Calculus- Early Transcendentals, 2021a

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Calculus: Early Transcendentals an Open Text Version 2021 — Revision A Be a champion of OER! Contribute suggestions for improvements, new content, or errata: • Anewtopic • A new example • An interesting new question • Any other suggestions to improve the material Contact Lyryx at info@lyryx.com with your ideas. CONTRIBUTIONS Jim Bailey, College of the Rockies Mark Blenkinsop, Carleton University Michael Cavers, University of Calgary David Guichard, Whitman College APEX Calculus: Gregory Hartman, Virginia Military Institute Joseph Ling, University of Calgary Lyryx Learning Creative Commons License (CC BY-NC-SA): This text, including the art and illustrations, are available under the Creative Commons license (CC BY-NC-SA), allowing anyone to reuse, revise, remix and redistribute the text. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/
Page 1: with Open Texts Calculus Early Tran
Page 7: Attribution Calculus: Early Transce
Page 10 and 11: ii 2014 A • M. Cavers: Addition o
Page 12 and 13: iv Table of Contents 2.8 Additional
Page 14 and 15: vi Table of Contents 9.2 Series . .
Page 16 and 17: viii Table of Contents Table of Int
Page 19 and 20: Chapter 1 Review Success in calculu
Page 21 and 22: 1.1. Algebra 5 8 is equal to 2. Def
Page 23 and 24: 1.1. Algebra 7 Suppose we want to s
Page 25 and 26: 1.1. Algebra 9 Note that the circle
Page 27 and 28: 1.1. Algebra 11 Solution. We need b
Page 29 and 30: 1.1. Algebra 13 Now we have the num
Page 31 and 32: 1.1. Algebra 15 • |x| < a means x
Page 33 and 34: 1.1. Algebra 17 Exercises for 1.1 E
Page 35 and 36: 1.2. Analytic Geometry 19 We use th
Page 37 and 38: 1.2. Analytic Geometry 21 The slope
Page 39 and 40: 1.2. Analytic Geometry 23 (e) The l
Page 41 and 42: 1.2. Analytic Geometry 25 Example 1
Page 43 and 44: 1.2. Analytic Geometry 27 • b is
Page 45 and 46: 1.2. Analytic Geometry 29 Notice th
Page 47 and 48: 1.2. Analytic Geometry 31 the form
Page 49 and 50: 1.3. Trigonometry 33 From now on, u
Page 51 and 52: 1.3. Trigonometry 35 For angles gr
Page 53 and 54: 1.3. Trigonometry 37 Mnemonic The f
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1.3. Trigonometry 39 To determine s
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1.3. Trigonometry 41 1. Shifts and
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1.4. Additional Exercises 43 1.4 Ad
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Chapter 2 Functions 2.1 What is a F
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2.1. What is a Function? 47 Example
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2.2. Transformations and Compositio
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2.2. Transformations and Compositio
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2.3. Exponential Functions 53 (i) f
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2.3. Exponential Functions 55 Three
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2.4. Inverse Functions 57 2.4 Inver
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2.5. Logarithms 59 Exercises for
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2.5. Logarithms 61 We also give it
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2.6. Inverse Trigonometric Function
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2.7. Hyperbolic Functions 69 e x =
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2.8. Additional Exercises 71 Figure
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2.8. Additional Exercises 73 (a) e
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76 Limits One-sided limits Consider
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78 Limits As is often the case in m
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80 Limits Exercises for Section 3.2
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82 Limits 10 8 6 4 2 K4 K2 0 2 4 6
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84 Limits should, be interpreted he
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86 Limits Exercises for 3.4 Exercis
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88 Limits A shortcut technique is t
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90 Limits Example 3.23: Vertical As
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92 Limits approaches the slant asym
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94 Limits The easiest way to prove
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96 Limits Another function that gro
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98 Limits We start by analyzing the
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100 Limits This limit is just as ha
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102 Limits When solving problems us
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104 Limits On the other hand, if f
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106 Limits Example 3.44: Continuous
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108 Limits This definition can be e
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110 Limits Example 3.53: Motivation
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112 Limits f (x) is continuous on [
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Chapter 4 Derivatives 4.1 The Rate
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4.1. The Rate of Change of a Functi
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4.1. The Rate of Change of a Functi
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4.2. The Derivative Function 121 4.
