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97909_FrontEP_FrontEP_pF2_RefPage1-2_97909_FrontEP_FrontEP_pF2_RefPage1-2 9/24/10 5:36 PM Page 1 REFERENCE PAGE 1 ALGEBRA GEOMETRY e Arithmetic Operations Geometric Formulas c n ere a(cid:2)b(cid:4)c(cid:3)(cid:2)ab(cid:4)ac a (cid:4) c (cid:2) ad(cid:4)bc Formulas for area A, circumference C, and volume V: ef b d bd ep for r a(cid:4)c (cid:2) a (cid:4) c ab (cid:2) a (cid:2) d (cid:2) ad TAri(cid:2)an12gbleh CAir(cid:2)cle(cid:3)r2 SAe(cid:2)cto12rr o2f(cid:6) Circle e b b b c b c bc (cid:2)1absin (cid:6) C(cid:2)2(cid:3)r s(cid:2)r(cid:6) (cid:2)(cid:6)in radians(cid:3) k 2 d d n a ere Exponents and Radicals a h r s h ut xm ¨ r C xmxn(cid:2)xm(cid:4)n (cid:2)xm(cid:5)n b ¨ xn r 1 (cid:2)xm(cid:3)n(cid:2)xmn x(cid:5)n(cid:2) xn (cid:4) (cid:5) x n xn Sphere Cylinder Cone (cid:2)xy(cid:3)n(cid:2)xnyn (cid:2) y yn V(cid:2)4(cid:3)r3 V(cid:2)(cid:3)r2h V(cid:2)1(cid:3)r2h 3 3 x1(cid:8)n(cid:2)snx xm(cid:8)n(cid:2)snxm(cid:2)(snx)m A(cid:2)4(cid:3)r2 A(cid:2)(cid:3)rsr2(cid:4)h2 (cid:7) snxy(cid:2)snxsny n x (cid:2) snx r y sny r h h Factoring Special Polynomials r x2(cid:5)y2(cid:2)(cid:2)x(cid:4)y(cid:3)(cid:2)x(cid:5)y(cid:3) x3(cid:4)y3(cid:2)(cid:2)x(cid:4)y(cid:3)(cid:2)x2(cid:5)xy(cid:4)y2(cid:3) x3(cid:5)y3(cid:2)(cid:2)x(cid:5)y(cid:3)(cid:2)x2(cid:4)xy(cid:4)y2(cid:3) Distance and Midpoint Formulas Binomial Theorem Distance between P(cid:2)x, y(cid:3)and P(cid:2)x, y(cid:3): 1 1 1 2 2 2 (cid:2)x(cid:4)y(cid:3)2(cid:2)x2(cid:4)2xy(cid:4)y2 (cid:2)x(cid:5)y(cid:3)2(cid:2)x2(cid:5)2xy(cid:4)y2 (cid:2)x(cid:4)y(cid:3)3(cid:2)x3(cid:4)3x2y(cid:4)3xy2(cid:4)y3 d(cid:2)s(cid:2)x2(cid:5)x1(cid:3)2(cid:4)(cid:2)y2(cid:5)y1(cid:3)2 (cid:2)x(cid:5)y(cid:3)3(cid:2)x3(cid:5)3x2y(cid:4)3xy2(cid:5)y3 (cid:4) (cid:5) (cid:2)x(cid:4)y(cid:3)n(cid:2)xn(cid:4)nxn(cid:5)1y(cid:4) n(cid:2)n(cid:5)1(cid:3) xn(cid:5)2y2 Midpoint of P1P2: x1(cid:4)2 x2, y1(cid:4)2 y2 2 (cid:4) (cid:5) n (cid:4)(cid:10)(cid:10)(cid:10)(cid:4) xn(cid:5)kyk(cid:4)(cid:10)(cid:10)(cid:10)(cid:4)nxyn(cid:5)1(cid:4)yn k (cid:4) (cid:5) Lines n n(cid:2)n(cid:5)1(cid:3)(cid:10)(cid:10)(cid:10)(cid:2)n (cid:5)k(cid:4)1(cid:3) where (cid:2) Slope of line through P(cid:2)x, y(cid:3)and P(cid:2)x, y(cid:3): k 1(cid:2)2(cid:2)3(cid:2)(cid:10)(cid:10)(cid:10)(cid:2)k 1 1 1 2 2 2 y (cid:5)y Quadratic Formula m(cid:2) 2 1 x (cid:5)x 2 1 (cid:5)b(cid:9)sb2(cid:5)4ac If ax2(cid:4)bx(cid:4)c(cid:2)0, then x(cid:2) . 2a Point-slope equation of line through P(cid:2)x, y(cid:3)with slope m: 1 1 1 Inequalities and Absolute Value y(cid:5)y (cid:2)m(cid:2)x(cid:5)x(cid:3) 1 1 If a(cid:7)band b(cid:7)c, then a(cid:7)c. Slope-intercept equation of line with slope mand y-intercept b: If a(cid:7)b, then a(cid:4)c(cid:7)b(cid:4)c. If a(cid:7)band c(cid:8)0, then ca(cid:7)cb. y(cid:2)mx(cid:4)b If a(cid:7)band c(cid:7)0, then ca(cid:8)cb. If a(cid:8)0, then Circles (cid:6)x(cid:6)(cid:2)a means