GEOMETRIC FORMULAS Formulas for area A, perimeter P, circumference C, volume V

1078 Pages·2007·31.08 MB·English

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EXPONENTS AND RADICALS GEOMETRIC FORMULAS
xm
xmxn(cid:1)xm(cid:2)n (cid:3)(cid:3)(cid:1)xm(cid:5)n Formulas for area A,perimeter P,circumference C,volume V:
xn
xmn (cid:1)xmn x(cid:5)n (cid:1)(cid:3)1(cid:3) Rectangle Box
1 2 xn
xyn (cid:1)xnyn (cid:3)x(cid:3) n (cid:1)(cid:3)x(cid:3)n A(cid:1)l„ V(cid:1)l„h
1 2 ayb yn P(cid:1)2l (cid:2)2„
x1(cid:3)n(cid:1)(cid:1)nx(cid:2) xm(cid:3)n(cid:1)(cid:1)nx(cid:2)(cid:2)m(cid:1) (cid:1)nx(cid:2) m
Q R
(cid:1)nx(cid:2)y (cid:1)(cid:1)nx(cid:2)(cid:1)ny(cid:2) (cid:4)n(cid:3)(cid:5)x(cid:3) (cid:1)(cid:3)(cid:1)n(cid:3)x(cid:2) „ h
(cid:1)m(cid:1)(cid:2)n(cid:2)x(cid:2)(cid:1)(cid:1)n(cid:1)(cid:2)m(cid:2)x(cid:2)(cid:1)(cid:1)mnx(cid:2) y (cid:1)n(cid:2)y
„
l l
SPECIAL PRODUCTS
Triangle Pyramid
1x (cid:2)y22(cid:1)x2 (cid:2)2xy (cid:2)y2 A(cid:1)(cid:3)1(cid:3)bh V(cid:1)(cid:3)1(cid:3)ha2
2 3
x (cid:5)y2(cid:1)x2 (cid:5)2xy (cid:2)y2
1 2
x (cid:2)y3(cid:1)x3 (cid:2)3x2y (cid:2)3xy2 (cid:2)y3
1 2
x (cid:5)y3(cid:1)x3 (cid:5)3x2y (cid:2)3xy2 (cid:5)y3 h
1 2
h
a
FACTORING FORMULAS b a
x2 (cid:5)y2(cid:1) x (cid:2)y x (cid:5)y
1 21 2 Circle Sphere
x2 (cid:2)2xy (cid:2)y2(cid:1)1x (cid:2)y22 A(cid:1)(cid:4)r2 V(cid:1)(cid:3)4(cid:3)(cid:4)r3
3
x2 (cid:5)2xy (cid:2)y2(cid:1) x (cid:5)y2 C(cid:1)2(cid:4)r A(cid:1)4(cid:4)r2
1 2
x3 (cid:2)y3(cid:1) x (cid:2)y x2 (cid:5)xy (cid:2)y2
1 21 2
x3 (cid:5)y3(cid:1) x (cid:5)y x2 (cid:2)xy (cid:2)y2 r r
1 21 2
QUADRATIC FORMULA
If ax2 (cid:2)bx (cid:2)c(cid:1)0,then Cylinder Cone
x(cid:1)(cid:3)(cid:5)b(cid:8)(cid:1)(cid:3)b(cid:2)(cid:2)2(cid:5)(cid:2)(cid:2)4a(cid:2)c(cid:2) V(cid:1)(cid:4)r2h V(cid:1)(cid:3)13(cid:3)(cid:4)r2h
2a
r
INEQUALITIES AND ABSOLUTE VALUE
h h
If a(cid:6)band b(cid:6)c,then a(cid:6)c. r
If a(cid:6)b,then a(cid:2)c(cid:6)b(cid:2)c.
If a(cid:6)band c(cid:7)0,then ca(cid:6)cb.
