Thomas Calculus 13th [Solutions]

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Section 4.5 Indeterminate Forms and LHôpitals Rule 283 40. 3 1 1 (3 1)(sin ) 3sin (3 1)(cos ) 1 3cos 3cos (3 1)( sin ) lim x x x x x x x x x x x lim lim lim x sin x xsin x sin x x cos x cos x cos x xsin x x 0 x 0 x 0 x 0 3 3 (1)(0) 6 1 1 0 2 3 41. 1 ln x ( x 1) 1 x 1 x 1 ln ( 1)(ln ) 1 (ln x) ( x 1) ( ln ) 1 x lim 1 1 lim lim lim x x x x x x x x 1 x 1 x 1 x 1 lim x 1 1 1 1 (ln x 1) 1 (0 1) 1 2 42. 1 cos (1 cos x) (sin x)(cos x) lim (csc x cot x cos x) lim x cos x lim sin x sin x sin x x 0 x 0 x 0 2 2 lim sin x cos x sin x 0 1 0 cos x 1 1 x 0 43. lim cos 1 lim sin lim cos 1 0 e 1 0 e 1 0 e h h h e 44. (1 h lim ) lim e 1 lim e 1 2 h 0 h h 0 2h h 0 2 2 45. t 2 t t t lim e t lim e 2t lim e 2 lim e 1 t t t t t e 1 t e t e t e 46. 2 2 x lim x e lim x lim 2x lim 2 0 x x x x x e x e x e lim x sin x 1 cos sin 0 tan lim x lim x 0 x x 0 sec tan 0 2 sec tan 2sec 2 0 x x x x x x x x x x 47. 2 2 2 48. x 2 x x lim e 1 2 1 2 2 2 2 4 2 2 sin lim e e 0 0 cos sin lim x x 0 cos sin lim x x e e e e x x x x x x x x 0 xsin x 2cos x 2 1 x x x x 49. 2 2 2 lim sin cos 1 sin cos 2sin 2 tan lim lim lim 2cos 2 2 2 0 0 sec 1 0 tan 0 50. 2 lim sin 3x 3x x lim 3cos3x 3 2x lim 3cos3x 3 2x lim 9sin 3x 2 x 0 sin xsin 2x x 0 2sin x cos 2x cos xsin 2x x 0 sin x cos 2x sin 3x x 0 2sin xsin 2x cos xcos 2x 3cos3x 2 1 4 2 1/(1 ) 1/(1 ) 51. The limit leads to the indeterminate form 1 . Let ( ) x ln ( ) ln x f x x f x x ln x 1 x . Now lim ln f ( x ) lim ln x 1 x lim 1 1. Therefore x 1 x 1 x 1 1 x 1/(1 x) ln f ( x) 1 lim x lim f ( x) lim e e 1 e x 1 x 1 x 1 Copyright 2014 Pearson Education, Inc.
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1 Functions 1 TABLE OF CONTENTS 1.1
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8.3 Trigonometric Integrals 569 8.4
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2 Chapter 1 Functions 15. The domai
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4 Chapter 1 Functions 3 0 0 ( 1) 1
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6 Chapter 1 Functions 43. Symmetric
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8 Chapter 1 Functions 68. (a) From
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10 Chapter 1 Functions The complete
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12 Chapter 1 Functions 35. 36. 37.
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14 Chapter 1 Functions (c) domain:
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16 Chapter 1 Functions 69. 3 y f (
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18 Chapter 1 Functions (c) (d) 80.
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20 Chapter 1 Functions 21. 22. peri
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22 Chapter 1 Functions 40. sin(2 x)
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24 Chapter 1 Functions 65. A 2, B 2
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26 Chapter 1 Functions 1.4 GRAPHING
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28 Chapter 1 Functions 17. [ 5, 1]
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30 Chapter 1 Functions 35. 36. 37.
