Calculus With Analytic Geometry

1984·98.16 MB·English

by  Mustafa Munem

CALCULUS
Trigonometric Identities
Fundamental Identities Even-Odd Identities Addition Formulas
1 ^ sin(-0) = -sin 6 fsin(a + P) = sin acos/3(•^n^^osa
(?) CSC sin ^ cos(-e) =cos e cos(a+ P) = cos acos/3— sin asin/3
@ sec e cos1 6 1112 tcaont((--00)) == --tcaont 6 1ta-n a+ tan/3
(3) cot = J 13 sec(-0)= sece
tan = Subtraction Formulas
14 csc(-0) -esc e
^ tan e sin e 22 sin(a-j8) = sin acos/3- sin/3cosa
cos Double-Angle Formulas' 23 cos(a- /3) =cos acos)8+ sinasin^
®^cocots- e+cssoiinsn-ee= 1 1156 csions2206===221sc-ion2s-6scion-^s 1 24 tan(a - /3) = tana— tan 6
7 1 + tan- = sec- =cos- 6- sin^ Product Formulas
8 1 + cot- Q= csc^ e 17 cos- =^(1 +cos20) 25 sin acos^=5[sin(a + /3) + sin(a- /3)1
18 sin^ =5(1 -cos20) 26 cos acos /3 =|[cos(a +^) +cos(a - /3))
27 sinasin/3=5[cos (a-/3) - cos (a + ^)]
Geometry
(see Appendix A, page 1015)
Calculus
Differentiation
1 D,u" = nu"~^ D^ 11 DvsinhM = cosh uDji Vl -M^
2 D^u + v) =D^+D^v 12 D;^cosh u = sinh mD^m
3 D,{uv) = mO,v+ vD^M 13 D(tanh « = sech- uD^m 22 D,cos"' M
uD^v 14 Dtcoth M = -csch- uD^
23 D,tan"' m
15 D, sech M = —sech htanh uD^u 1 + M^
56 DD^,scionsuM==c—ossiunDM,Du,u 1167 DDj^"cs=cheM"=D^—csch wcoth «D^w 24 D, cot"' M : 1 + u^
7 D,tan u = sec^ mD^m 18 D,a" = a" InaD,m 25 D,sec"' M |m|Vm^ - 1
8 Dj cot M = -csc^ uD,M
9 D^sec M = sec u tanu D^ 19 D, log,,« = u Ina 26 DjCSC"' « u\Vu^ - 1
10 D^CSC u = —CSCMcot«D^M 20 D. in w = 27 D f(t)dt=f(u)D,u
J
nsw BOOK $51. 45 ijcrn
Tables of.Integrals (constantso^n^^gf'"'^l^^^^ '*^'' omitted tosave space)
Elementary forms
1 du = n + 1 (n^-l) 10 cotudu= \n |sin u| 19 coshudu = sinh m
— 11 secudu = \n |sec m + tan u\ 20 sech" M^M = tanhu
=
In |m|
J u 12 CSC MJm =In |csc M — cotm| 21 csch" udu = -cothm
sinMdu = —cos «
J 13 sin^ udu = 2U —1sin 2m 22 sech M tanh udu = —sechu
cos M<iM = sinu
I 14 COS" udu = 2U + 4 sin 2m 23 csch Mcoth udu = —csch u
sec"Mc/u = tanu
csc^ M^« = —cot u IS rfw , M 24 udv= uv— \ Vdu
du 25 e"du = e"
7 sec Mtan udu = sec m 16
a^+ M- 26 a"du = a"
8 CSC Mcotudu = —CSC m du Ina
9 jI tan Mt/w = —In |cos m| 1178 smiVnhm1—"u^d—ua=~1 c~oshau^'
IWgonometric Forms
27 tan" udu = ta.nu — u
I
2289 J^rcsoint"^uudduu== —costin"M—'"uMcos « 1 n —« \ '{sin"_,- udu 34 rCSC" udu= "-i1 esc" ^HM"/cIo--t2M1 Jrsec„_," MJm
30 Jrcos"Mdu = cos-InM si•n M 1 w —n ,1 JIrcos"_-,~ udu-t ^ . -^ + n -- 2 CSC - Mdu
—— n 1
31 J tan"Mdu = —tan""' m - I tan"~^ udu 35 I u"sinudu = -u"cos u + n \ u" cosMc/«
32 J cot"udu = M- 1 cot""' u—J cot""- udu m"cosudu= u"sinm —n «" ' sinmdu
;3a^
Inverse Trigonometric Forms
37 sin"' udu = u sin"' u + Vl — m" 40 fM sin_, udu = lu^- 1 sin m+ mVi -
J 4
38 cos"' //rfw = Mcos"' M — Vl —M^ 41 fM cos_, udu = 2u- - 1 cos_, M -
J 4 4
39 I tan"' udu = u tan"' u In (1 + u^) 42 \rutan- udu =—1 (m"- + 1) tan-. m —«—
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SECOND EDITION
CALCULUS
with Analytic Geometry