If a(cid:6)band c(cid:6)0,then ca(cid:7)cb. HERON’S FORMULA
If a(cid:7)0,then
B
⏐x⏐(cid:1)a means x(cid:1)a or x(cid:1)(cid:5)a. A(cid:1)(cid:1)(cid:2)s s(cid:2)(cid:2)(cid:5)(cid:2)a(cid:2)s(cid:2)(cid:5)(cid:2)(cid:2)b(cid:2)s(cid:2)(cid:5)(cid:2)(cid:2)c(cid:2)
1 21 21 2 c a
⏐x⏐(cid:6)a means (cid:5)a(cid:6)x(cid:6)a. a(cid:2)b(cid:2)c
where s(cid:1)(cid:3)(cid:3)
⏐x⏐(cid:7)a means x(cid:7)a or x(cid:6)(cid:5)a. 2 A C
b
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DISTANCE AND MIDPOINT FORMULAS GRAPHS OF FUNCTIONS
Distancebetween P x ,y and P x ,y : Linear functions: f x (cid:1)mx(cid:2)b
11 1 12 21 2 22 1 2
d(cid:1)(cid:1)(cid:2)x(cid:2)(cid:5)(cid:2)(cid:2)x(cid:2)(cid:2)2(cid:2)(cid:2)(cid:2)(cid:2)y(cid:2)(cid:5)(cid:2)y(cid:2)(cid:2)2 y y
1 2 12 1 2 12
x (cid:2)x. y (cid:2)y.
Midpointof PP: (cid:3)1 (cid:3)2, (cid:3)1 (cid:3)2 b
1 2
a 2 2 b
b
LINES x x
Slope of linethrough m(cid:1)(cid:3)y2(cid:5)(cid:3)y1 Ï=b Ï=mx+b
x (cid:5)x
P x ,y and P x ,y 2 1
11 1 12 21 2 22 Power functions: f x (cid:1)xn
1 2
Point-slope equationof line y(cid:5)y (cid:1)m x(cid:5)x
1 1 12 y y
through P x ,y with slopem
11 1 12
Slope-intercept equationof y(cid:1)mx(cid:2)b
line with slopemand y-intercept b x
Two-intercept equationof line (cid:3)x(cid:3)(cid:2)(cid:3)y(cid:3)(cid:1)1 x
with x-intercept aand y-intercept b a b Ï=≈ Ï=x£
LOGARITHMS Root functions: f x (cid:1)(cid:1)nx(cid:6)
1 2
y(cid:1)log x means ay(cid:1)x y y
a
log ax(cid:1)x alogax(cid:1)x
a
loga1(cid:1)0 logaa(cid:1)1 x x
logx(cid:1)log x lnx(cid:1)log x
10 e
log xy(cid:1)log x (cid:2)log y log (cid:3)x(cid:3) (cid:1)log x (cid:5)log y Ï=œx∑ Ï=£œx∑
a a a aayb a a
log xb(cid:1)blog x log x(cid:1) logax Reciprocal functions: f x (cid:1)1/xn
a a b log b 1 2
a
y y
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
y y
x x
y=a˛ y=a˛
a>1 0<a<1
1 1
Ï= Ï=
1 1 x ≈
0 x 0 x Absolute value function Greatest integer function
y y
y y=loga x y y=loga x
a>1 0<a<1
1
1 x
0 x 0 x x
1 1
Ï=|x| Ï=“x‘
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COMPLEX NUMBERS POLAR COORDINATES
For the complex number z(cid:1)a(cid:2)bi y
x(cid:1)rcos(cid:9)
the conjugateis z(cid:1)a(cid:5)bi P(x, y)
the modulusis ⏐z⏐(cid:1)(cid:1)a(cid:2)2(cid:2)(cid:2)(cid:2)b(cid:6)2 P(r, ¨) y(cid:1)rsin(cid:9)
the argumentis (cid:9),where tan (cid:9)(cid:1)b/a r y r2(cid:1)x2(cid:2)y2
y
Im ¨ tan(cid:9)(cid:1)(cid:3)(cid:3)
x
bi a+bi 0 x x
|z|
SUMS OF POWERS OF INTEGERS
¨
0 a Re (cid:7)n 1 (cid:1)n (cid:7)n k(cid:1)(cid:3)n1n(cid:2)(cid:3)12
2
k(cid:1)1 k(cid:1)1
Polar form of a complex number
For z(cid:1)a(cid:2)bi,the polar formis (cid:7)n k2(cid:1)(cid:3)n1n(cid:2)162(cid:3)12n(cid:2)12 (cid:7)n k3(cid:1)(cid:3)n21n4(cid:2)(cid:3)122
k(cid:1)1 k(cid:1)1
z(cid:1)r cos(cid:9)(cid:2)isin(cid:9)
1 2
THE DERIVATIVE