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32 Chapter 1 Functions 9. 10. 11. 2
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34 Chapter 1 Functions 35. After t
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36 Chapter 1 Functions 24. Step 1:
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38 Chapter 1 Functions y 1 36. y =
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40 Chapter 1 Functions 56. (a) (b)
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42 Chapter 1 Functions 1 50 50 50 (
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44 Chapter 1 Functions 10. 5 3 5 y(
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46 Chapter 1 Functions 37. (a) ( f
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48 Chapter 1 Functions 50. (a) Shif
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50 Chapter 1 Functions 65. Let h he
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52 Chapter 1 Functions 81. (a) No (
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54 Chapter 1 Functions 11. If f is
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56 Chapter 1 Functions 21. (a) y =
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58 Chapter 1 Functions Copyright 20
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60 Chapter 2 Limits and Continuity
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116 Chapter 3 Derivatives 9. m 3 3
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118 Chapter 3 Derivatives 32. (2 )
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120 Chapter 3 Derivatives 45. (a) T
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122 Chapter 3 Derivatives 2. 2 2 F(
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124 Chapter 3 Derivatives 18. 19. (
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126 Chapter 3 Derivatives (b) The t
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128 Chapter 3 Derivatives 48. (a) f
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130 Chapter 3 Derivatives 59. The g
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132 Chapter 3 Derivatives 6. y 3 2
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134 Chapter 3 Derivatives 33. 34. 3
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136 Chapter 3 Derivatives 56. (a) 3
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138 Chapter 3 Derivatives 74. (a) W
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140 Chapter 3 Derivatives 10 . (a)
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142 Chapter 3 Derivatives 25. 6 4 3
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144 Chapter 3 Derivatives 36. (a) v
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146 Chapter 3 Derivatives 25. r sec
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148 Chapter 3 Derivatives 44. We wa
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150 Chapter 3 Derivatives 64. cos(
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152 Chapter 3 Derivatives 3.6 THE C
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154 Chapter 3 Derivatives y 1 6 1 1
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156 Chapter 3 Derivatives 58. sin(
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158 Chapter 3 Derivatives 82. g( x)
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160 Chapter 3 Derivatives (b) y sin
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162 Chapter 3 Derivatives (c) df dt
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168 Chapter 3 Derivatives 47. 2 2 x
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170 Chapter 3 Derivatives Mathemati
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172 Chapter 3 Derivatives 13. y 2 l
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174 Chapter 3 Derivatives 44. 1 1/2
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178 Chapter 3 Derivatives 100. Supp
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180 Chapter 3 Derivatives 3.9 INVER
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184 Chapter 3 Derivatives 60. (a) D
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186 Chapter 3 Derivatives 65. The v
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188 Chapter 3 Derivatives 2 2 2 2 2
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190 Chapter 3 Derivatives 35. 1 dy
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192 Chapter 3 Derivatives 3.11 LINE
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194 Chapter 3 Derivatives x x x x 3
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196 Chapter 3 Derivatives 2 dA dt d
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198 Chapter 3 Derivatives As one zo
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200 Chapter 3 Derivatives 12. y 1 2
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202 Chapter 3 Derivatives 43. 1 4x
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204 Chapter 3 Derivatives 72. 2 1/2
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206 Chapter 3 Derivatives 90. 2 t 2
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208 Chapter 3 Derivatives 104. y si
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210 Chapter 3 Derivatives 124. rabb
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212 Chapter 3 Derivatives 147. Give
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214 Chapter 3 Derivatives CHAPTER 3
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216 Chapter 3 Derivatives 2 (b) On
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218 Chapter 3 Derivatives k k k k d
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220 Chapter 4 Applications of Deriv
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Section 4.8 Antiderivatives 309 4.8
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Section 4.8 Antiderivatives 311 42.
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85. (a) Wrong: (b) (c) d dx 3 (2x 1
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Section 4.8 Antiderivatives 315 2 2
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Section 4.8 Antiderivatives 317 128
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Chapter 4 Practice Exercises 319 12
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45. (a) y 6 x( x 1)( x 2) (b) 3 2 6
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52. The graph of the first derivati
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92. (a) The distance between the pa
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Chapter 4 Practice Exercises 335 x
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Chapter 4 Additional and Advanced E
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CHAPTER 5 INTEGRATION 5.1 AREA AND
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Section 5.1 Area and Estimating wit
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Section 5.1 Area and Estimating wit
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Section 5.2 Sigma Notation and Limi
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Section 5.3 The Definite Integral 3
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65. Consider the partition P that s
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74. See Exercise 73 above. On [0, 0
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Section 5.4 The Fundamental Theorem
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93. Example CAS commands: Maple: f
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Section 5.5 Indefinite Integrals an
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80. Let 2 u t du dt du dt Section 5
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1 1 11. (a) Let u 4 5 t t ( u 4), d
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Section 5.6 Substitution and Area B
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61. AREA A1 A2 A3 A1: For the sketc
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Chapter 5 Practice Exercises 407 CH
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80. Let u 7 5r du 5 dr 1 du dr; r 0
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29. (a) (b) 1 g(1) f ( t) dt 0 1 3
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CHAPTER 6 APPLICATIONS OF DEFINITE
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Section 6.1 Volumes Using Cross-Sec
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Section 6.2 Volumes Using Cylindric
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Section 6.3 Arc Length 455 2. dy 3
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Section 6.3 Arc Length 457 9. dx 1
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Section 6.3 Arc Length 459 19. (a)
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Section 6.3 Arc Length 461 32. (a)
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Section 6.4 Areas of Surfaces of Re
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Section 6.5 Work and Fluid Forces 4
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(d) In a location where water weigh
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38. Using the coordinate system giv
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Section 6.6 Moments and Centers of
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mass: 2 dm dA 3 1 x dx . The moment
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Chapter 6 Practice Exercises 493 6.