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4.2. The Derivative Function 123 So
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4.2. The Derivative Function 125 Ex
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4.3. Derivative Rules 131 ♣ It tu
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4.3. Derivative Rules 133 Example 4
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4.4. Derivative Rules for Trigonome
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4.5. The Chain Rule 137 From our ru
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Now we need the derivative of So th
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4.5. The Chain Rule 141 Exercise 4.
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4.6. Derivatives of Exponential & L
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4.6. Derivatives of Exponential & L
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4.7. Implicit Differentiation 147 E
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4.7. Implicit Differentiation 149 w
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4.7. Implicit Differentiation 151 E
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4.7. Implicit Differentiation 153 l
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4.8. Derivatives of Inverse Functio
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dy dx = − 1 siny 4.8. Derivatives
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4.9. Additional Exercises 159 (k) y
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162 Applications of Derivatives To
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164 Applications of Derivatives Sol
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166 Applications of Derivatives Exe
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168 Applications of Derivatives . A
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170 Applications of Derivatives It
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172 Applications of Derivatives 5.2
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178 Applications of Derivatives
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180 Applications of Derivatives the
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182 Applications of Derivatives (b)
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) 1 2 188 Applications of Der
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c 190 Applications of Derivatives
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192 Applications of Derivatives Def
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194 Applications of Derivatives Thi
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204 Applications of Derivatives So
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208 Applications of Derivatives . .
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210 Applications of Derivatives so
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212 Applications of Derivatives √
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216 Integration interval number i,
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218 Integration Even better, we now
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220 Integration Note how in the fir
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222 Integration = 1 2 + 3 2 + 5 2 +
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224 Integration Solution. Using six
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226 Integration Figure 6.
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228 Integration Notice in the previ
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230 Integration Now find the exact
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232 Integration Exercises for Secti
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234 Integration is a function: plug
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236 Integration can be interpreted
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238 Integration Properties of Defin
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240 Integration n ( ) 2i 2 = lim n
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242 Integration Solution. We will u
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244 Integration where C is a consta
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246 Integration Solution. ∫ ( 1 x
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Chapter 7 Techniques of Integration
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7.1. Substitution Rule 251 Example
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7.1. Substitution Rule 253 This is
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7.1. Substitution Rule 255 Exercise
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7.2. Powers of Trigonometric Functi
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7.3. Trigonometric Substitutions 26
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7.4. Integration by Parts 275 Examp
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7.4. Integration by Parts 277 To co
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7.5. Rational Functions 279 Exercis
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7.5. Rational Functions 281 If x 2
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7.5. Rational Functions 283 The fin
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7.6. Numerical Integration 285 pict
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7.6. Numerical Integration 287 a se
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7.7. Improper Integrals 289 Exercis
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7.7. Improper Integrals 291 To get
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7.7. Improper Integrals 293 Again,
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7.7. Improper Integrals 295 ∫ =
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7.7. Improper Integrals 297 Example
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7.8. Additional Exercises 299 Exerc
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Chapter 8 Applications of Integrati
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8.2. Area Between Curves 303 t = 7/
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8.2. Area Between Curves 305 More f
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8.2. Area Between Curves 307 Thus,
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8.3. Volume 309 From the sketch c =
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8.3. Volume 311 As you may know, th
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8.3. Volume 313 Note that we can in
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8.3. Volume 315 4 . 4 . 3 3 2 2 1 1
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8.4. Average Value of a Function 31
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8.5. Work 319 We can interpret this
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8.5. Work 321 As D increases W of c
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8.6. Center of Mass 323 Exercises f
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8.6. Center of Mass 325 = = n−1
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8.6. Center of Mass 327 Since the p
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8.7. Arc Length 329 (x . 1 ,y 1 )
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8.8. Surface Area 331 Exercise 8.7.
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8.8. Surface Area 333 frustum is π
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8.8. Surface Area 335 Solution. We
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338 Sequences and Series 9.1 Sequen
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340 Sequences and Series 5 4 3 2 1
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342 Sequences and Series Example 9.
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344 Sequences and Series Solution.
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346 Sequences and Series Theorem 9.
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348 Sequences and Series so to make
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350 Sequences and Series The proble
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352 Sequences and Series Example 9.
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354 Sequences and Series Theorem 9.
