 90.
(b) The original function, f(x), has an in
ection point when the second derivative
changes sign. Since we are shown the ▯rst derivative, we need to think about
when its derivative changes sign. This appears to happen twice, once near
x = 60 and once near x = 80.
Page 3 of ?? 3. Consider the function
p(t) = t ▯ 6t:
(a) What is the average rate of change of p(t) between t = 2 and t = 3? (4)
(b) Is there any time t0, between t = 2 and t = 3, where the instantaneous rate of (6)
change is equal to the average rate of change between t = 2 and t = 3? If so, solve