x(cid:2)a or x(cid:2)(cid:5)a (cid:6)x(cid:6)(cid:7)a means (cid:5)a(cid:7)x(cid:7)a Equation of the circle with center (cid:2)h, k(cid:3)and radius r: (cid:6)x(cid:6)(cid:8)a means x(cid:8)a or x(cid:7)(cid:5)a (cid:2)x(cid:5)h(cid:3)2(cid:4)(cid:2)y(cid:5)k(cid:3)2(cid:2)r2 97909_FrontEP_FrontEP_pF2_RefPage1-2_97909_FrontEP_FrontEP_pF2_RefPage1-2 9/24/10 5:36 PM Page 2 REFERENCE PAGE 2 TRIGONOMETRY Angle Measurement Fundamental Identities (cid:2)radians(cid:2)180(cid:3) 1 1 s csc (cid:4)(cid:2) sec (cid:4)(cid:2) r sin (cid:4) cos (cid:4) (cid:2) 180(cid:3) 1(cid:3)(cid:2) rad 1rad(cid:2) ¨ 180 (cid:2) sin (cid:4) cos (cid:4) s(cid:2)r(cid:4) r tan (cid:4)(cid:2) cos (cid:4) cot (cid:4)(cid:2) sin (cid:4) (cid:3)(cid:4)in radians(cid:4) cot (cid:4)(cid:2) 1 sin2(cid:4)(cid:5)cos2(cid:4)(cid:2)1 tan (cid:4) Right Angle Trigonometry 1(cid:5)tan2(cid:4)(cid:2)sec2(cid:4) 1(cid:5)cot2(cid:4)(cid:2)csc2(cid:4) opp hyp sin (cid:4)(cid:2) hyp csc (cid:4)(cid:2) opp hyp sin(cid:3)(cid:6)(cid:4)(cid:4)(cid:2)(cid:6)sin (cid:4) cos(cid:3)(cid:6)(cid:4)(cid:4)(cid:2)cos (cid:4) opp (cid:5) (cid:6) adj hyp (cid:2) cos (cid:4)(cid:2) sec (cid:4)(cid:2) ¨ tan(cid:3)(cid:6)(cid:4)(cid:4)(cid:2)(cid:6)tan (cid:4) sin (cid:6)(cid:4) (cid:2)cos (cid:4) hyp adj 2 adj (cid:5) (cid:6) (cid:5) (cid:6) opp adj tan (cid:4)(cid:2) cot (cid:4)(cid:2) (cid:2) (cid:2) adj opp cos (cid:6)(cid:4) (cid:2)sin (cid:4) tan (cid:6)(cid:4) (cid:2)cot (cid:4) 2 2 Trigonometric Functions The Law of Sines B sin (cid:4)(cid:2) y csc (cid:4)(cid:2) r y r y sin A (cid:2) sin B (cid:2) sin C a x r (x, y) a b c cos (cid:4)(cid:2) sec (cid:4)(cid:2) r C r x c tan (cid:4)(cid:2) xy cot (cid:4)(cid:2) yx ¨ x The Law of Cosines b a2(cid:2)b2(cid:5)c2(cid:6)2bccos A Graphs of Trigonometric Functions b2(cid:2)a2(cid:5)c2(cid:6)2accos B y y y y=tan x c2(cid:2)a2(cid:5)b2(cid:6)2abcos C A y=sin x y=cos x 1 1 π 2π 2π Addition and Subtraction Formulas x π 2πx π x sin(cid:3)x(cid:5)y(cid:4)(cid:2)sin x cos y(cid:5)cos x sin y _1 _1 sin(cid:3)x(cid:6)y(cid:4)(cid:2)sin x cos y(cid:6)cos x sin y cos(cid:3)x(cid:5)y(cid:4)(cid:2)cos x cos y(cid:6)sin x sin y y y=csc x y y=sec x y y=cot x cos(cid:3)x(cid:6)y(cid:4)(cid:2)cos x cos y(cid:5)sin x sin y tan x(cid:5)tan y tan(cid:3)x(cid:5)y(cid:4)(cid:2) 1 1 1(cid:6)tan x tan y tan x(cid:6)tan y π 2πx π 2πx π 2πx tan(cid:3)x(cid:6)y(cid:4)(cid:2) 1(cid:5)tan x tan y _1 _1 Double-Angle Formulas sin 2x(cid:2)2 sin x cos x Trigonometric Functions of Important Angles cos 2x(cid:2)cos2x(cid:6)sin2x(cid:2)2 cos2x(cid:6)1(cid:2)1(cid:6)2 sin2x (cid:4) radians sin (cid:4) cos (cid:4) tan (cid:4) 2 tan x tan 2x(cid:2) 0(cid:3) 0 0 1 0 1(cid:6)tan2x 30(cid:3) (cid:2)(cid:2)6 1(cid:2)2 s3(cid:2)2 s3(cid:2)3 45(cid:3) (cid:2)(cid:2)4 s2(cid:2)2 s2(cid:2)2 1 Half-Angle Formulas 60(cid:3) (cid:2)(cid:2)3 s3(cid:2)2 1(cid:2)2 s3 1(cid:6)cos 2x 1(cid:5)cos 2x 90(cid:3) (cid:2)(cid:2)2 1 0 — sin2x(cid:2) 2 cos2x(cid:2) 2 98678_FMSVET_FMSVET_pi-xxiii.qk_98678_FMSVET_FMSVET_pi-xxiii 9/24/10 10:11 AM Page i S I N G L E V A R I A B L E C A L C U L U S E A R L Y T R A N S C E N D E N T A L S SEVENTH EDITION JAMES STEWART McMASTER UNIVERSITY AND UNIVERSITY OF TORONTO Australia.Brazil.Japan.Korea.Mexico.Singapore.Spain.UnitedKingdom.United States This is an electronic version of the print textbook. 