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INTRODUCTORY
HAEUSSLER JR.
ERNEST F. HAEUSSLER JR.
MATHEMATICAL
WOOD
PAUL
RICHARD S. PAUL
RICHARD J. WOOD
F O U RT EE N TH E D IT I ON
application of many areas of mathematics in many ways. Readers of
the linear programming chapter (7) of this book may find it useful to
glance at the cover while contemplating routes, via edges, between
the vertices of similar structures.
www.pearsoncanada.ca 90000
ISBN 978-0-13-414110-7
9 780134 141107
Haeussler-50501 A01_HAEU1107_14_SE_FM November 27, 2017 14:16
For Bronwen
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Contents
Preface ix
v
Haeussler-50501 A01_HAEU1107_14_SE_FM November 27, 2017 14:16
vi Contents
Contents vii
viii Contents
Index I-1
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Preface
T
he fourteenth edition of Introductory Mathematical Analysis for Business, Econo-
mics, and the Life and Social Sciences (IMA) continues to provide a mathematical
foundation for students in a variety of fields and majors, as suggested by the title.
As begun in the thirteenth edition, the book has three parts: College Algebra, Chapters 0–4;
Finite Mathematics, Chapters 5–9; and Calculus, Chapters 10–17.
Schools that have two academic terms per year tend to give Business students a term
devoted to Finite Mathematics and a term devoted to Calculus. For these schools we rec-
ommend Chapters 0 through 9 for the first course, starting wherever the preparation of the
students allows, and Chapters 10 through 17 for the second, including as much as the stu-
dents’ background allows and their needs dictate.
For schools with three quarter or three semester courses per year there are a number
of possible uses for this book. If their program allows three quarters of Mathematics, well-
prepared Business students can start a first course on Finite Mathematics with Chapter 1
and proceed through topics of interest up to and including Chapter 9. In this scenario, a
second course on Differential Calculus could start with Chapter 10 on Limits and Continu-
ity, followed by the three “differentiation chapters”, 11 through 13 inclusive. Here, Section
12.6 on Newton’s Method can be omitted without loss of continuity, while some instructors
may prefer to review Chapter 4 on Exponential and Logarithmic Functions prior to study-
ing them as differentiable functions. Finally, a third course could comprise Chapters 14
through 17 on Integral Calculus with an introduction to Multivariable Calculus. Note that
Chapter 16 is certainly not needed for Chapter 17 and Section 15.8 on Improper Integrals
can be safely omitted if Chapter 16 is not covered.
Approach
Introductory Mathematical Analysis for Business, Economics, and the Life and Social
Sciences (IMA) takes a unique approach to problem solving. As has been the case in ear-
lier editions of this book, we establish an emphasis on algebraic calculations that sets this
text apart from other introductory, applied mathematics books. The process of calculating
with variables builds skill in mathematical modeling and paves the way for students to use
calculus. The reader will not find a “definition-theorem-proof” treatment, but there is a sus-
tained effort to impart a genuine mathematical treatment of applied problems. In particular,
our guiding philosophy leads us to include informal proofs and general calculations that
shed light on how the corresponding calculations are done in applied problems. Emphasis
on developing algebraic skills is extended to the exercises, of which many, even those of
the drill type, are given with general rather than numerical coefficients.
We have refined the organization of our book over many editions to present the content
in very manageable portions for optimal teaching and learning. Inevitably, that process
tends to put “weight” on a book, and the present edition makes a very concerted effort to
pare the book back somewhat, both with respect to design features—making for a cleaner
approach—and content—recognizing changing pedagogical needs.
ix
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x Preface
Finance, Chapter 5, we explicitly discuss negative interest rates and ask, somewhat rhetor-
ically, why banks do not use continuous compounding (given that for a long time now
continuous compounding has been able to simplify calculations in practice as well as in
theory).
Whenever possible, we have tried to incorporate the extra ideas that were in the “Explore
and Extend” chapter-closers into the body of the text. For example, the functions tax rate t.i/
and tax paid T.i/ of income i, are seen for what they are: everyday examples of case-defined
functions. We think that in the process of learning about polynomials it is helpful to include
Horner’s Method for their evaluation, since with even a simple calculator at hand this makes
the calculation much faster. While doing linear programming, it sometimes helps to think
of lines and planes, etcetera, in terms of intercepts alone, so we include an exercise to show
that if a line has (nonzero) intercepts x0 and y0 then its equation is given by
x y
C D1
x0 y0
and, moreover, (for positive x0 and y0 ) we ask for a geometric interpretation of the equivalent
equation y0 x C x0 y D x0 y0 .
But, turning to our “paring” of the previous IMA, let us begin with Linear Program-
ming. This is surely one of the most important topics in the book for Business students. We
now feel that, while students should know about the possibility of Multiple Optimum Solu-
tions and Degeneracy and Unbounded Solutions, they do not have enough time to devote
an entire, albeit short, section to each of these. The remaining sections of Chapter 7 are
already demanding and we now content ourselves with providing simple alerts to these
possibilities that are easily seen geometrically. (The deleted sections were always tagged
as “omittable”.)
