Schaum's outline of theory and problems of college mathematics : algebra, discrete mathematics, precalculus, introduction to caculus

431 Pages·2003·5.47 MB·English

SCHAUM’S
OUTLINE OF
Theory and Problems of
COLLEGE
MATHEMATICS
THIRD EDITION
Algebra
Discrete Mathematics
Precalculus
Introduction to Calculus
FRANK AYRES, Jr., Ph.D.
Formerly Professor and Head
Department of Mathematics, Dickinson College
PHILIP A. SCHMIDT, Ph.D.
Program Coordinator, Mathematics and Science Education
The Teachers College, Western Governors University
Salt Lake City, Utah
Schaum’s Outline Series
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PREFACE
In the Third Edition of College Mathematics, I have maintained the point-of-view of
the first two editions.Students who are engaged in learning mathematics in the
mathematical range from algebra to calculus will find virtually all major topics from
those curricula in this text.However, a substantial number of important changes have
been made in this edition.First, there is more of an emphasis now on topics in discrete
mathematics.Second, the graphing calculator is introduced as an important problem-
solving tool.Third, material related to manual and tabular computations of logarithms
has been removed, and replaced with material that is calculator-based.Fourth, all
material related to the concepts of locus has been modernized.Fifth, tables and graphs
have been changed to reflect current curriculum and teaching methods.Sixth, all
material related to the conic sections has been substantially changed and modernized.
Additionally, much of the rest of the material in the third edition has been changed
to reflect current classroom methods and pedagogy, and mathematical modeling is
introduced as a problem-solving tool.Notation has been changed as well when
necessary.
My thanks must be expressed to Barbara Gilson and Andrew Littell of
McGraw-Hill.They have been supportive of this project from its earliest stages.I
also must thank Dr.Marti Garlett, Dean of the Teachers College at Western Governors
University, for her professional support as I struggled to meet deadlines while
beginning a new position at the University.I thank Maureen Walker for her handling
of the manuscript and proofs.And finally, I thank my wife, Dr.Jan Zlotnik Schmidt,
for putting up with my frequent need to work at home on this project.Without her
support, this edition would not have been easily completed.
PHILIP A.S CHMIDT
New Paltz, NY
iii
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CONTENTS
PART I Review of Algebra 1
1. Elements of Algebra 3
2. Functions 8
3. Graphs of Functions 13
4. Linear Equations 19
5. Simultaneous Linear Equations 24
6. Quadratic Functions and Equations 33
7. Inequalities 42
8. The Locus of an Equation 47
9. The Straight Line 54
10. Families of Straight Lines 60
11. The Circle 64
PART II Topics in Discrete Mathematics 73
12. Arithmetic and Geometric Progressions 75
13. Infinite Geometric Series 84
14. Mathematical Induction 88
15. The Binomial Theorem 92
16. Permutations 98
17. Combinations 104
18. Probability 109
19. Determinants of Orders Two and Three 117
20. Determinants of Order n 122
21. Systems of Linear Equations 129
22. Introduction to Transformational Geometry 136
PART III Topics in Precalculus 153
23. Angles and Arc Length 155
24. Trigonometric Functions of a General Angle 161
25. Trigonometric Functions of an Acute Angle 169
26. Reduction to Functions of Positive Acute Angles 178
27. Graphs of the Trigonometric Functions 183
28. Fundamental Relations and Identities 189
29. Trigonometric Functions of Two Angles 195
30. Sum, Difference, and Product Formulas 207
31. Oblique Triangles 211
32. Inverse Trigonometric Functions 222
33. Trigonometric Equations 232
34. Complex Numbers 242
v
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vi CONTENTS
35. The Conic Sections 254
36. Transformation of Coordinates 272
37. Points in Space 283
38. Simultaneous Equations Involving Quadratics 294
39. Logarithms 303
40. Power, Exponential, and Logarithmic Curves 307
41. Polynomial Equations, Rational Roots 312
42. Irrational Roots of Polynomial Equations 319
43. Graphs of Polynomials 329
44. Parametric Equations 336
PART IV Introduction to Calculus 343
45. The Derivative 345
46. Differentiation of Algebraic Expressions 355
47. Applications of Derivatives 360
48. Integration 371
49. Infinite Sequences 377
50. Infinite Series 383
51. Power Series 389
52. Polar Coordinates 394
APPENDIX A Introduction to the Graphing Calculator 410
APPENDIX B The Number System of Algebra 414
APPENDIX C Mathematical Modeling 421
INDEX 424
PART I
REVIEW OF ALGEBRA
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Chapter 1
Elements of Algebra
IN ARITHMETIC the numbers used are always known numbers; a typical problem is to con-
vert 5 hours and 35 minutes to minutes.This is done by multiplying 5 by 60 and adding 35; thus,
5 · 60 þ 35 ¼ 335 minutes.
In algebra some of the numbers used may be known but others are either unknown or not specified;
that is, they are represented by letters.For example, convert h hours and m minutes into minutes.This is
done in precisely the same manner as in the paragraph above by multiplying h by 60 and adding m; thus,
h · 60 þ m ¼ 60h þ m.We call 60 h þ m an algebraic expression.(See Problem 11. ).
Since algebraic expressions are numbers, they may be added, subtracted, and so on, following the
same laws that govern these operations on known numbers.For example, the sum of 5 · 60 þ 35 and
2 · 60 þ 35 is ð5 þ 2Þ · 60 þ 2 · 35; similarly, the sum of h · 60 þ m and k · 60 þ m is ðh þ kÞ · 60 þ 2m.(See
Problems 1.2–1.6.)
POSITIVE INTEGRAL EXPONENTS. If a is any number and n is any positive integer, the product
of the n factors a · a · a · · · a is denoted by an.To distinguish between the letters, a is called the base and n
is called the exponent.
If a and b are any bases and m and n are any positive integers, we have the following laws of
exponents:
(1) am · an ¼ amþn
(2) ðamÞn ¼ amn
(3) a
amn ¼ amÿn; a ¼6 0; m > n; amn ¼ an1ÿm ; a ¼6 0; m < n
(4) ða · bÞn¼ anbn

