Algebra and Trigonometry with Analytic Geometry, Classic Edition

921 Pages·2009·18.31 MB·English

by  Earl Swokowski

ALGEBRA
QUADRATIC FORMULA SPECIAL PRODUCT FORMULAS SPECIAL FACTORING FORMULAS
If a(cid:1)0,the roots of (cid:1)x(cid:4)y(cid:2)(cid:1)x(cid:6)y(cid:2)(cid:1)x2(cid:6)y2 x2(cid:6)y2(cid:1)(cid:1)x(cid:4)y(cid:2)(cid:1)x(cid:6)y(cid:2)
ax2(cid:4)bx(cid:4)c(cid:1)0are
(cid:1)x(cid:4)y(cid:2)2(cid:1)x2(cid:4)2xy(cid:4)y2 x2(cid:4)2xy(cid:4)y2(cid:1)(cid:1)x(cid:4)y(cid:2)2
(cid:6)b(cid:7) 2b2(cid:6)4ac
x(cid:1) (cid:1)x(cid:6)y(cid:2)2(cid:1)x2(cid:6)2xy(cid:4)y2 x2(cid:6)2xy(cid:4)y2(cid:1)(cid:1)x(cid:6)y(cid:2)2
2a
(cid:1)x(cid:4)y(cid:2)3(cid:1)x3(cid:4)3x2y(cid:4)3xy2(cid:4)y3 x3(cid:6)y3(cid:1)(cid:1)x(cid:6)y(cid:2)(cid:1)x2(cid:4)xy(cid:4)y2(cid:2)
(cid:1)x(cid:6)y(cid:2)3(cid:1)x3(cid:6)3x2y(cid:4)3xy2(cid:6)y3 x3(cid:4)y3(cid:1)(cid:1)x(cid:4)y(cid:2)(cid:1)x2(cid:6)xy(cid:4)y2(cid:2)
EXPONENTS AND RADICALS BINOMIAL THEOREM INEQUALITIES
(cid:6) (cid:7) (cid:6) (cid:7)
aman (cid:1)am(cid:4)n a1/n(cid:1) 2n a n n If a(cid:3)band b(cid:3)c,then a(cid:3)c
(cid:1)x(cid:4)y(cid:2)n(cid:1)xn(cid:4) xn(cid:6)1y(cid:4) xn(cid:6)2y2(cid:4)
(cid:1)am(cid:2)n(cid:1)amn am/n (cid:1) 2n am (cid:6)1 (cid:7) 2 If a(cid:3)b,then a(cid:4)c(cid:3)b(cid:4)c
(cid:6)(cid:1)ab(cid:2)(cid:7)n(cid:1)anbn am/n (cid:1)(cid:1)2n a(cid:2)m (cid:2)(cid:2)(cid:2)(cid:4) n xn(cid:6)kyk(cid:4) (cid:2)(cid:2)(cid:2)(cid:4)yn, If a(cid:3)band c(cid:3)0,then ac(cid:3)bc
(cid:6) (cid:7) k
a n an If a(cid:3)band c(cid:5)0,then ac(cid:5)bc
(cid:1) 2n ab(cid:1) 2n a 2n b n n!
b bn (cid:5) where (cid:1)
k k!(cid:1)n(cid:6)k(cid:2)!
