Math 1100-001
Summer 2008
A. McDonald
Final Exam Cover Sheet
Monday, July 28
The exam is broken into three sections. You are to complete any nine questions from Section
1, any four questions from Section 2, and any two questions from Section 3. Each problem in Section 1 is worth 5 points, in Section 2 is worth 10 points, and in Section 3 is worth
15 points. The total points you can achieve on this exam is 115 points. There is no extra credit.
You are more than welcome to use any of the materials handed out in class and your textbook in completing this exam. You are also welcome to work with your fellow classmates. If
you do decide to work in teams, make sure that everyone writes individual solutions. You are
not permitted to use the Mathematics Tutoring Center or any other tutoring service. Copying
a classmate’s work or using outside help will automatically result in a zero score on the final
exam.
Some of you have requested that I send an email to you detailing your performance on this
exam and in the course. In consideration of these requests, please fill out the form below and
attach it to your exam solutions.
1. Would you like me to email you your final exam and course grades when I have them
computed? Please specify yes or no.
2. Would you like me to email you comments on your final review packet? Please specify yes
or no.
If you answered yes to either question above, please provide me with your email address in the
space below.
Thanks for an incredible summer semester. I really enjoyed having each of you in class. I
look forward to hearing about your many successes in the coming years. Good luck on this
exam and in your future endeavors.
Math 1100-001
Summer 2008
A. McDonald
Final Exam
Due Thursday, July 31 by 5:00 PM
Name:
University ID:
Read all directions carefully. Make sure that you clearly indicate which problems you would
like graded from each section. Solutions to your chosen problems should be well-written and
in your own words. Remember that it is important to show all of your work (if applicable).
Copying is not allowed. Please use pencil.
Section 1: You are to complete any nine of the twelve problems presented in this section.
Each problem will be scored out of 5 points.
5(x+h)11 −(x+h)22 −5x11 +x22
.
x+h−x
1.1: Compute limh→0
1.2: Compute
d at
( )
dt t+b
for any constants a and b.
1.3: Find an expression for
1.4: Compute
1.5: Find
1.6: Find
R
R
R
using the equation ey + x2 y 3 = π via implicit differentiation.
((x3 (x − 1)2 )4 ).
x(2x2 −1)
(x4 −x2 )6
dx.
4zx5 dz.
1.7: Compute
1.8: Find
d
dx
dy
dx
d
dx
x2 −1
x2 +x
1.9: Compute
R1
0
¡R x
0
¢
t4 dt .
dx.
(x2 − x + 7) dx.
1.10: For what value of k does
R∞
k
10 x3
dx = 1?
1.11: Find all critical points for the curve y = f (x) = e(x
2 −100)3
.
1.12: Find all inflection points for the curve y = f (x) = x4 − 2x3 − 12x2 + 6.
Section 2: You are to complete any four of the five problems presented in this section. Each
problem will be scored out of 10 points.
2.1: Find the derivative of q(p) = q 3 − q + 1 using the definition of the derivative.
2.2: In a 100-unit apartment building, when the price charged per apartment rental is 830+30x
dollars, then the number of apartments rented is 100 − x and the total revenue for the
building is
R(x) = (830 + 30x)(100 − x),
where x is the number of $30 rent increases (and also the resulting number of unrented
apartments). Find the marginal revenue when x = 10. Does this tell you that the rent
should be raised (causing more vacancies) or lowered? Explain.
2.3: Find the particular solution to the differential equation
2xy
dy
= y2 + 2
dx
if the solution y = y(x) satisfies the condition y(1) = 2. Does your solution define a
function? Why or why not?
f eet
2.4: A large red balloon is rising at the rate of 20 second
. The balloon is 10 f eet above the
ground at the point in time that the back end of a green car is directly below the bottom
f eet
of the balloon. The car is traveling at 40 second
. What is the rate of change of the distance
between the bottom of the balloon and the point on the ground directly below the back
of the car one second after the back of the car is directly below the balloon?
2.5: Find the area of the planar region bounded between the curves y = l(x) = x2 + 6 and
y = m(x) = 5x on the domain 1 ≤ x ≤ 3.
Section 3: You are to complete any two of the three problems presented in this section. Each
problem will be scored out of 15 points.
3.1: Suppose that the oxygen level P (for purity) in a body of water t months after an oil spill
is given by
¶
µ
16
4
+
.
P (t) = 500 1 −
t + 4 (t + 4)2
Find how long it will be before the oxygen level reaches its minimum. Next, find how long
it will before the rate of change of oxygen level is maximized. That is, find the point of
diminishing returns.
3.2: Sketch the graph of a function y = f (x) which has first derivative
f 0 (x) = x(x − 4)(x + 6)
and satisfies the conditions f (0) = 5, f (4) = −3, and f (−6) = 0. In doing so, you should
identify all critical values and then classify each as the position of either a relative maximum or relative minimum.
3.3: Suppose that the marginal cost for a certain product is given by
C 0 (x) = 1.02(x + 200)0.02
and marginal revenue is given by
R0 (x) = √
2
+ 1.75,
4x + 1
where x is in thousands of units and revenue and cost are in thousands of dollars. Suppose further that fixed costs are $150, 000 and production is limited to at most 200, 000
units. Determine what level of production yields maximum profit (on the domain described
above), and find the maximum profit.