an an
(5)
b ¼ bn ; b ¼6 0
(See Problem 1.7.)
LET n BE A POSITIVE INTEGER and a and b be two numbers such that bn ¼ a; then b is called an nth
root of a.Every number a ¼6 0 has exactly n distinct nth roots.
If a is imaginary, all of its nth roots are imaginary; this case will be excluded here and treated later.
(See Chapter 35.)
3
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4 ELEMENTS OF ALGEBRA [CHAP.1
If a is real and n is odd, then exactly one of the nth roots of a is real.For example, 2 is the real cube

3 5
root of 8, 2 ¼ 8 , and ÿ3 is the real fifth root of ÿ243½ðÿ3Þ ¼ ÿ243Š.
If a is real and n is even, then there are exactly two real nth roots of a when a > 0, but no real nth
roots of a when a < 0.For example, þ3 and ÿ3 are the square roots of 9; þ2 and ÿ2 are the real sixth
roots of 64.
THE PRINCIPAL nth ROOT OF a is the positive real nth root of a when a is positive and the real nth
pffiffi
n
root of a, if any, when a is negative.The principal nth root of a is denoted by a, called a radical.The
integer n is called the index of the radical and a is called the radicand.For example,
pffiffi pffiffiffi pffiffiffiffiffiffiffi
6 5
9 ¼ 3 64 ¼ 2 ÿ243 ¼ ÿ3
(See Problem 1.8.)
ZERO, FRACTIONAL, AND NEGATIVE EXPONENTS. When s is a positive integer, r is any
n
integer, and p is any rational number, the following extend the definition of a in such a way that the
laws (1)-(5) are satisfied when n is any rational number.
DEFINITIONS EXAMPLES

0
0 0 1 0
(6) a ¼ 1; a ¼6 0 2 ¼ 1; ¼ 1; ðÿ8Þ ¼ 1
100
r=s ps ffiffiffir ps ffiffi r 1=2 pffiffi 5=6 p6 ffiffiffi5 5 ÿ2=1 ÿ2 1
(7) a ¼ a ¼ a 3 ¼ 3; ð64Þ ¼ 64 ¼ 2 ¼ 32; 3 ¼ 3 ¼
9
pffiffi
ÿp p ÿ1 1 ÿ1=2
(8) a ¼ 1=a ; a ¼6 0 2 ¼ ; 3 ¼ 1 3
2
pffi
2 p
[ NOTE: Without attempting to define them, we shall assume the existence of numbers such as a ; a ; . . . ; in which
the exponent is irrational.We shall also assume that these numbers have been defined in such a way that
the laws (1)–(5) are satisfied.] (See Problem 1.9–1.10.)
Solved Problems
1.1 For each of the following statements, write the equivalent algebraic expressions: ðaÞ the sum of x and 2,
ðbÞ the sum of a and ÿb, ðcÞ the sum of 5a and 3b, ðd Þ the product of 2a and 3a, ðeÞ the product of 2a and 5b,