am a 2n a
(cid:1)am(cid:6)n n (cid:1)
an b 2n b
1
a(cid:6)n(cid:1) (cid:4)m2n a(cid:1) 2mna
an
ABSOLUTE VALUE (cid:1)d(cid:3) 0(cid:2) SEQUENCES EXPONENTIALS AND LOGARITHMS
(cid:3)x(cid:3)(cid:5)dif and only if nth term of an arithmetic sequence with first y(cid:1)log x means ay(cid:1)x
a
(cid:6)d(cid:5)x(cid:5)d term a1and common difference d log xy(cid:1)log x(cid:4)log y
a a a
(cid:3)x(cid:3)(cid:3)dif and only if either an(cid:1)a1(cid:4)(cid:1)n(cid:6)1(cid:2)d x
log (cid:1)log x(cid:6)log y
x(cid:3)d or x(cid:5)(cid:6)d Sum S of the first nterms of an arithmetic a y a a
n
sequence
log xr(cid:1)rlog x
MEANS S (cid:1) n (cid:1)a (cid:4)a(cid:2) a a
n 2 1 n alogax(cid:1)x
Arithmetic mean Aofnnumbers n log ax(cid:1)x
or S (cid:1) (cid:8)2a (cid:4)(cid:1)n(cid:6)1(cid:2)d(cid:9) a
a (cid:4)a (cid:4)(cid:2)(cid:2)(cid:2)(cid:4)a n 2 1
A(cid:1) 1 2 n log 1(cid:1)0
a
n
nth term of a geometric sequence with first
log a(cid:1)1
term a and common ratio r a
Geometric mean Gofnnumbers 1 logx(cid:1)log x
a (cid:1)arn(cid:6)1 10
G(cid:1)(cid:1)a1a2(cid:2)(cid:2)(cid:2)an(cid:2)1/n,ak(cid:3)0 n 1 lnx(cid:1)log x
Sum S of the first nterms of a geometric e
n
sequence log u
log u(cid:1) a
a(cid:1)1(cid:6)rn(cid:2) b logab
S (cid:1) 1
n 1(cid:6)r
FORMULAS FROM GEOMETRY
area A perimeterP circumference C volume V curved surface area S altitude h radius r
RIGHT TRIANGLE TRIANGLE EQUILATERAL TRIANGLE
c a c a s s
h h
b
b s
23 23
Pythagorean Theorem:c2(cid:1) a2(cid:4) b2 A (cid:1) 1bh P (cid:1) a (cid:4) b (cid:4) c h (cid:1) s A (cid:1) s2
2 2 4
RECTANGLE PARALLELOGRAM TRAPEZOID
a
w h h
l b b
A (cid:1) lw P (cid:1) 2l(cid:4) 2w A (cid:1) bh A (cid:1) 1(cid:1)a (cid:4) b(cid:2)h
2
CIRCLE CIRCULAR SECTOR CIRCULAR RING
r u s r
r R
A (cid:1) (cid:8)r2 C (cid:1) 2(cid:8)r A (cid:1) 1r2(cid:9) s (cid:1) r(cid:9) A (cid:1) (cid:8)(cid:1)R2(cid:6) r2(cid:2)
2
RECTANGULAR BOX SPHERE RIGHT CIRCULAR CYLINDER
h h
r
w
l
r
V (cid:1) lwh S (cid:1) 2(cid:1)hl(cid:4) lw (cid:4) hw(cid:2) V (cid:1) 4(cid:8)r3 S (cid:1) 4(cid:8)r2 V (cid:1) (cid:8)r2h S (cid:1) 2(cid:8)rh
3
RIGHT CIRCULAR CONE FRUSTUM OF A CONE PRISM
r
h
h h
r
R
V (cid:1) 1(cid:8)r2h S (cid:1) (cid:8)r2r2 (cid:4) h2 V (cid:1) 1(cid:8)h(cid:1)r2(cid:4) rR (cid:4) R2(cid:2) V (cid:1) BhwithBthe area of the base
3 3
ANALYTIC GEOMETRY
DISTANCE FORMULA EQUATION OF A CIRCLE
d(cid:1)P,P(cid:2)(cid:1) 2(cid:1)x (cid:6)x(cid:2)2(cid:4)(cid:1)y (cid:6)y(cid:2)2 (cid:1)x(cid:6)h(cid:2)2(cid:4)(cid:1)y(cid:6)k(cid:2)2(cid:1)r2
1 2 2 1 2 1 y
y
r
(h,k)
P1(x1,y1) P2(x2,y2)
x
x
SLOPE mOF A LINE GRAPH OF A QUADRATIC FUNCTION
y (cid:6)y y(cid:1)ax2,a(cid:3)0 y(cid:1)ax2(cid:4)bx(cid:4)c,a(cid:3)0
y m(cid:1) 2 1
l x2(cid:6)x1 y y
(x,y ) (x ,y ) c
1 1 2 2
x
x (cid:6) b x
2a
POINT-SLOPE FORM OF A LINE CONSTANTS
y y(cid:6)y (cid:1)m(cid:1)x(cid:6)x(cid:2) (cid:8)(cid:10)3.14159
1 1
l e(cid:10)2.71828
(x,y )
1 1 CONVERSIONS
x
1centimeter(cid:10)0.3937inch
SLOPE-INTERCEPT FORM OF A LINE 1meter(cid:10)3.2808feet
1kilometer(cid:10)0.6214mile
y y(cid:1)mx(cid:4)b
1gram(cid:10)0.0353ounce
l
(0,b) 1kilogram(cid:10)2.2046pounds
1liter(cid:10)0.2642gallon
x
1milliliter(cid:10)0.0381fluid ounce
INTERCEPT FORM OF A LINE 1joule(cid:10)0.7376foot-pound
1newton (cid:10)0.2248pound
x y
l y a (cid:4) b (cid:1)1 (cid:1)a(cid:1)0,b(cid:1)0(cid:2) 1lumen(cid:10)0.0015watt
(0,b) 1acre(cid:1)43,560square feet