We think further that, in Integral Calculus, it is far more important for Applied Mathe-
matics students to be adept at using tables to evaluate integrals than to know about Integra-
tion by Parts and Partial Fractions. In fact, these topics, of endless joy to some as recre-
ational problems, do not seem to fit well into the general scheme of serious problem solving.
It is a fact of life that an elementary function (in the technical sense) can easily fail to have
an elementary antiderivative, and it seems to us that Parts does not go far enough to rescue
this difficulty to warrant the considerable time it takes to master the technique. Since Par-
tial Fractions ultimately lead to elementary antiderivatives for all rational functions, they
are part of serious problem solving and a better case can be made for their inclusion in an
applied textbook. However, it is vainglorious to do so without the inverse tangent function
at hand and, by longstanding tacit agreement, applied calculus books do not venture into
trigonometry.
After deleting the sections mentioned above, we reorganized the remaining material of
the “integration chapters”, 14 and 15, to rebalance them. The first concludes with the Funda-
mental Theorem of Calculus while the second is more properly “applied”. We think that the
formerly daunting Chapter 17 has benefited from deletion of Implicit Partial Differentia-
tion, the Chain Rule for partial differentiation, and Lines of Regression. Since Multivariable
Calculus is extremely important for Applied Mathematics, we hope that this more manage-
able chapter will encourage instructors to include it in their syllabi.
Preface xi
xii Preface
Preface xiii
Supplements
MyLab Math Online Course (access code required) Built around Pearson’s best-
selling content, MyLab™ Math, is an online homework, tutorial, and assessment
program designed to work with this text to engage students and improve results. MyLab
Math can be successfully implemented in any classroom environment—lab-based,
hybrid, fully online, or traditional. By addressing instructor and student needs, MyLab
Math improves student learning. Used by more than 37 million students worldwide,
MyLab Math delivers consistent, measurable gains in student learning outcomes, reten-
tion and subsequent course success. Visit www.mymathlab.com/results to learn more.
Student Solutions Manual includes worked solutions for all odd-numbered problems.
ISBN 0-134-77040-4 j 978-0-134-77040-6
These instructor supplements are available for download from a password-protected
section of Pearson Canada’s online catalogue (catalogue.pearsoned.ca). Navigate to your
book’s catalogue page to view a list of those supplements that are available. Speak to your
local Pearson sales representative for details and access.
Instructor’s Solution Manual has worked solutions to all problems, including those in
the Apply It exercises. It is downloadable from a password-protected section of Pearson
Canada’s online catalogue (catalogue.pearsoned.ca).
– Computerized Test Bank. Pearson’s computerized test banks allow instructors to
filter and select questions to create quizzes, tests, or homework. Instructors can revise
questions or add their own, and may be able to choose print or online options. These
questions are also available in Microsoft Word format.
– PowerPoint® Lecture Slides. The chapter-by-chapter PowerPoint lecture slides
include key concept, equations, and worked examples from the text.
– Learning Solutions Managers. Pearson’s Learning Solutions Managers work with
faculty and campus course designers to ensure that Pearson technology products,
assessment tools, and online course materials are tailored to meet your specific needs.
This highly qualified team is dedicated to helping schools take full advantage of a
wide range of educational resources, by assisting in the integration of a variety of
instructional materials and media formats. Your local Pearson Canada sales repre-
sentative can provide you with more details on this service program.
– Pearson eText. The Pearson eText gives students access to their textbook anytime,
anywhere. In addition to note taking, highlighting, and bookmarking, the Pearson
eText offers interactive and sharing features. Instructors can share their comments
or highlights, and students can add their own, creating a tight community of learners
within the class.
Haeussler-50501 A01_HAEU1107_14_SE_FM November 27, 2017 14:16
xiv Preface
Acknowledgments
We express our appreciation to the following colleagues who contributed comments and
suggestions that were valuable to us in the evolution of this text. (Professors marked with
an asterisk reviewed the fourteenth edition.)
Preface xv
Some exercises are taken from problem supplements used by students at Wilfrid Laurier
University. We wish to extend special thanks to the Department of Mathematics of Wilfrid
Laurier University for granting Prentice Hall permission to use and publish this material,
and also to Prentice Hall, who in turn allowed us to make use of this material.
We again express our sincere gratitude to the faculty and course coordinators of The
Ohio State University and Columbus State University who took a keen interest in this and
other editions, offering a number of invaluable suggestions.
Special thanks are due to MPS North America, LLC. for their careful work on the solu-
tions manuals. Their work was extraordinarily detailed and helpful to us. We also appreciate
the care that they took in checking the text and exercises for accuracy.
