Schaum's outlines: precalculus

409 Pages·2011·6.919 MB·English

by  Safier, Fred

SCHAUM'S
outlines
Precalculus
Second Edition
Fred Safier
Professor of Mathematics
City College of San Francisco
Schaum’s Outline Series
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DOI: 10.1036/0071508643
Preface to the
Second Edition
This edition has been expanded by material on average rate of change, price/demand, polar form of complex
numbers, conic sections in polar coordinates, and the algebra of the dot product. An entire chapter (Chapter 45)
is included as an introduction to differential calculus, which now appears in many precalculus texts. More
than 30 solved and more than 110 supplementary problems have been added.
Thanks are due to Anya Kozorez and her staff at McGraw-Hill, and to Madhu Bhardwaj and her staff at
International Typesetting and Composition. Also, the author would like to thank the users who sent him
(mercifully few) corrections, in particular D. Mehaffey and B. DeRoes.
Most of all he owes thanks once again to his wife Gitta, whose careful checking eliminated
numerous errors. Any further errors that users spot would be gratefully received at [email protected] or
[email protected].
Fred Safier
iii
Copyright © 2009, 1998 by The McGraw-Hill Companies, Inc. Click here for terms of use.
Preface to the
First Edition
Acourse in precalculus is designed to prepare college students for the level of algebraic skills and knowl-
edge that is expected in a calculus class. Such courses, standard at two-year and four-year colleges, review
the material of algebra and trigonometry, emphasizing those topics with which familiarity is assumed in
calculus. Key unifying concepts are those of functions and their graphs.
The present book is designed as a supplement to college courses in precalculus. The material is divided
into forty-four chapters, and covers basic algebraic operations, equations, and inequalities, functions and
graphs, and standard elementary functions including polynomial, rational, exponential, and logarithmic func-
tions. Trigonometry is covered in Chapters 20 through 29, and the emphasis is on trigonometric functions as
defined in terms of the unit circle. The course concludes with matrices, determinants, systems of equations,
analytic geometry of conic sections, and discrete mathematics.
Each chapter starts with a summary of the basic definitions, principles, and theorems, accompanied by
elementary examples. The heart of the chapter consists of solved problems, which present the material in
logical order and take the student through the development of the subject. The chapter concludes with
supplementary problems with answers. These provide drill on the material and develop some ideas further.
The author would like to thank his friends and colleagues, especially F. Cerrato, G. Ling, and J. Morell,
for useful discussions. Thanks are also due to the staff of McGraw-Hill and to the reviewer of the text for their
invaluable help. Most of all he owes thanks to his wife Gitta, whose careful line-by-line checking of the
manuscript eliminated numerous errors. Any errors that remain are entirely his responsibility, and students
and teachers who find errors are invited to send him email at [email protected].
iv
Copyright © 2009, 1998 by The McGraw-Hill Companies, Inc. Click here for terms of use.
For more information about this title, click here
Contents
CHAPTER 1 Preliminaries 1
CHAPTER 2 Polynomials 7
CHAPTER 3 Exponents 15
CHAPTER 4 Rational and Radical Expressions 20
CHAPTER 5 Linear and Nonlinear Equations 29
CHAPTER 6 Linear and Nonlinear Inequalities 41
CHAPTER 7 Absolute Value in Equations and Inequalities 49
CHAPTER 8 Analytic Geometry 54
CHAPTER 9 Functions 68
CHAPTER 10 Linear Functions 79
CHAPTER 11 Transformations and Graphs 87
CHAPTER 12 Quadratic Functions 95
CHAPTER 13 Algebra of Functions;Inverse Functions 104
CHAPTER 14 Polynomial Functions 114
CHAPTER 15 Rational Functions 132
CHAPTER 16 Algebraic Functions;Variation 146
CHAPTER 17 Exponential Functions 154
CHAPTER 18 Logarithmic Functions 162
CHAPTER 19 Exponential and Logarithmic Equations 168
CHAPTER 20 Trigonometric Functions 176
CHAPTER 21 Graphs of Trigonometric Functions 187
CHAPTER 22 Angles 197
v
vi Contents
CHAPTER 23 Trigonometric Identities and Equations 211
CHAPTER 24 Sum,Difference,Multiple,and Half-Angle Formulas 220
CHAPTER 25 Inverse Trigonometric Functions 230
CHAPTER 26 Triangles 240
CHAPTER 27 Vectors 252
CHAPTER 28 Polar Coordinates;Parametric Equations 261
CHAPTER 29 Trigonometric Form of Complex Numbers 270
CHAPTER 30 Systems of Linear Equations 279
CHAPTER 31 Gaussian and Gauss-Jordan Elimination 287
CHAPTER 32 Partial Fraction Decomposition 294
CHAPTER 33 Nonlinear Systems of Equations 302
CHAPTER 34 Introduction to Matrix Algebra 309
CHAPTER 35 Matrix Multiplication and Inverses 313
CHAPTER 36 Determinants and Cramer’s Rule 322
CHAPTER 37 Loci;Parabolas 330
CHAPTER 38 Ellipses and Hyperbolas 337
CHAPTER 39 Rotation of Axes 349
CHAPTER 40 Conic Sections 356
CHAPTER 41 Sequences and Series 362
CHAPTER 42 The Principle of Mathematical Induction 368
CHAPTER 43 Special Sequences and Series 374
CHAPTER 44 Binomial Theorem 381
CHAPTER 45 Limits,Continuity,Derivatives 387
Index 399
CHAPTER 1
Preliminaries
The Sets of Numbers Used in Algebra
The sets of numbers used in algebraare, in general, subsets of R,the set of real numbers.