f the number which is 4 more than 3 times x, g the number which is 5 less than twice y, ðhÞ the time
required to travel 250 miles at x miles per hour, ði Þ the cost (in cents) of x eggs at 65¢ per dozen.

2
ðaÞ x þ 2 ðd Þ ð2aÞð3aÞ ¼ 6a g 2y ÿ 5
ðbÞ a þ ðÿbÞ ¼ a ÿ b ðeÞ ð2aÞð5bÞ ¼ 10ab ðhÞ 250=x

ðcÞ 5a þ 3b f 3x þ 4 ði Þ 65 x=12
1.2 Let x be the present age of a father. ðaÞ Express the present age of his son, who 2 years ago was one-third his
father’s age. ðbÞ Express the age of his daughter, who 5 years from today will be one-fourth her father’s age.
ðaÞ Two years ago the father’s age was x ÿ 2 and the son’s age was ðx ÿ 2Þ=3.Today the son’s age is
2 þ ðx ÿ 2Þ=3.
1
ðbÞ Five years from today the father’s age will be x þ 5 and his daughter’s age will be ðx þ 5Þ.Today the
4
1
daughter’s age is ðx þ 5Þ ÿ 5.
4
CHAP.1] ELEMENTS OF ALGEBRA 5
1.3 A pair of parentheses may be inserted or removed at will in an algebraic expression if the first parenthesis of
the pair is preceded by a þ sign.If, however, this sign is ÿ, the signs of all terms within the parentheses must
be changed.
1 1 1 3 1
ðaÞ 5a þ 3a ÿ 6a ¼ ð5 þ 3 ÿ 6Þa ¼ 2a ðbÞ a þ b ÿ a þ b ¼ a þ b
2 4 4 4 4

2 2 2 2 2 2 2 2 2 2 2 2 2 2
ðcÞ 13a ÿ b þ ÿ4a þ 3b ÿ 6a ÿ 5b ¼ 13a ÿ b ÿ 4a þ 3b ÿ 6a þ 5b ¼ 3a þ 7b
ðd Þ ð2ab ÿ 3bcÞ ÿ ½5 ÿ ð4ab ÿ 2bcފ ¼ 2ab ÿ 3bc ÿ ½5 ÿ 4ab þ 2bcŠ
¼ 2ab ÿ 3bc ÿ 5 þ 4ab ÿ 2bc ¼ 6ab ÿ 5bc ÿ 5

2
ðeÞ 2x þ 5y ÿ 4 3x ¼ ð2xÞð3xÞ þ 5y ð3xÞ ÿ 4ð3xÞ ¼ 6x þ 15xy ÿ 12x

2
f 5a ÿ 2 g 2x ÿ 3y ðhÞ 3a þ 2a ÿ 1
3a þ 4 5x þ 6y 2a ÿ 3
2 2 3 2
15a ÿ 6a 10x ÿ 15xy 6a þ 4a ÿ 2a
2 2
ðþÞ 20a ÿ 8 ðþÞ 12xy ÿ 18y ðþÞ ÿ9a ÿ 6a þ 3
2 2 2 3 2
15a þ 14a ÿ 8 10x ÿ 3xy ÿ 18y 6a ÿ 5a ÿ 8a þ 3
x2 þ 4x ÿ 2 x2 ÿ 2x ÿ 1
ðiÞ
x ÿ 3 x3 þ x2 ÿ 14x þ 6 ðjÞ x2 þ 3x ÿ 2 x4 þ x3 ÿ 9x2 þ x þ 5
ðÿÞx3 ÿ 3x2 ðÿÞ x4 þ 3x3 ÿ 2x2
4x2 ÿ 14x ÿ2x3 ÿ 7x2 þ x
2 3 2
ðÿÞ 4x ÿ 12x ðÿÞ ÿ2x ÿ 6x þ 4x
2
ÿ2x þ 6 ÿx ÿ 3x þ 5
ðÿÞÿ2x þ 6 ðÿÞ ÿx2 ÿ 3x þ 2
3
3 2 4 3 2
x þ x ÿ 14x þ 6 x þ x ÿ 9x þ x þ 5 3
2 2
¼ x þ 4x ÿ 2 ¼ x ÿ 2x ÿ 1 þ
2 2
x ÿ 3 x þ 3x ÿ 2 x þ 3x ÿ 2
1.4 The problems below involve the following types of factoring:
2 2 2
ab þ ac ÿ ad ¼ aðb þ c ÿ dÞ a 6 2ab þ b ¼ ða 6 bÞ

