Final Exam, Math 30, Fall 2008, 12/13/08
Instructions: Write your name and section number. Draw grading table on the cover.
Read each problem carefully and follow all of its instructions. For each of the problems
below, write a clear and concise solution in your blue book. Solutions must be simplified
as much as possible, no full credit for partially completed problems. Blue books with
torn or missing pages will not be accepted !
1. (10 pts) Find the average of )(cos on interval
[]
)(sin)( 32 xxxf =
π
π
, −
2. (10 pts) Derive the given formula where n and a are constants.
dxaxx
a
n
axx
a
dxaxx nnn )sin()sin(
1
)cos( 1
∫∫ −
−=
3. (10 pts) Sketch the region bounded by y=x3, x=0 and y = 1. Find the volume of
revolution when the region is revolved about the x-axis.
4. (10 pts) Find the length of the curve described by the function
)ln(
8
2x
x
y−= from x =1 to x = 4.
5. (10 pts) Solve the differential equation y’ + cos(x)y=cos(x), where y(0)=2
6. (10 pts) A manager of a fast food restaurant advertises that any customer waiting
for more than X minutes will get a free meal. The mean waiting time is 5 min.
What should she set X to so that no more than 1% of customers get a free meal?
7. Virions (virus particles) in an infected patient increase at the rate proportional to
the virion number. kV
dt
dV =. Suppose that at t=0 (Measured in days) the patient
begins to take antivirus medication that eliminates virions at the rate r. The
elimination rate is related to the daily medicine dose by equation r = aD. Let k
=.1/day, a = 200 /(day mg) , V(0)=100000.
a. (5 pts) Solve the equation rkV
dt
dV −=
b. (5 pts) What minimum dose does the patient need to take so that virion
number decreases over time? (Hint: Write the answer as inequality D > ?)
8. A climate model for average annual global temperature(in Fahrenheit) is given
by: )104)(86)(68(
2TTTT
dt
dT −−−=
a. (8 pts) Find and identify by type all equilibrium points.
b. (7 pts) Suppose that the current average annual global temperature is 77 F.
Suppose that current CO2 emissions are projected to increase this
temperature by 11F. Is there a major risk? Using equilibrium points,
explain what might happen.
9. Suppose population of wolves and rabbits are modeled with the following Lotka-
Voltera equations.