where r(cid:1)⏐z⏐is the modulus of zand (cid:9)is the argument of z
The average rate of changeof fbetween aand bis
DeMoivre’s Theorem
f b (cid:5)f a
zn(cid:1)(cid:8)r1cos(cid:9)(cid:2)isin(cid:9)2(cid:9)n(cid:1)rn1cosn(cid:9)(cid:2)isinn(cid:9)2 (cid:3)1 b2 (cid:5)(cid:3)a1 2
(cid:1)nz(cid:2)(cid:1)(cid:8)r cos(cid:9)(cid:2)isin(cid:9)(cid:9)1(cid:3)n
1 2 The derivativeof fat ais
(cid:1)r1(cid:3)n cos(cid:3)(cid:9)(cid:2)(cid:3)2k(cid:4)(cid:2)isin(cid:3)(cid:9)(cid:2)(cid:3)2k(cid:4) f x (cid:5)f a
a n n b f(cid:10)a (cid:1)lim(cid:3)1 2 (cid:3)1 2
1 2 x(cid:1)a x (cid:5)a
where k(cid:1)0,1,2,...,n(cid:5)1
f a(cid:2)h (cid:5)f a
f(cid:10)a (cid:1)lim(cid:3)1 (cid:3)2 1 2
ROTATION OF AXES 1 2 h(cid:1)0 h
y
AREA UNDER THE GRAPH OF f
Y P(x, y)
P(X, Y)
The area under the graph offon the interval [a,b] is the limit
X of the sum of the areas of approximating rectangles
n
A(cid:1)lim (cid:7)f x (cid:11)x
ƒ n(cid:1)(cid:1)k(cid:1)1 1 k2
0 x where
b(cid:5)a
(cid:11)x(cid:1)(cid:3)(cid:3)
Rotation of axes formulas n
x(cid:1)Xcos(cid:12)(cid:5)Ysin(cid:12) x (cid:1)a(cid:2)k (cid:11)x
k
y(cid:1)Xsin(cid:12)(cid:2)Ycos(cid:12) y
Îx
Angle-of-rotation formula for conic sections
A(cid:5)C
cot2(cid:12)(cid:1)(cid:3)(cid:3)
B
f(x)
k
0 a x⁄ x¤ x‹ x x b x
k-1 k
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PRECALCULUS
Mathematics for Calculus
F I F T H E D I T I O N
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PRECALCULUS
Mathematics for Calculus
F I F T H E D I T I O N
James Stewart
McMaster University
Lothar Redlin
The Pennsylvania State University
Saleem Watson
California State University, Long Beach
Australia • Brazil • Canada • Mexico • Singapore • Spain
United Kingdom • United States
00-W4189-FM 1/5/07 8:50 AM Page iv
Precalculus Enhanced Review Edition
Stewart,Redlin,Watson
Publisher:Bob Pirtle Text Designer:John Edeen
Assistant Editor:Stacy Green Photo Researcher:Stephen Forsling
Editorial Assistant:Beth Rodio Copy Editor:Luana Richards
Technology Project Manager:Sam Subity Art Editor:Martha Emry
Senior Marketing Manager:Joe Rogove Illustrator:Jade Myers,Matrix
Marketing Communications Manager:Darlene Amidon-Brent Cover Designer:Roy E. Neuhaus
Senior Content Project Manager:Janet Hill Cover Image:Bill Ralph
Senior Art Director:Vernon Boes Cover Printer:R R Donnelley/Willard
Senior Print/Media Buyer:Judy Inouye Compositor:Newgen
Production Service:Martha Emry Printer:R R Donnelley/Willard
©2007 Thomson Brooks/Cole,a part of The Thomson Corpora- Thomson Arts and Sciences
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Library of Congress Control Number:2006939524
Student Edition
ISBN-13:978-0-495-39276-7
ISBN-10:0-495-39276-6
00-W4189-FM 12/13/06 9:46 AM Page v
To our students,
from whom we have learned so much.