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356 Sequences and Series Sometimes,
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358 Sequences and Series 9.6 Absolu
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360 Sequences and Series ∞ ∑ n=
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362 Sequences and Series Exercises
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364 Sequences and Series |x + 2| <
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366 Sequences and Series Because th
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368 Sequences and Series f (n) (x)=
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370 Sequences and Series (e) lnx, a
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372 Sequences and Series It may not
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374 Sequences and Series since f (n
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376 Differential Equations Definiti
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378 Differential Equations This tec
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380 Differential Equations Exercise
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382 Differential Equations Solution
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384 Differential Equations Finally
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386 Differential Equations Example
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388 Differential Equations Figure 1
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390 Differential Equations You shou
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392 Differential Equations Exercise
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394 Differential Equations equation
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396 Differential Equations C cos(4t
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398 Differential Equations The firs
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400 Differential Equations v ′ =
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402 Polar Coordinates, Parametric E
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420 Three Dimensions z . . . . . .
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422 Three Dimensions Exercise 12.1.
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.. .. 424 Three Dimensions We also
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426 Three Dimensions z . . . . . .
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428 Three Dimensions v . .. . . . .
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430 Three Dimensions v .. . . . . .
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432 Three Dimensions Theorem 12.6:
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434 Three Dimensions w 1 c 1 + w 2
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436 Three Dimensions If a > 0andc >
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438 Three Dimensions Working backwa
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.. 440 Three Dimensions and a vecto
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.. 442 Three Dimensions . . . . . .
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444 Three Dimensions Exercise 12.5.
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446 Three Dimensions 2 1 0 -1 -2 -2
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448 Three Dimensions Example 12.16
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Chapter 13 Partial Differentiation
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13.1. Functions of Several Variable
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13.2. Limits and Continuity 455 tha
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13.2. Limits and Continuity 457 -3
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13.3. Partial Differentiation 459 8
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13.3. Partial Differentiation 461 i
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13.3. Partial Differentiation 463 S
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13.4. The Chain Rule 465 Exercise 1
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13.4. The Chain Rule 467 Exercises
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13.5. Directional Derivatives 469 a
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13.5. Directional Derivatives 471 E
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13.6. Higher Order Derivatives 473
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13.7. Maxima and Minima 475 so ther
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13.7. Maxima and Minima 477 Recall
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13.8. Lagrange Multipliers 479 Exer
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13.8. Lagrange Multipliers 481 50 4
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13.8. Lagrange Multipliers 483 Solv
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Chapter 14 Multiple Integration 14.
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14.1. Volume and Average Height 487
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14.2. Double Integrals in Polar Coo
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14.2. Double Integrals in Polar Coo
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14.3. Moment and Center of Mass 499
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14.4. Surface Area 501 Exercise 14.
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14.5. Triple Integrals 503 using th
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14.5. Triple Integrals 505 Figure 1
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14.6. Cylindrical and Spherical Coo
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14.6. Cylindrical and Spherical Coo
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14.7. Change of Variables 511 0.5 0
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14.7. Change of Variables 513 are
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14.7. Change of Variables 515 Subst
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Chapter 15 Vector Functions 15.1 Cu
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15.2. Calculus with Vector Function
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15.3. Arc Length 527 15.3 Arc Lengt
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15.4. Curvature 529 and take the de
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15.4. Curvature 531 and |T ′ (t)|
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15.5. Acceleration Vectors 533 15.5
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Chapter 16 Vector Calculus 16.1 Vec
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16.2. Divergence and Curl 537 that
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16.3. Line Integrals 539 16.3.1 Lin
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16.3. Line Integrals 541 the projec
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∫ Exercise 16.3.11 Compute 〈1/x
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16.3. Line Integrals 545 closed pat
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16.4. Green’s Theorem 547 The pro
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16.4. Green’s Theorem 549 (0,b)
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16.4. Green’s Theorem 551 Exercis
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16.5. The Divergence Theorem 553 in
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16.6. Vector Functions for Surfaces
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16.7. Surface Integrals 561 In prac
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16.7. Surface Integrals 563 = = ∫
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16.8. Stokes’ Theorem 565 are dif
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16.8. Stokes’ Theorem 567 Exercis
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570 Selected Exercise Answers 1.3.7
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572 Selected Exercise Answers 2.
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576 Selected Exercise Answers 3. x
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588 Selected Exercise Answers 10.2.
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600 INDEX dot product, 428 properti
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602 INDEX non-increasing, 342 prope
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∫ 23. cos 3 udu= 1 3 (2 + cos2 u)
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∫ 61. tan −1 udu= utan −1 u
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∫ 92. ∫ 93. (a 2 − u 2 ) 3/2
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