SECOND EDITION
CALCULUS
with Analytic Geometry
M. A. Muneni Macomb Community College
D. J. Foulis University of Massachusetts
Worth Publishers, Inc.
Calculus with AnalyticGeometry, secondedition
Copyright © 1984 by Worth Publishers. Inc.
All rights reserved.
Printed in the United StatesofAmerica
Library ofCongress Catalog No. 83-50583
ISBN: 0-87901-236-6
First printing, January 1984
editor: Anne Vinnicombe
production: GeorgeTouloumes and Pat Lawson
designer: MalcolmGrearDesigners
illustrator: Mel Erikson Art Services
typographer: YorkGraphic Services. Inc.
printer and binder: Kingsport Press
cover: Sinusoidalsurface. Computergraphics byThomas Banchoff. David Laidlaw. and
David Margolis.
Textfigureandpicturecreditsappearonpage 1033.whichconstitutesacontinuationofthe
copyright page.
Wortli Publishers, Inc.
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New York, New York 10016
11
CONTENTS
Preface
Introduction
CHAPTER 1 FUNCTIONS AND LIMITS 1
1.1 Real Numbers, Inequalities, and Absolute Value 1
1.2 The Cartesian Coordinate System 10
1.3 Straight Lines and Their Slopes 16
1.4 Functions 22
1.5 Types ofFunctions 31
1.6 Trigonometric Functions 41
1.7 Limits ofFunctions 52
1.8 Properties ofLimits ofFunctions 60
1.9 Continuous Functions 69
CHAPTER 2 THE DERIVATIVE 85
2.1 Rates ofChange and Slopes ofTangent Lines 85
2.2 The Derivative ofa Function 92
2.3 Basic Algebraic Rules for Differentiation 98
2.4 Tangent and Normal Lines 109
2.5 Rules for Differentiating Trigonometric Functions 114
2.6 Function Composition 120
2.7 The Chain Rule 124
2.8 Implicit Functions and Implicit Differentiation 132
2.9 The Rational-Power Rule 138
2.10 Higher-Order Derivatives 143
2.1 Algebraic Signs and Zeros ofFunctions 150
CHAPTER 3 APPLICATIONS OF THE DERIVATIVE 165
3. The Mean-ValueTheorem 165
3.2 Monotonicity and the First-Derivative Test 172
1
3.3 Concavity and the Second-Deiivative Test 180
3.4 Absolute Extrema 188
3.5 Asymptotes and Limits Involving Infinity 194
3.6 Graph Sketching 204
3.7 Applied Maximum and Minimum Problems 211
3.8 Related Rates 221
3.9 Applications to Economics and Business 228
(HAPTiiR 4 ANTIDIFFERENTIATION AND DIFFERENTIAL EQUATIONS 243
4.1 Differentials and Linear Approximation 243
4.2 Antiderivatives 251
4.3 The Method ofSubstitution, or Change ofVariable 258
4.4 Differential Equations 264
4.5 Applications ofDifferential Equations 274
4.6 The Harmonic-Oscillator Equation 281
4.7 Areas by the Method of Slicing 290
CHAPTER 5 THE DEFINITE INTEGRAL 299
5.1 The Sigma Notation for Sums 299
5.2 The Definite Integral 307
5.3 Basic Properties ofthe Definite Integral 316
5.4 The Fundamental Theorem ofCalculus 327
—
5.5 Approximation ofDefinite Integrals The Trapezoidal Rule and
Simpson's Rule 338
5.6 Areas ofRegions in the Plane 346
CHAPTER 6 APPLICATIONS OF THE DEFINITE INTEGRAL 357
6. Volumes ofSolids ofRevolution 357
6.2 The Method ofCylindrical Shells 365
6.3 Volumes by the Method ofSlicing 369
6.4 Arc Length and Surface Area 375
6.5 Work, Force, and Energy 383
6.6 Applications to Economics and the Life Sciences 393
CHAPTER 7 TRANSCENDENTAL FUNCTIONS 401
7.1 Inverse Functions 401
7.2 Inverse Trigonometric Functions 409
7.3 Derivatives ofand Integrals Yielding Inverse Trigonometric Functions 418
7.4 The Natural Logarithm Function 426
7.5 Properties ofthe Natural Logarithms 431
7.6 The Exponential Function 438
7.7 Exponential and Logarithmic Functions with Base OtherThan e 445
7.8 Hyperbolic Functions 452
7.9 Exponential Growth and Decay 458
7.10 Mathematical Models for Biological Growth 468
7.11 First-Order Linear Differential Equations 474