0 Review of Algebra
L
esley Griffith worked for a yacht supply company in Antibes, France. Often,
0.1 Sets of Real Numbers
she needed to examine receipts in which only the total paid was reported and
0.2 Some Properties of Real then determine the amount of the total which was French “value-added tax”.
Numbers It is known as TVA for “Taxe à la Value Ajouté”. The French TVA rate was
19.6% (but in January of 2014 it increased to 20%). A lot of Lesley’s business came
0.3 Exponents and Radicals
from Italian suppliers and purchasers, so she also had to deal with the similar problem
0.4 Operations with of receipts containing Italian sales tax at 18% (now 22%).
Algebraic Expressions A problem of this kind demands a formula, so that the user can just plug in a tax
rate like 19.6% or 22% to suit a particular place and time, but many people are able
0.5 Factoring
to work through a particular case of the problem, using specified numbers, without
0.6 Fractions knowing the formula. Thus, if Lesley had a 200-Euro French receipt, she might have
reasoned as follows: If the item cost 100 Euros before tax, then the receipt total would
0.7 Equations, in Particular be for 119.6 Euros with tax of 19.6, so tax in a receipt total of 200 is to 200 as 19.6 is
Linear Equations
to 119.6. Stated mathematically,
0.8 Quadratic Equations tax in 200 19:6
D 0:164 D 16:4%
Chapter 0 Review 200 119:6
If her reasoning is correct then the amount of TVA in a 200-Euro receipt is about 16.4%
of 200 Euros, which is 32.8 Euros. In fact, many people will now guess that
p
tax in R D R
100 C p
gives the tax in a receipt R, when the tax rate is p%. Thus, if Lesley felt confident about
18
her deduction, she could have multiplied her Italian receipts by 118 to determine the tax
they contained.
Of course, most people do not remember formulas for very long and are uncom-
fortable basing a monetary calculation on an assumption such as the one we italicized
above. There are lots of relationships that are more complicated than simple proportion-
ality! The purpose of this chapter is to review the algebra necessary for you to construct
your own formulas, with confidence, as needed. In particular, we will derive Lesley’s
formula from principles with which everybody is familiar. This usage of algebra will
appear throughout the book, in the course of making general calculations with variable
quantities.
In this chapter we will review real numbers and algebraic expressions and the basic
operations on them. The chapter is designed to provide a brief review of some terms and
methods of symbolic calculation. Probably, you have seen most of this material before.
However, because these topics are important in handling the mathematics that comes
later, an immediate second exposure to them may be beneficial. Devote whatever time
is necessary to the sections in which you need review.
1
Haeussler-50501 M01_HAEU1107_14_SE_C00 November 27, 2017 14:19
1
-r -1.5 2 2 r Positive
-3 -2 -1 0 1 2 3 direction
Origin
Positions to the right of the origin are considered positive .C/ and positions to the left
are negative . /. For example, with the point 12 unit to the right of the origin there
corresponds the number 12 , which is called the coordinate of that point. Similarly, the
coordinate of the point 1.5 units to the left of the origin is 1:5. In Figure 0.1, the
coordinates of some points are marked. The arrowhead indicates that the direction to
the right along the line is considered the positive direction.
To each point on the line there corresponds a unique real number, and to each
real number there corresponds a unique point on the line. There is a one-to-one cor-
respondence between points on the line and real numbers. We call such a line, with
coordinates marked, a real-number line. We feel free to treat real numbers as points
on a real-number line and vice versa.
PROBLEMS 0.1
p
In Problems 1–12, determine the truth of each statement. If the 7. 25 is not a positive integer.
statement is false, give a reason why that is so. p
p 8. 2 is a real number.
1. 13 is an integer. 0
2 9. is rational.
2. is rational. 0
7 10. is a positive integer.
3. 3 is a positive integer. p
11. 0 is to the right of 2 on the real-number line.
4. 0 is not rational.
p 12. Every integer is positive or negative.
5. 3 is rational.
13. Every terminating decimal number can be regarded as a
1 repeating decimal number.
6. is a rational number. p
0
14. 1 is a real number.
Thus, two numbers that are both equal to a third number are equal to each other.
For example, if x D y and y D 7, then x D 7.
Haeussler-50501 M01_HAEU1107_14_SE_C00 November 27, 2017 14:19
This means that any two numbers can be added and multiplied, and the result in
each case is a real number.
This means that two numbers can be added or multiplied in any order. For example,
3 C 4 D 4 C 3 and .7/. 4/ D . 4/.7/.
This means that, for both addition and multiplication, numbers can be grouped in
any order. For example, 2 C .3 C 4/ D .2 C 3/ C 4; in both cases, the sum is 9. Simi-
larly, 2x C .x C y/ D .2x C x/ C y, and observe that the right side more obviously sim-
plifies to 3x C y than does the left side. Also, .6 13 / 5 D 6. 13 5/, and here the left side
obviously reduces to 10, so the right side does too.
1
For each real number a, except 0, there is a unique real number denoted a such
that
1
aa D1
1
The number a is called the reciprocal of a.
Zero does not have a reciprocal because Thus, all numbers except 0 have a reciprocal. Recall that a 1 can be written 1a . For
there is no number that when multiplied example, the reciprocal of 3 is 13 , since 3. 13 / D 1. Hence, 13 is the reciprocal of 3. The
by 0 gives 1. This is a consequence of
0 a D 0 in 7. The Distributive Properties. reciprocal of 13 is 3, since . 13 /.3/ D 1. The reciprocal of 0 is not defined.