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Math 141, Fall 2008
Name:
Exam 1
1. [15 points (5 pts each)] Evaluate the following limits. Each of your answers should be either a number,
infinity, or minus infinity.
B
BÄ_ &  B#
(a) lim
(b) lim
BÄ%
B'
B%
a#  2b&  $#
2Ä!
2
(c) lim
Math 141, Fall 2008
Exam 1
2. [26 points] A particularly fast-moving slug is crawling straight across an 8-foot-wide driveway. The
following graph shows the distance crawled by the slug as a function of time:
(a) [5 pts] What was the average speed of the slug between 1:00 pm and 6:00 pm? What was his average
speed between 4:00 pm and 6:00 pm?
(b) [5 pts] At what time was the slug crawling the fastest? Explain.
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Math 141, Fall 2008
Exam 1
(c) [8 pts] What was the slug's crawl speed at 6:00 pm? Assuming he can maintain this pace, when will he
reach the other end of the 8-foot-wide driveway?
(d) [8 pts] Sketch a graph of the slug's speed as a function of time:
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Math 141, Fall 2008
Exam 1
3. [20 points (5 pts each)]
(a) Find 0 w aBb if 0 aBb œ )B# 
"!
 $.
ÈB
(b) Find 0 w aBb if 0 aBb œ B# a&B  "b) .
(c) Find
.Z
.Z
%
(not
) if Z œ 1<$ .
.>
.<
$
(d) Suppose that 0 aBb œ 1aBb$ . Given that 1a#b œ & and 1w a#b œ (, find 0 w a#b.
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Math 141, Fall 2008
Exam 1
5
4. [7 points] Water is draining out of the bottom of a 5000-gallon tank. The volume Z of water (in gallons)
remaining in the tank after > minutes is given by the following formula:
Z œ &!!!Œ" 
>

%!
#
How quickly is water draining out of the tank when > œ "! minutes?
5. [8 points (4 pts each)] Suppose that 0 a#b œ & and 0 w a#b œ (.
(a) Find the equation of the tangent line to the graph of C œ 0 aBb at the point a#ß &b.
(b) Use a linear approximation to estimate 0 a#Þ!$b.
Math 141, Fall 2008
Exam 1
6
6. [8 points] Define a function 0 as follows:
0 a8b œ the area of a regular polygon with 8 sides and perimeter "
For example, 0 a$b is the area of an equilateral triangle with perimeter ":
1/3
1/3
0 a$b œ the area of this triangle ¸ !Þ!%)"
1/3
and 0 a%b is the area of a square with perimeter ":
1/4
1/4
1/4
1/4
What is lim 0 a8b? Explain your answer.
8Ä_
0 a%b œ the area of this square œ
"
œ !Þ!'#&
"'
Math 141, Fall 2008
Exam 1
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7. [16 points] In economics and finance, the GDP (or “gross domestic product”) of a country is a measure of
its total annual economic output. The GDP per capita is defined to be the total GDP divided by the size of
the population. In equation form:
Gœ
X
T
where G is the GDP per capita, X is the total GDP, and T is the population size.
Currently, the United States has a total GDP of $"$Þ) trillion dollars, and a population of 300 million, so
the GDP per capita is $46,000 per person.
Note: " million œ ",!!!,!!! œ "!'
" billion œ "!!! million œ ",!!!,!!!,!!! œ "!*
" trillion œ "!!! billion œ ",!!!,!!!,!!!,!!! œ "!"#
(a) [6 pts] Find a formula for
.G
.X
.T
in terms of X , T ,
, and
.
.>
.>
.>
(b) [10 pts] The total GDP of the United States is currently increasing at a rate of about $690 billion/year,
while the population of the United States is increasing at rate of 3 millionÎyear. How quickly is the
GDP per capita increasing? (Note: You must simplify your answer.)