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

432 Pages·2003·5.616 MB·English

by  Frank Ayres, Philip A Schmidt

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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DOI: 10.1036/0071425888
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
thosecurriculainthistext.However,asubstantialnumberofimportantchangeshave
beenmadeinthisedition.First,thereismoreofanemphasisnowontopicsindiscrete
mathematics. Second, the graphing calculator is introduced as an important problem-
solvingtool.Third,materialrelatedtomanualandtabularcomputationsoflogarithms
has been removed, and replaced with material that is calculator-based. Fourth, all
materialrelatedtotheconceptsoflocushasbeenmodernized.Fifth,tablesandgraphs
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
alsomustthankDr.MartiGarlett,DeanoftheTeachersCollegeatWesternGovernors
University, for her professional support as I struggled to meet deadlines while
beginninganewpositionattheUniversity.IthankMaureenWalkerforherhandling
ofthemanuscript andproofs.Andfinally,Ithankmywife, Dr.JanZlotnikSchmidt,
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. SCHMIDT
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,
5·60þ35¼335minutes.
Inalgebrasomeofthenumbersusedmaybeknownbutothersareeitherunknownornotspecified;
thatis,theyarerepresentedbyletters.Forexample,converthhoursandmminutesintominutes.Thisis
doneinpreciselythesamemannerasintheparagraphabovebymultiplyinghby60andaddingm;thus,
h·60þm¼60hþm. We call 60hþm an algebraic expression. (See Problem 1.1.)
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 5·60þ35 and
2·60þ35isð5þ2Þ·60þ2·35;similarly,thesumofh·60þmandk·60þmisðhþkÞ·60þ2m.(See
Problems 1.2–1.6.)
POSITIVEINTEGRALEXPONENTS. Ifaisanynumberandnisanypositiveinteger,theproduct
ofthenfactorsa·a·a···aisdenotedbyan.Todistinguishbetweentheletters,aiscalledthebaseandn
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 ¼amþn
(2) ðamÞn¼amn
am am 1
(3) ¼am(cid:5)n; a6¼0; m>n; ¼ ; a6¼0; m<n
an an an(cid:5)m
(4) ða·bÞn¼anbn
(cid:1) (cid:2)
a n an
(5) ¼ ; b6¼0
b bn
(See Problem 1.7.)
LETnBEAPOSITIVEINTEGERandaandbbetwonumberssuchthatbn¼a;thenbiscalledannth
root of a. Every number a6¼0 has exactly n distinct nth roots.
Ifaisimaginary,allofitsnthrootsareimaginary;thiscasewillbeexcludedhereandtreatedlater.
(See Chapter 35.)
3
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4 ELEMENTSOFALGEBRA [CHAP.1
Ifais(cid:3)realand(cid:4) nisodd,thenexactlyoneofthenthrootsofaisreal.Forexample,2istherealcube
root of 8, 23¼8 , and (cid:5)3 is the real fifth root of (cid:5)243½ð(cid:5)3Þ5¼(cid:5)243(cid:7).
Ifaisrealandniseven,thenthereareexactlytworealnthrootsofawhena>0,butnorealnth
rootsofawhena<0.Forexample,þ3and(cid:5)3arethesquarerootsof9;þ2and(cid:5)2aretherealsixth
roots of 64.
THEPRINCIPALnthROOTOFaisthepositiverealnthrootofawhenaisppffioffisitiveandtherealnth
rootofa,ifany,whenaisnegative.Theprincipalnthrootofaisdenotedby na,calledaradical.The
integer n is called the index of the radical and a is called the radicand. For example,
pffiffi pffiffiffi pffiffiffiffiffiffiffi
9¼3 664¼2 5(cid:5)243¼(cid:5)3
(See Problem 1.8.)
ZERO, FRACTIONAL, AND NEGATIVE EXPONENTS. When s is a positive integer, r is any
integer, and p is any rational number, the following extend the definition of an in such a way that the
laws (1)-(5) are satisfied when n is any rational number.
DEFINITIONS EXAMPLES
(cid:6) (cid:7)
(6) a0¼1;a6¼0 20¼1; 1 0¼1;ð(cid:5)8Þ0¼1
(7) ar=s¼psffiaffiffir¼(cid:3)psffiaffi(cid:4)r 31=2¼p1ffi3ffi0;0ð64Þ5=6¼(cid:6)p6ffi6ffi4ffi(cid:7)5¼25¼32;3(cid:5)2=1 ¼3(cid:5)2 ¼1
9
pffiffi
(8) a(cid:5)p¼1=ap;a6¼0 2(cid:5)1¼1;3(cid:5)1=2¼1 3
2
ffiffi
p
[NOTE: Withoutattemptingtodefinethem,weshallassumetheexistenceofnumberssuchasa 2;ap;...;inwhich
theexponentisirrational.Weshallalsoassumethatthesenumbershavebeendefinedinsuchawaythat
thelaws(1)–(5)aresatisfied.](SeeProblem1.9–1.10.)
Solved Problems
1.1 For each of the following statements, write the equivalent algebraic expressions: ðaÞ the sum of x and 2,
(cid:3)ðbÞ(cid:4)thesumofaand(cid:5)b,ðcÞthesumof5aand3b,ð(cid:3)d(cid:4)Þtheproductof2aand3a,ðeÞtheproductof2aand5b,
f the number which is 4 more than3 times x, g the number which is 5 less thantwice y, ðhÞ the time
requiredtotravel250milesatxmilesperhour,ðiÞthecost(incents)ofxeggsat65¢perdozen.
(cid:3) (cid:4)
ðaÞ xþ2 ðdÞ ð2aÞð3aÞ¼6a2 g 2y(cid:5)5
ðbÞ aþð(cid:5)bÞ¼a(cid:5)b ðeÞ ð2aÞð5bÞ¼10ab ðhÞ 250=x
(cid:3) (cid:4) (cid:3) (cid:4)
ðcÞ 5aþ3b f 3xþ4 ðiÞ 65x=12
1.2 Letxbethepresentageofafather.ðaÞExpressthepresentageofhisson,who2yearsagowasone-thirdhis
father’sage.ðbÞExpresstheageofhisdaughter,who5yearsfromtodaywillbeone-fourthherfather’sage.
ðaÞ Two years ago the father’s age was x(cid:5)2 and the son’s age was ðx(cid:5)2Þ=3. Today the son’s age is
2þðx(cid:5)2Þ=3.
ðbÞ Fiveyearsfromtodaythefather’sagewillbexþ5andhisdaughter’sagewillbe1ðxþ5Þ.Todaythe
4
daughter’sageis1ðxþ5Þ(cid:5)5.
4