3 3 2 2 2 2
a þ b ¼ ða þ bÞ a ÿ ab þ b a ÿ b ¼ ða ÿ bÞða þ bÞ

2 3 3 2 2
acx þ ðad þ bcÞx þ bd ¼ ðax þ bÞðcx þ dÞ a ÿ b ¼ ða ÿ bÞ a þ ab þ b

2
ðaÞ 5x ÿ 10y ¼ 5 x ÿ 2y ðeÞ x ÿ 3x ÿ 4 ¼ ðx ÿ 4Þðx þ 1Þ

1 2 1 2 1 2 2
ðbÞ gt ÿ g t ¼ gt t ÿ g f 4x ÿ 12x þ 9 ¼ ð2x ÿ 3Þ
2 2 2

2 2 2
ðcÞ x þ 4x þ 4 ¼ ðx þ 2Þ g 12x þ 7x ÿ 10 ¼ ð4x þ 5Þð3x ÿ 2Þ

2 3 2
ðd Þ x þ 5x þ 4 ¼ ðx þ 1Þðx þ 4Þ ðhÞ x ÿ 8 ¼ ðx ÿ 2Þ x þ 2x þ 4

4 3 2 2 2 2
ðiÞ 2x ÿ 12x þ 10x ¼ 2x x ÿ 6x þ 5 ¼ 2x ðx ÿ 1Þðx ÿ 5Þ
1.5 Simplify.
8 4 · 2 2 4x ÿ 12 4ðx ÿ 3Þ 4ðx ÿ 3Þ 4
ðaÞ ¼ ¼ ðd Þ ¼ ¼ ¼ ÿ
12x þ 20 4 · 3x þ 4 · 5 3x þ 5 15 ÿ 5x 5ð3 ÿ xÞ ÿ5ðx ÿ 3Þ 5
2 2
9x 3x · 3x 3x x ÿ x ÿ 6 ðx þ 2Þðx ÿ 3Þ x ÿ 3
ðbÞ ¼ ¼ ðeÞ ¼ ¼
2
12xy ÿ 15xz 3x · 4y ÿ 3x · 5z 4y ÿ 5z x þ 7x þ 10 ðx þ 2Þðx þ 5Þ x þ 5
2
5x ÿ 10 5ðx ÿ 2Þ 5 6x þ 5x ÿ 6 ð2x þ 3Þð3x ÿ 2Þ 3x ÿ 2
ðcÞ ¼ ¼ f ¼ ¼
2
7x ÿ 14 7ðx ÿ 2Þ 7 2x ÿ 3x ÿ 9 ð2x þ 3Þðx ÿ 3Þ x ÿ 3
2 2
3a ÿ 11a þ 6 4 ÿ 4a ÿ 3a ð3a ÿ 2Þða ÿ 3Þð2 ÿ 3aÞð2 þ aÞ 3a ÿ 2
g · ¼ ¼ ÿ
2 2
a ÿ a ÿ 6 36a ÿ 16 ða ÿ 3Þða þ 2Þ4ð3a þ 2Þð3a ÿ 2Þ 4ð3a þ 2Þ
1.6 Combine as indicated.
2a þ b a ÿ 6b 3ð2a þ bÞ þ 2ða ÿ 6bÞ 8a ÿ 9b
ðaÞ þ ¼ ¼
10 15 30 30
2 3 5 2 · 4 ÿ 3 · 2 þ 5 · x 2 þ 5x
ðbÞ ÿ þ ¼ ¼
x 2x 4 4x 4x