Download Name MAT 131 – Calculus for Non-Science Majors March 1, 2012

Name ___________________________________
MAT 131 – Calculus for Non-Science Majors
Professor Pestieau
Multiple-Choice Questions
March 1, 2012
Exam 1 – Models & Limits
[5 pts each]
Circle the correct answer for the following questions.
For questions 1, 2 and 3 below, let the functions f and g be defined as
f ( x)  3x 2  3x  18
g ( x)  x 2  x  2 .
and
Compute the following:
1.
 f ( x) 
lim  g ( x) 
x 2
2.
a.
-5
b.
0
c.
5
d.
The limit does not exist.
 f ( x) 
lim  g ( x) 
x 
a.
3
b.
1
3
c.

d.

3.
 1 
lim  g ( x) 
x 2
4.
a.
-3
b.
0
c.

d.

 4.2 
2t 

lim  2  3
t
a.
0
b.
2.1
c.
4.2
d.

For questions 4 and 5 below, assume that you are in charge of your college newspaper, The East
Enders, which sells for 50¢ a copy, has fixed production costs of $70 per edition and has
variable printing & distribution costs of 40¢ per copy.
5.
6.
What profit/loss results from the sale of 500 copies of The East Enders?
a.
$50 profit
b.
$180 profit
c.
$20 loss
d.
$70 loss
After selling how many copies of The East Enders will you break-even?
a.
400 copies
b.
500 copies
c.
600 copies
d.
700 copies
Show all your work on the following problems to receive full credit.
Problem 1
a)
[15 pts]
If you invest $380,000 in a mutual fund with an annual yield of 8.9% and the earnings are
compounded monthly, how long will it take for you to become a millionaire? Round
your answer to the nearest month.


The formula for discrete compounding is given by A(t )  P  1 
b)
mt
r
 .
m
Your good friend who is a savvy investor suggests that you switch to a different mutual
fund with a lower annual yield of 8.7%. He argues that because this mutual fund
compounds your earnings continuously, it will actually speed up your process of
becoming a millionaire. Is your friend correct?
rt
The formula for continuous compounding is given by A(t )  Pe .
Problem 2
[20 pts]
In 2005, the Las Vegas Monorail Company (LVMC) charged $3 per ride and had an average
ridership of about 28,000 people per day. In December of 2005, the LVMC raised the fare to
$5 per ride and, as a result, ridership in 2006 plunged to around 19,000 people per day.
a)
Using the given information, find a linear demand equation of the average ridership of
the monorail per day ( q ) as a function of the fare price ( p ).
b)
Based on this linear demand, find the price the LVMC should have charged to maximize
revenue from ridership. What would have been the corresponding daily revenue with
this optimal price?
c)
The LVMC would have needed $44.9 million in revenues from ridership to break even
in 2006. Would it have been possible for the company to break even in 2006 by
charging this optimal price?
Problem 3
[10 pts]
The following table shows approximate daily oil production by Pemex, Mexico’s national oil
company, for the years 2001 – 2009 ( t  1 represents the start of 2001):
t
(year since 2000)
P (t )
(in millions of
barrels)
1
2
3
4
5
6
7
8
9
3.1
3.3
3.4
3.4
3.4
3.3
3.2
3.1
3.0
a)
Compute the average rate of change of P (t ) , the approximate daily oil production by
Pemex (in millions of barrels), over the period 2002 – 2007. Interpret your result.
b)
Which of the following is true? From 2001 to 2008, the one-year average rates of
change of daily oil production by Pemex…
(A)
…increased in value.
(B)
…decreased in value.
(C)
…never increased in value.
(D)
…never decreased in value.
Problem 4
[15 pts]
There are currently 10,000 cases of a nasty flu going around in a total susceptible population of
120,000. The number of cases, P (t ) , is increasing by 20% each day at the early stages of this flu
epidemic.
a)
Find a logistic model of the form P (t ) 
N
, where N is the limiting value and
1  Ab  t
A, b are nonzero positive constants.
b)
Use your model to predict the number of flu cases 2 weeks from now.
c)
According to your model, how long will it take for 40% of the susceptible population to
become infested with this flu? [Bonus – 5pts]
Problem 5
[15 pts]
The Richter scale is used to measure the intensity of earthquakes. This scale rating of an
earthquake is given by the formula
R
2
 log E  11.8  ,
3
where E is the energy released by the earthquake (measured in ergs).
a)
The San Francisco earthquake of 1989 registered R  7.1 on the Richter scale. How
many ergs of energy were released?
b)
The recent Chilean earthquake of 2010 registered R  8.8 on the Richter scale. How
much more intense was this earthquake in comparison with the San Francisco
earthquake of 1989?