Calculus- Early Transcendentals, 2021a

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Calculus: Early Transcendentals an Open Text Version 2021 — Revision A Extensive edits, additions, and revisions have been completed by the editorial team at Lyryx Learning. All new content (text and images) is released under the same license as noted above. 2021 A • Lyryx: Front matter has been updated including cover, Lyryx with Open Texts, copyright, and revision pages. Attribution page has been added. • Lyryx: Typo and other minor fixes have been implemented throughout. 2017 A • Lyryx: Front matter has been updated including cover page, copyright, and revision pages. • Lyryx: Examples 15.6, 16.11, 16.13, 16.3, 15.10, 16.20 and Exercises 15.4.3, 15.5.1, 16.6.1 have been rewritten. Several exercises from Vector Calculus have been removed. • Lyryx: Order and name of topics in Chapter 15 and Chapter 16 have been revised. 2016 B • D. Guichard: New content developed for the Three Dimensions, Vector Functions, and Vector Calculus chapters. • Lyryx: Exercise numbering has been updated to restart with each section. • G. Hartman: New content on Riemann Sums is included, Section 6.1.1. This section was adapted by Lyryx from the section of the same name in APEX Calculus, Version 3.0, written by G. Hartman. T. Siemers and D. Chalishajar of the Virginia Military Instititue and B. Heinold of Mount Saint Mary’s University also contributed to APEX Calculus. This material is released under Creative Commons license CC BY-NC (https://creativecommons.org/licenses/by-nc/4.0/ ). See http://www.apexcalculus.com/ for more information and original version. 2016 A • Lyryx: The layout and appearance of the text has been updated, including the title page and newly designed back cover. 2015 A • J. Ling: Addition of new exercises and proofs throughout. • J. Ling: Revised arrangement of topics in the Application of Derivatives chapter. • J. Ling: Continuity section has been revised to include additional explanations of content and additional examples.
Page 1: with Open Texts Calculus Early Tran
Page 5: Calculus: Early Transcendentals an
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
Page 55 and 56: 1.3. Trigonometry 39 To determine s
Page 57 and 58: 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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