2013 Math for Business (Fin) -- Prof. Tsang
Assignment 1 (due: Mar 15, 2013)
Chapter 1
§1. Functions
P10
In Problems 9 and 14, compute the indicated values of the given function.
9. If f (t )  (2t  1) 3 / 2 ; f (1), f (5), f (13), .
3
if t  5

14. If f (t )  t  1 if  5  t  5 ; f (6), f (5), f (0), f (16) .

 t if t  5
t 1
22. Find the range and domain of function f (t )  2
.
t t 2
50 Find the indicated composite function
f ( x 2  2 x  9) where f ( x)  2 x  20
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56. If f ( x)  x  4 
find the functions h(x) and g (u ) such that
( x  4) 4
f ( x)  g (h( x))
65. IMMUNIZATION Suppose that during a nationwide program to immunize the
population against a certain form of influenza, public health officials found that the cost
of inoculating x% of
the population was approximately C ( x) 
150 x
million dollars.
200  x
a. What is the domain of the function C?
b. For what values of x does C(x) have a practical interpretation in this context?
c. What was the cost of inoculating the first 50% of the population?
d. What was the cost of inoculating the second 50% of the population?
e. What percentage of the population had been inoculated by the time 37.5 million dollars
had been spent?
68. EXPERIMENTAL PSYCHOLOGY To study the rate at which animals learn, a
psychology student performed an experiment in which a rat was sent repeatedly through a
laboratory maze. Suppose that the time required for the rat to traverse the maze on the nth
trial was approximately
12
T ( n)  3 
minutes.
n
a. what is the domain of the function T?
b. For what values of n does T(n) have meaning in the context of the psychology
experiment?
c. How long did it take the rat to traverse the maze on the 3rd trial?
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2013 Math for Business (Fin) -- Prof. Tsang
d. On which trial did the rat first traverse the maze in 4 minutes or less?
e. According to the function T, what will happen to the time required for the rat to
traverse the maze as the number of trials increases? Will the rat ever be able to traverse
the maze in less than 3 minutes?
70. POSITION OF A MOVING OBJECT A ball has been dropped from the top of a
building. Its height (in feet) after t seconds is given by the function H (t )  16t 2  256 .
a. What is the height of the ball after 2 seconds?
b. How far will the ball travel during the third second?
c. How tall is the building?
d. When will the ball hit the ground?
72. POPULATION DENSITY Observations suggest that for herbivorous mammals, the
91.2
number of animals N per square kilometer can be estimated by the formula N  0.73 ,
m
where m is the average mass of the animal in kilograms.
a. Assuming that the average elk on a particular reserve has mass 300 kg, approximately
how many elk would you expect to find per square kilometer in the reserve?
b. Using this formula, it is estimated that there is less than one animal of a certain species
per square kilometer. How large can the average animal of this species be?
c. One species of large mammal has twice the average mass as a second species. If a
particular reserve contains 100 animals of the large species, how many animals of the
smaller species would you expect to find there?
§2. The Graph of a Function
P24 -29
53 AIR POLLUTION Lead emissions are a major source of air pollution. Using data
gathered by the U.S. Environmental Protection Agency in the 1990s, it can be shown that
the formula
N (t )  35t 2  299t  3347
estimates the total amount of lead emission N (in thousands of tons) occurring in the
United States t years after the base year 1990.
a. Sketch the graph of the pollution function N(t).
b. Approximately how much lead emission did the formula predict for the year 1995?
(The actual amount was about 3924 thousand tons.)
c. Based on this formula, when during the decade 1990-2000 would you expect the
maximum lead emission to have occurred?
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2013 Math for Business (Fin) -- Prof. Tsang
d. Can this formula be used to predict the current level of lead emission? Explain.
68. Graph f ( x) 
8x 2  9 x  3
. Determine the values of x for which the function is
x 2  x 1
defined.
§3. Linear Functions
P39-44
In Problems 22 throught 36, write an equation for the line with the given properties
22.Through (-1, 2) with slope 2/3.
30. Through (-2,3) and (0,5).
33. Through (4,1) and parallel to the line 2 x  y  3 .
35. Through (3,5) and perpendicular to the line x  y  4 .
49. GROWTH OF A CHILD The average height H is certimeters of a child of age A
years can be estimated by the linear function H  6.5 A  50 . Use this formula to answer
questions.
a. What is the average height of a 7-year-old child?
b. How old must a child be to have an average heigh of 150 cm?
c. What is tha average height of a nowborn baby? Does this answer seem reaonable?
d. What is th aaverage height of a 20-year-old? Does this answer seem reaonable?
55. COLLEGE ADMISSIONS The average scores of incoming students at an eastern
liberal arts college in the SAT mathematics examination have been declining at a
constant rate in recent years. In 1995, the average SAT score was 575, while in 2000 it
was 545.
a. Express the average SAT score as a function of time.
b. If the trend continues, what will the average SAT score of incoming students be in
2005?
c. If the trend continues, when will the average SAT score be 527?
§4. Functional Models
P56-62
31. INCOME TAX The accompanying table represents the 2004 federal income tax rate
schedule for single taxpayers.
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2013 Math for Business (Fin) -- Prof. Tsang
a. Express an individual’s income tax as a function of the taxable income x for
0  x  146750 and draw the graph.
b. The graph in part(a) should consist of four line segments. Compute the slope of each
segment. What happens to these slopes as the taxable income increase? Explain the
behavior of the slopes in practical terms.
If the Taxable Income Is
Over
The Income Tax Is
But not Over
Of the Amout Over
0
$7,150
10%
0
$7,150
$29,050
$715+15%
$7,150
$29,050
$70,350
$4,000+25%
$29,050
$70,350
$146,750
$14,325+28%
$70,350
38. CONSTRUCTION COST An open box is to be made from a square piece of
cardboard, 18 inches by 18 inches, by removing a small square from each corner and
folding up the flaps to form the sides. Express the volume of the resulting box moved
squares.Draw the graph and estimate the value of x for which the volume of the resulting
box is greatest.
x
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§5. Limits
P73-77
In Problem 11 through 25 , find the indicated limit if it exists.
11. lim ( x  1) 2 ( x  1)
x 3
x3
x 5 5  x
9  x2
lim
18.
x 3 x  3
x 2  3x  10
19. lim
x 5
x 5
15. lim
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2013 Math for Business (Fin) -- Prof. Tsang
x2  x  6
x 2 x 2  3 x  2
x 2
25. lim
x4 x  4
For problems 27, 31 and 35, find lim f ( x) and lim f ( x) . If the limiting value is
23. lim
x
x 
infinite, indicate whether it is   or   .
27. f ( x)  x 3  4 x 2  4
31.
35.
x 2  2x  3
f ( x)  2
2 x  5x  1
3x 2  6 x  2
f ( x) 
2x  9
52. If $1,000 is invested at 5% compounded n times per year, the balance after 1 year will
1
be 1000(1  0.05x)1/ x , where x  is the length of the compounding peroid. For example,
n
if n=4 the compounding period is 1/4 year long. For what is called continuous
compounding of interest, the balance after 1 year is given by the limit
B  lim 1000(1  0.05x)1/ x
x0
Estimte the value of this limit by filling in the second line of the following table:
x
1
0.1
0.01
0.001
0.0001
1/ x
1000(1  0.05x)
65. Evaluate limit
a x n  an1 x n1    a1 x  a0
lim n m
m1
x b x  b
   b1 x  b0
m
m1 x
for constants a0 , a1 ,, an and b0 , b1 ,, bm in each of the following cases:
a. n  m
b. n  m
c. n  m
[Note: There are two possible answers, depending on the signs of a n and bm .]
§6. One-Sided Limits and Continuity
P86-89
In Problems 13 and 15, find the indicated one-sided limit. If the limiting value is infinite,
indicate whether it is   or   .
13. lim
x 3
x 1  2
x3
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2013 Math for Business (Fin) -- Prof. Tsang
2 x 2  x if x  3
15. lim f ( x ) and lim f ( x ) , where f ( x)  
x 3
x 3
if x  3
3  x
In problems 21 and 28, decide if the given function is continuous at the specified value of
x.
x 1
21. f ( x) 
at x  1 .
x 1
 x2 1
if x  1

