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1.1
Lesson 1: Mathematical Modeling, Review of Functions, and Elementary Functions
Homework
Mathematical Modeling Problems
Problem 1: A business that sells school supplies decides to sell notebooks for $1.50 per notebook. Based on
sales from the previous year, the business believes that between 2000 and 3100 notebooks will be sold before the
school year starts. Assuming the revenue generated by the sale of notebooks is proportional to the number of
notebooks sold, develop a mathematical model that can be used to determine the revenue generated by the sale
of these notebooks. Include a description of the domain and range of the resulting revenue function.
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Problem 2: The model of depreciation in the notes was determined to be the following.
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v(t) = √
1+t
This represents the value of a single truck. The freight company thinks this model is not a good enough
approximation and believes the value of a single truck is inversely proportional to the square root of the time
squared (t2 ) plus one. Modify this model to reflect the change in the assumption about the proportionality and
compare the old model to the new model at one year, two years, and five years. Use the absolute value of the
function values to compute the difference.
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Proportionality Problems
Problem 3: Write a mathematical relationship that represents the statement in the problem description as
shown in the following example.
Statement: The variable, x, is inversely proportional to the variable y with proportionality constant 5.1.
Solution: Since the proportionality is
x∝
1
y
the statement can be represented by
x=
C
y
→
x=
5.1
y
2
a. The variable, z, is directly proportional to the variable, y.
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b. The variable, f , is inversely proportional to the square of the variable, y. That is, y 2 .
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c. The variable, y, is inversely proportional to the square of the variable, t, plus 42.
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Definition of Function Problems These problems are included as a review of some of the concepts
related to functions.
Problem 4: Determine the implied domain and the range of each of the functions.
Example: For the function
V (t) = √
250
1 − t2
The argument of the square root function in the denominator of V (t) must be greater than or equal to zero.
In addition, since this term is in the denominator, the value must be nonzero. So, the following restriction
must be met.
1 − t2 > 0
This means that t2 < 1 or t ∈ (−1, 1). This means the implied domain is the interval (−1, 1). The range is
positive and as t gets closer to t = ±1 the denominator tends to zero which means that V (t) is unbounded.
This translates to the range being [250, ∞).
a. f (x) = ln(x + 1)
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b. g(z) =
1
z+1
+
z
1−z
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Discrete Function Definition
Problem 5: The following table of data is taken out of the book. The table has two columns that have been
left blank. Fill in the blanks with the median income determined by the arithmetic mean of the left and right
endpoints of the interval.
How to compute an arithmetic average If x ∈ [349, 1001) then the arithmetic average is
349 + 1001
1350
=
= 675
2
2
Use the formula to fill in the two columns of data to complete the table.
Tax Bracket Number
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2
3
4
5
6
Tax Rate
10%
15%
25%
28%
33%
35%
Single
$0-$8,375
$8,376-$34,000
$34,001-$82,400
$82,401-$171,850
$171,851-$373,650
$373,651+
Median Income
Filing Jointly
$0-$16,750
$16,751-$68,000
$68,001-$137,300
$137,301-$209,250
$209,251-$373,650
$373,651+
Median Income
a. Use a discrete function definition to represent the relationship between the first column and the fourth column.
of the table. Do the same for the first and fifth columns and finally, for the first and last columns in the table.
Determine the domain and range of each of the three functions your have described.
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b. Use a discrete function definition to represent the relationship between the fourth and second column where
the fourth column is the input and the second column is the output. Define the domain and range for this
function. Repeat using the last column and the second column.
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Solution of Linear Systems These problems should provide a review of methods for the solution of two
linear equations in two variables.
Problem 6: Suppose that the price vs. production curves on page 18 of the textbook are not quite correct.
The company decides to modify the two equations as follows.
ps
=
357 − 11xs
pd
=
217 + 6xd
Compute the equilibrium price and production levels. Compute the difference in the solution of the original
model and the modified model in this problem. Use the absolute value of the difference.
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Qualitative Behavior of Functions These problems are included to review how to sketch the graphical
behavior of functions.
Problem 7: Sketch the graph of each of these functions. You may choose to graph the function on a laptop
computer or using a graphing calculator. Then sketch the output for the graph in the space provided. Make
sure that the graph includes vertical asymptotes if they exist.
a.f (x) = 3 · x2 − 2 · x3
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Sketch the graph in the space provided
b.f (x) = ex−1 and g(x) = ex
Sketch the graph in the space provided
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c.f (x) =
x
x
and g(x) =
x−1
x+1
Sketch the graph in the space provided
Compositions of Functions These problems should provide a brief review of compositions.
Problem 8: Write out the compositions indicated in the following problems.
a. Use the following functions to create f (g(x)) and g(f (x)).
f (x) =
x
x−1
and
g(y) = 6 · y + 7
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b. Use the following functions to create f (g(x)) and g(f (x)).
f (x) = e2·x
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and
g(y) = ln(y)
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