Natural Numbers N
The counting numbers, e.g., 1, 2, 3, 4, ...
Integers Z
The counting numbers, together with their opposites and 0, e.g., 0, 1, 2, 3, ... (cid:3)1, (cid:3)2, (cid:3)3, ...
Rational Numbers Q
The set of all numbers that can be written as quotients a/b,b(cid:4)0,aand bintegers, e.g., 3/17, 10/3, (cid:3)5.13, ...
Irrational Numbers H
All real numbers that are not rational numbers, e.g., p, 2, 35,(cid:3)π/3,...
EXAMPLE 1.1 The number (cid:3)5 is a member of the sets Z,Q,R.The number 156.73 is a member of the sets Q,R.The
number 5pis a member of the sets H,R.
Axioms for the Real Number System
There are two fundamental operations, addition and multiplication, that have the following properties (a,b,c
arbitrary real numbers):
Closure Laws
The sum a(cid:5)band the product a(cid:6)bor abare unique real numbers.
Commutative Laws
a(cid:5)b(cid:7)b(cid:5)a:order does not matter in addition.
ab(cid:7)ba:order does not matter in multiplication.
Associative Laws
a(cid:5)(b(cid:5)c)(cid:7)(a(cid:5)b)(cid:5)c:grouping does not matter in repeated addition.
a(bc)(cid:7)(ab)c:grouping does not matter in repeated multiplication.
Note(removing parentheses): Since a(cid:5)(b(cid:5)c)(cid:7)(a(cid:5)b)(cid:5)c,a(cid:5)b(cid:5)ccan be written to mean either
quantity
Also, since a(bc)(cid:7)(ab)c,abccan be written to mean either quantity.
Distributive Laws
a(b(cid:5)c)(cid:7)ab(cid:5)ac;also (a(cid:5)b)c(cid:7)ac(cid:5)bc:multiplication is distributive over addition.
Identity Laws
There is a unique number 0 with the property that 0 (cid:5)a(cid:7)a(cid:5)0 (cid:7)a.
There is a unique number 1 with the property that 1 (cid:6)a(cid:7)a(cid:6)1 (cid:7)a.
1
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2 CHAPTER 1 Preliminaries
Inverse Laws
For any real number a, there is a real number (cid:3)asuch that a(cid:5)((cid:3)a) (cid:7)((cid:3)a) (cid:5)a(cid:7)0.
For any nonzero real number a, there is a real number a(cid:3)1such that aa(cid:3)1(cid:7)a(cid:3)1a(cid:7)1.
(cid:3)ais called the additive inverse, or negative, of a.
a(cid:3)1is called the multiplicative inverse, or reciprocal, of a.
EXAMPLE 1.2 Associative and commutative laws: Simplify (3 (cid:5)x) (cid:5)5.
(3 (cid:5)x) (cid:5)5 (cid:7)(x(cid:5)3) (cid:5)5 Commutative law
(cid:7)x(cid:5)(3 (cid:5)5) Associative law
(cid:7)x(cid:5)8
EXAMPLE 1.3 FOIL (First Outer Inner Last). Show that (a(cid:5)b) (c(cid:5)d)(cid:7)ac(cid:5)ad(cid:5)bc(cid:5)bd.