(a, 0)
x
CLASSIC TWELFTH EDITION
ALGEBRA AND
TRIGONOMETRY
WITH ANALYTIC GEOMETRY
EARL W. SWOKOWSKI
JEFFERY A. COLE
Anoka Ramsey Community College
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Algebra and Trigonometry with © 2010, 2006 Brooks/Cole, Cengage Learning
Analytic Geometry,
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To the memory of Earl W. Swokowski
CONTENTS
Preface viii
CHAPTER 1 Fundamental Concepts of Algebra 1
1.1 Real Numbers 2
1.2 Exponents and Radicals 16
1.3 Algebraic Expressions 27
1.4 Fractional Expressions 40
Chapter 1 Review Exercises 49
Chapter 1 Discussion Exercises 51
CHAPTER 2 Equations and Inequalities 53
2.1 Equations 54
2.2 Applied Problems 61
2.3 Quadratic Equations 73
2.4 Complex Numbers 87
2.5 Other Types of Equations 94
2.6 Inequalities 102
2.7 More on Inequalities 111
Chapter 2 Review Exercises 119
Chapter 2 Discussion Exercises 122
CHAPTER 3 Functions and Graphs 123
3.1 Rectangular Coordinate Systems 124
3.2 Graphs of Equations 130
3.3 Lines 140
3.4 Definition of Function 155
3.5 Graphs of Functions 171
3.6 Quadratic Functions 185
3.7 Operations on Functions 197
Chapter 3 Review Exercises 205
Chapter 3 Discussion Exercises 211
iv
Contents v
CHAPTER 4 Polynomial and Rational Functions 213
4.1 Polynomial Functions of Degree Greater Than 2 214
4.2 Properties of Division 222
4.3 Zeros of Polynomials 229
4.4 Complex and Rational Zeros of Polynomials 241
4.5 Rational Functions 248
4.6 Variation 265
Chapter 4 Review Exercises 272
Chapter 4 Discussion Exercises 275
CHAPTER 5 Inverse, Exponential, and Logarithmic Functions 277
5.1 Inverse Functions 278
5.2 Exponential Functions 287
5.3 The Natural Exponential Function 299
5.4 Logarithmic Functions 308
5.5 Properties of Logarithms 323
5.6 Exponential and Logarithmic Equations 330
Chapter 5 Review Exercises 342
Chapter 5 Discussion Exercises 345
CHAPTER 6 The Trigonometric Functions 347
6.1 Angles 348
6.2 Trigonometric Functions of Angles 358
6.3 Trigonometric Functions of Real Numbers 375
6.4 Values of the Trigonometric Functions 393
6.5 Trigonometric Graphs 400
6.6 Additional Trigonometric Graphs 412
6.7 Applied Problems 420
Chapter 6 Review Exercises 433
Chapter 6 Discussion Exercises 439
vi CONTENTS
CHAPTER 7 Analytic Trigonometry 441
7.1 Verifying Trigonometric Identities 442
7.2 Trigonometric Equations 447
7.3 The Addition and Subtraction Formulas 457
7.4 Multiple-Angle Formulas 467
7.5 Product-to-Sum and Sum-to-Product Formulas 477
7.6 The Inverse Trigonometric Functions 482
Chapter 7 Review Exercises 496
Chapter 7 Discussion Exercises 499
CHAPTER 8 Applications of Trigonometry 501
8.1 The Law of Sines 502
8.2 The Law of Cosines 512
8.3 Vectors 522
8.4 The Dot Product 536
8.5 Trigonometric Form for Complex Numbers 546
8.6 De Moivre’s Theorem and nth Roots of
Complex Numbers 552
Chapter 8 Review Exercises 557
Chapter 8 Discussion Exercises 560
CHAPTER 9 Systems of Equations and Inequalities 563
9.1 Systems of Equations 564
9.2 Systems of Linear Equations in Two Variables 573
9.3 Systems of Inequalities 582
9.4 Linear Programming 590
9.5 Systems of Linear Equations in More Than Two
Variables 598
9.6 The Algebra of Matrices 614
9.7 The Inverse of a Matrix 623
9.8 Determinants 628
9.9 Properties of Determinants 634
9.10 Partial Fractions 642
Chapter 9 Review Exercises 648
Chapter 9 Discussion Exercises 651