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(b) shell method: Chapter 6 Practic
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Chapter 6 Practice Exercises 497 x
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(a) Chapter 6 Additional and Advanc
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10. Converting to pounds and feet,
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CHAPTER 7 INTEGRALS AND TRANSCENDEN
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Section 7.1 The Logarithm Defined a
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Section 7.1 The Logarithm Defined a
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61. y = ln kx y = ln x + ln k; thus
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Section 7.2 Exponential Change and
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Section 7.2 Exponential Change and
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37. 0 kt A A e and Section 7.2 Expo
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Section 7.3 Hyperbolic Functions 52
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Section 7.4 Relative Rates of Growt
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Chapter 7 Practice Exercises 535 CH
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Chapter 7 Practice Exercises 537 (d
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CHAPTER 8 TECHNIQUES OF INTEGRATION
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Section 8.1 Using Basic Integration
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Section 8.2 Integration by Parts 55
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Section 8.3 Trigonometric Integrals
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Section 8.4 Trigonometric Substitut
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Section 8.5 Integration of Rational
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Section 8.6 Integral Tables and Com
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Section 8.7 Numerical Integration 6
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Section 8.8 Improper Integrals 617
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1 sin x 56. 2 x 1 sin x dx; 0 2 2 2
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Section 8.9 Probability 629 In orde
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Section 8.9 Probability 631 20. 3/2
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Section 8.9 Probability 633 36. For
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Section 8.9 Probability 635 52 . 20
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Chapter 8 Practice Exercises 639 (3
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Chapter 8 Practice Exercises 643 d
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Chapter 8 Practice Exercises 647 3/
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Chapter 8 Practice Exercises 649 li
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t b t t Chapter 8 Additional and Ad
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CHAPTER 9 FIRST-ORDER DIFFERENTIAL
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Section 9.1 Solutions, Slope Fields
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Section 9.2 First-Order Linear Equa
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Section 9.2 First-Order Linear Equa
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Section 9.3 Applications 675 6. 2 2
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14. (a) dV (5 3) 2 V 100 2t dt The
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Section 9.4 Graphical Solutions of
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Section 9.4 Graphical Solution of A
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Implies coexistence is not possible
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Section 9.5 Systems of Equations an
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Section 9.5 Systems of Equations an
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CHAPTER 10 INFINITE SEQUENCES AND S
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Section 10.1 Sequences 703 40. 1 n
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Section 10.1 Sequences 705 70. 1 n
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95. Since a n converges Section 10.
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111. 3( n 1) 1 3 1 3 4 3 1 2 2 a n
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Section 10.1 Sequences 711 133. a2k
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Section 10.2 Infinite Series 713 12
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6. f ( x ) 1 is positive, continuou
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Section 10.3 The Integral Test 723
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Section 10.4 Comparison Tests 729 4
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Section 10.5 Absolute Convergence;
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Section 10.6 Alternating Series and
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Section 10.7 Power Series 753 6. n
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Section 10.7 Power Series 755 17. l
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Section 10.7 Power Series 757 25. n
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Section 10.7 Power Series 759 1 2 1
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Section 10.7 Power Series 761 inter
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Section 10.7 Power Series 763 3 5 1
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Section 10.8 Taylor and Maclaurin S
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Section 10.8 Taylor and Maclaurin S
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Section 10.9 Convergence of Taylor
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Section 10.10 The Binomial Series a
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Chapter 10 Additional and Advanced
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CHAPTER 11 PARAMETRIC EQUATIONS AND
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Section 11.1 Parametrizations of Pl
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Section 11.2 Calculus with Parametr
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Section 11.3 Polar Coordinates 819
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Section 11.4 Graphing Polar Coordin
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9. r 2cos and r 2sin 2cos 2sin cos
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17. r sec and A r 2 4cos 4cos sec c
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Section 11.5 Areas and Lengths in P
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Section 11.6 Conic Sections 839 9.