(a(cid:5)b) (c(cid:5)d)(cid:7)a(c(cid:5)d)(cid:5)b(c(cid:5)d) by the second form of the distributive law
(cid:7)ac(cid:5)ad(cid:5)bc(cid:5)bd by the first form of the distributive law
Zero Factor Laws
1. For every real number a,a(cid:6)0 (cid:7)0.
2. If ab(cid:7)0, then either a(cid:7)0 or b(cid:7)0.
Laws for Negatives
1. (cid:3)((cid:3)a) (cid:7)a
2. ((cid:3)a)((cid:3)b) (cid:7)ab
3. (cid:3)ab(cid:7)((cid:3)a)b(cid:7)a((cid:3)b)(cid:7)(cid:3)((cid:3)a)((cid:3)b)
4. ((cid:3)l)a(cid:7)(cid:3)a
Subtraction and Division
Definition of Subtraction: a(cid:3)b(cid:7)a(cid:5)((cid:3)b)
a 1
Definition of Division: (cid:7)a(cid:8)b(cid:7)a(cid:6)b(cid:3)1.Thus, b(cid:3)1(cid:7)1 (cid:6)b(cid:3)1(cid:7)1 (cid:8)b(cid:7) .
b b
Note:Since 0 has no multiplicative inverse, a(cid:8)0 is not defined.
Laws for Quotients
a (cid:3)a a (cid:3)a
(cid:3) (cid:7) (cid:7) (cid:7)(cid:3)
1.
b b (cid:3)b (cid:3)b
(cid:3)a a
(cid:7)
2.
(cid:3)b b
a c
3. (cid:7) if and only if ad(cid:7)bc.
b d
a ka
4. (cid:7) , for kany nonzero real number. (Fundamental principle of fractions)
b kb
Ordering Properties
The positive real numbers, designated by R(cid:2),are a subset of the real numbers with the following properties:
1. If aand bare in R(cid:2),then so are a(cid:5)band ab.
2. For every real number a,either ais in R(cid:2),or ais zero, or (cid:3)ais in R(cid:2).
If ais in R(cid:2), ais called positive; if (cid:3)ais in R(cid:2), ais called negative.
3
CHAPTER 1 Preliminaries
The number a is less than b,written a(cid:9)b,if b(cid:3)ais positive. Then b is greater than a,written b(cid:10)a.If a
is either less than or equal to b,this is written a(cid:11)b.Then bis greater than or equal to a,written b(cid:12)a.
EXAMPLE 1.4 3 (cid:9)5 because 5 (cid:3)3 (cid:7)2 is positive. (cid:3)5 (cid:9)3 because 3 (cid:3)((cid:3)5) (cid:7)8 is positive.
The following may be deduced from these definitions:
1. a(cid:10)0 if and only if ais positive.
2. If a(cid:4)0, then a2(cid:10)0.
3. If a(cid:9)b,then a(cid:5)c(cid:9)b(cid:5)c.
{
4. If a(cid:9)b,then ac(cid:9)bc if c(cid:10)0
ac(cid:10)bc if c(cid:9)0
5. For any real number a,either a(cid:10)0, or a(cid:7)0, or a(cid:9)0.
6. If a(cid:9)band b(cid:9)c,then a(cid:9)c.
The Real Number Line
Real numbers may be represented by points on a line l such that to each real number a there corresponds
exactly one point on l, and conversely.
EXAMPLE 1.5 Indicate the set {3, (cid:3)5, 0, 2/3, 5, (cid:3)1.5, (cid:3)p} on a real number line.
Figure 1-1
Absolute Value of a Number
The absolute value of a real number a,written |a|,is defined as follows:
⎧ a if a(cid:12)0
|a|(cid:7)⎨
⎩(cid:3)a if a(cid:9)0
Complex Numbers
Not all numbers are real numbers. The set C of numbers of the form a (cid:5) bi, where a and b are real and
i2(cid:7)(cid:3)1, is called the complex numbers. Since every real number xcan be written as x(cid:5)0i, it follows that
every real number is also a complex number.
1 3
EXAMPLE 1.6 3(cid:5) (cid:3)4(cid:7)3(cid:5)2i,(cid:3)5i,2πi, + i are examples of nonreal complex numbers.
2 2
Order of Operations
In expressions involving combinations of operations, the following order is observed:
1. Perform operations within grouping symbols first. If grouping symbols are nested inside other grouping
symbols, proceed from the innermost outward.
2. Apply exponents before performing multiplications and divisions, unless grouping symbols indicate
otherwise.
3. Perform multiplications and divisions, in order from left to right, before performing additions and sub-
tractions (also from left to right), unless operation symbols indicate otherwise.