00-W4189-FM 12/13/06 9:46 AM Page vi
About the Cover
The art on the cover was created by Bill Ralph, a mathe-
matician who uses modern mathematics to produce vi-
sual representations of “dynamical systems.” Examples of
dynamical systems in nature include the weather, blood
pressure, the motions of the planets, and other phenom-
ena that involve continual change. Such systems, which
tend to be unpredictable and even chaotic at times, are
modeled mathematically using the concepts of composi-
tion and iteration of functions (see Section 2.7 and the
Discovery Project on pages 223–224). The basic idea is to
start with a particular function and evaluate it at some
point in its domain, yielding a new number. The function
is then evaluated at the new number. Repeating this pro-
cess produces a sequence of numbers called iterates of
the function. The original domain is “painted” by assign-
ing a color to each starting point; the color is determined
by certain properties of its sequence of iterates and the
mathematical concept of “dimension.” The result is a
picture that reveals the complex patterns of the dynami-
cal system. In a sense, these pictures allow us to look,
through the lens of mathematics, at exotic little uni-
verses that have never been seen before.
Professor Ralph teaches at Brock University in Canada.
He can be contacted by e-mail at
[email protected].
About the Authors
James Stewartwas educated at the Lothar Redlin grew up on Vancouver Saleem Watson received his
University of Toronto and Stanford Island, received a Bachelor of Bachelor of Science degree from
University, did research at the Science degree from the University Andrews University in Michigan. He
University of London, and now of Victoria, and a Ph.D. from did graduate studies at Dalhousie
teaches at McMaster University. His McMaster University in 1978. He University and McMaster University,
research field is harmonic analysis. subsequently did research and where he received his Ph.D. in 1978.
taught at the University of He subsequently did research at the
He is the author of a best-selling
Washington, the University of Mathematics Institute of the
calculus textbook series published by
Waterloo, and California State University of Warsaw in Poland. He
Brooks/Cole, including Calculus, 5th
University, Long Beach. also taught at The Pennsylvania
Ed., Calculus: Early Transcendentals,
State University.
5th Ed., and Calculus: Concepts and He is currently Professor of
Contexts,3rd Ed., as well as a series Mathematics at The Pennsylvania He is currently Professor of
of high-school mathematics State University, Abington College. Mathematics at California State
textbooks. His research field is topology. University, Long Beach. His research
field is functional analysis.
The authors have also published College Algebra, Fourth Edition(Brooks/Cole, 2004), Algebra and Trigonometry, Sec-
ond Edition(Brooks/Cole, 2007), and Trigonometry(Brooks/Cole, 2003).
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Contents
Preface xiii
To the Student xxi
Calculators and Calculations xxii
1 Fundamentals 1
■ Chapter Overview 1
1.1 Real Numbers 2
1.2 Exponents and Radicals 12
1.3 Algebraic Expressions 24
● DISCOVERY PROJECT Visualizing a Formula 34
1.4 Rational Expressions 35
1.5 Equations 44
1.6 Modeling with Equations 58
● DISCOVERY PROJECT Equations through the Ages 75
1.7 Inequalities 76
1.8 Coordinate Geometry 87
1.9 Graphing Calculators; Solving Equations and Inequalities
Graphically 101
1.10 Lines 111
1.11 Modeling Variation 123
Chapter 1 Review 130
Chapter 1 Test 135
■ FOCUS ON PROBLEM SOLVING General Principles 138
2 Functions 146
■ Chapter Overview 147
2.1 What is a Function? 148
2.2 Graphs of Functions 158
● DISCOVERY PROJECT Relations and Functions 171
vii