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K0 Printed in the United States of America 1 2 3 4 5 6 7 14 13 12 11 10 98678_FMSVET_FMSVET_pi-xxiii.qk_98678_FMSVET_FMSVET_pi-xxiii 9/24/10 10:11 AM Page iii To Bill Ralph and Bruce Thompson This page intentionally left blank 98678_FMSVET_FMSVET_pi-xxiii.qk_98678_FMSVET_FMSVET_pi-xxiii 9/24/10 10:11 AM Page v Contents Preface xi To the Student xxii Diagnostic Tests xxiv A PREVIEW OF CALCULUS 2 1 Functions and Models 9 1.1 Four Ways to Represent a Function 10 1.2 Mathematical Models: A Catalog of Essential Functions 23 1.3 New Functions from Old Functions 36 1.4 Graphing Calculators and Computers 44 1.5 Exponential Functions 51 1.6 Inverse Functions and Logarithms 58 Review 72 Principles of Problem Solving 75 2 Limits and Derivatives 81 2.1 The Tangent and Velocity Problems 82 2.2 The Limit of a Function 87 2.3 Calculating Limits Using the Limit Laws 99 2.4 The Precise Definition of a Limit 108 2.5 Continuity 118 2.6 Limits at Infinity; Horizontal Asymptotes 130 2.7 Derivatives and Rates of Change 143 Writing Project Early Methods for Finding Tangents 153 N 2.8 The Derivative as a Function 154 Review 165 Problems Plus 170 v 98678_FMSVET_FMSVET_pi-xxiii.qk_98678_FMSVET_FMSVET_pi-xxiii 9/24/10 10:11 AM Page vi vi CONTENTS 3 Differentiation Rules 173 3.1 Derivatives of Polynomials and Exponential Functions 174 Applied Project Building a Better Roller Coaster 184 N 3.2 The Product and Quotient Rules 184 3.3 Derivatives of Trigonometric Functions 191 3.4 The Chain Rule 198 Applied Project Where Should a Pilot Start Descent? 208 N 3.5 Implicit Differentiation 209 Laboratory Project Families of Implicit Curves 217 N 3.6 Derivatives of Logarithmic Functions 218 3.7 Rates of Change in the Natural and Social Sciences 224 3.8 Exponential Growth and Decay 237 3.9 Related Rates 244 3.10 Linear Approximations and Differentials 250 Laboratory Project Taylor Polynomials 256 N 3.11 Hyperbolic Functions 257 Review 264 Problems Plus 268 4 Applications of Differentiation 273 4.1 Maximum and Minimum Values 274 Applied Project The Calculus of Rainbows 282 N 4.2 The Mean Value Theorem 284 4.3 How Derivatives Affect the Shape of a Graph 290 4.4 Indeterminate Forms and l’Hospital’s Rule 301 Writing Project The Origins of l’Hospital’s Rule 310 N 4.5 Summary of Curve Sketching 310 4.6 Graphing with Calculus andCalculators 318 4.7 Optimization Problems 325 Applied Project The Shape of a Can 337 N 4.8 Newton’s Method 338 4.9 Antiderivatives 344 Review 351 Problems Plus 355