28. f ( x)   x  1
at x  1
2
x - 3
if x  1

In Problems 35 and 41, list all the value of x for which the given function is not
continuous.
3x  2
35. f ( x) 
( x  3)( x  6)
3x  2 if x  0
41. f ( x)   2
 x  x if x  0
In Problems 42, list all the value of x for which the given function is not continuous
if x  1
2  3x
42. f ( x)   2
 x  x  3 if x  1
46. WATER POLLUTION A ruptured pipe in a North Sea oil rig produces a circular
oil slick that is y meters thick at a distance x meters from the rupture. Turbulence
makes it diffucult to directly measure the thickness of the slick at the source (where
x  0 ), but for x  0 , it is found that
0.5( x 2  3x)
y 3
x  x 2  4x
Assuming the oil slick is continuously distributed, how thick would you espect it to be at
the source?
52. AIR POLLUTION It is estimated that t years from now the population of a certain
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suburban community will be p thousand people, where p(t )  20 
. An
t2
environmental study indicates that the average level of carbon monoxide in the air will be
c parts per million when the population is p thousand, where c( p)  0.4 p 2  p  21 .
What happens to the level of pollution c in the long run (as t  )
In Problem 53, find the values of the constant A such that the function f(x) will be
continuous for all x.
if x  2
 Ax  3
53. f ( x)  
2
3  x  2 x if x  2
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2013 Math for Business (Fin) -- Prof. Tsang
57. Show that the equation 3 x  8  9 x 2 / 3  29 has at least one solution for the interval
0  x  8.
58. Show that the equation 3 x  x 2  2 x  1 must have at least one solution on the
interval 0  x  1 .
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