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Section 11.7 Conics in Polar Coordi
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75. (a) Perihelion a ae a(1 e ), Ap
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CHAPTER 11 ADDITIONAL AND ADVANCED
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CHAPTER 12 VECTORS AND THE GEOMETRY
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Section 12.1 Three-Dimensional Coor
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Section 12.2 Vectors 881 66. 2 2 2
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Section 12.2 Vectors 883 25. length
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Section 12.2 Vectors 885 F 1 sin 40
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Section 12.3 The Dot Product 887 11
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Section 12.4 The Cross Product 893
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23. (a) u v 6, u w 81, v w 18 none
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Section 12.4 The Cross Product 897
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Section 12.5 Lines and Planes in Sp
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L1 & L3: x 3 2t 3 2r 2t 2r 0 t r 0
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Section 12.6 Cylinders and Quadric
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i j k Chapter 12 Practice Exercises
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Chapter 12 Practice Exercises 919 (
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CHAPTER 13 VECTOR-VALUED FUNCTIONS
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Section 13.1 Curves in Space and Th
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Section 13.1 Curves in Space and Th
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Section 13.2 Integrals of Vector Fu
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dr dt Section 13.2 Integrals of Vec
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t Section 13.3 Arc Length in Space
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Section 13.3 Arc Length in Space 94
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Section 13.4 Curvature and Normal V
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8. (a) Section 13.4 Curvature and N
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Section 13.5 Tangential and Normal
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Section 13.6 Velocity and Accelerat
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13. Assuming Earth has a circular o
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Chapter 13 Practice Exercises 963 i
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i j k 9. (a) ur u cos sin 0 k a rig
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CHAPTER 14 PARTIAL DERIVATIVES 14.1
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17. (a) Domain: all points in the x
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Section 14.1 Functions of Several V
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44. (a) (b) Section 14.1 Functions
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Section 14.2 Limits and Continuity
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Section 14.3 Partial Derivatives 99
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Section 14.4 The Chain Rule 999 92.
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Section 14.4 The Chain Rule 1001 w
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Section 14.4 The Chain Rule 1003 18
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Section 14.4 The Chain Rule 1005 y
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Section 14.4 The Chain Rule 1007 46
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Section 14.5 Directional Derivative
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Section 14.6 Tangent Planes and Dif
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Section 14.7 Extreme Values and Sad
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2 33. (i) On OA, f ( x, y) f (0, y)
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38. (i) On OA, f ( x, y) f (0, y) 2
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5. We optimize Section 14.8 Lagrang
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15. T (8x 4 y) i ( 4x 2 y) j and Se
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Section 14.8 Lagrange Multipliers 1
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34. Q( p, q, r) 2( pq pr qr ) and G
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Section 14.8 Lagrange Multipliers 1
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49-54. Example CAS commands: Maple:
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Section 14.9 Taylors Formula for Tw
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Section 14.10 Partial Derivatives w
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Section 14.10 Partial Derivatives w
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Chapter 14 Practice Exercises 1059
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74. (i) On OA, 2 f ( x, y) f (0, y)
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(iii) On CD, 3 f ( x, y) f (1, y) y
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96. Let Chapter 14 Practice Exercis
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21. (a) k is a vector normal to wil
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CHAPTER 15 MULTIPLE INTEGRALS 15.1
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Section 15.1 Double and Iterated In
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7. 8. Section 15.2 Double Integrals
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Section 15.2 Double Integrals Over
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Section 15.3 Area by Double Integra
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Section 15.3 Area by Double Integra
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Section 15.4 Double Integrals in Po
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6. The projection of D onto the xy
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Section 15.5 Triple Integrals in Re
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Section 15.5 Triple Integrals in Re
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Section 15.7 Triple Integrals in Cy
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66. average 1 2 3 Section 15.7 Trip
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Section 15.8 Substitutions in Multi
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53. x u y and y v x u v and y v 1 1
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CHAPTER 16 INTEGRALS AND VECTOR FIE
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Section 16.1 Line Integrals 1157 1
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Section 16.1 Line Integrals 1159 34
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Section 16.2 Vector Fields and Line
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(b) Section 16.2 Vector Fields and
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11. 2 2 Section 16.3 Path Independe
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Section 16.3 Path Independence, Pot
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Section 16.3 Path Independence, Pot
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Section 16.4 Greens Theorem in the
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Section 16.5 Surfaces and Area 1185
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(b) In a fashion similar to cylindr
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Section 16.6 Surface Integrals 1197
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Section 16.7 Stokes Theorem 1207 i
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Section 16.7 Stokes Theorem 1209 2
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Section 16.7 Stokes Theorem 1211 C
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Section 16.8 The Divergence Theorem
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30. By Exercise 29, 2 f g d f g f g
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CHAPTER 17 SECOND-ORDER DIFFERENTIA
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ww w w 2 2 Section 17.1 Second-Orde
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Section 17.1 Second-Order Linear Eq
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Section 17.2 Nonhomogeneous Linear
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Section 17.2 Nonhomogeneous Linear
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2 dy dx dy dx Section 17.2 Nonhomog
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dy w dx œc1sin x c2cos x cos xlnks
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Section 17.3 Applications 1017 ! !
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5 10 5È 2 2 5 dy 5È2 dy 1 16 $ È
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Section 17.4 Euler Equations 1021 "
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Section 17.5 Power-Series Soutions
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∞ ∞ ∞ # ww w # 5. x y 2xy 2
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# ww # 11. x 1y 6y 0 x 1 n2 nn 1c x
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∞ ∞ ∞ ww w 17. y xy 3y œ 0
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Thomas Calculus 13th [Solutions]

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