S OLUTIONS M ANUAL D UANE K OUBA J ENNIFER A. B LUE University of California, Davis SUNY Empire State College U NIVERSITY C ALCULUS E ARLY T RANSCENDENTALS F OURTH E DITION Joel Hass University of California, Davis Christopher Heil Georgia Institute of Technology Przemyslaw Bogacki Old Dominion University Maurice D. Weir Naval Postgraduate School George B. Thomas, Jr. Massachusetts Institute of Technology iii TABLE OF CONTENTS 1 Functions 1 1.1 Functions and Their Graphs 1 1.2 Combining Functions; Shifting and Scaling Graphs 9 1.3 Trigonometric Functions 19 1.4 Graphing with Software 27 1.5 Exponential Functions 32 1.6 Inverse Functions and Logarithms 35 2 Limits and Continuity 47 2.1 Rates of Change and Tangent Lines to Curves 47 2.2 Limit of a Function and Limit Laws 51 2.3 The Precise Definition of a Limit 61 2.4 One-Sided Limits 70 2.5 Continuity 75 2.6 Limits Involving Infinity; Asymptotes of Graphs 81 Practice Exercises 92 Additional and Advanced Exercises 98 3 Derivatives 107 3.1 Tangent Lines and the Derivative at a Point 107 3.2 The Derivative as a Function 113 3.3 Differentiation Rules 124 3.4 The Derivative as a Rate of Change 130 3.5 Derivatives of Trigonometric Functions 136 3.6 The Chain Rule 143 3.7 Implicit Differentiation 155 3.8 Derivatives of Inverse Functions and Logarithms 163 3.9 Inverse Trigonometric Functions 173 3.10 Related Rates 180 3.11 Linearization and Differentials 185 Practice Exercises 193 Additional and Advanced Exercises 207 iv 4 Applications of Derivatives 213 4.1 Extreme Values of Functions on Closed Intervals 213 4.2 The Mean Value Theorem 222 4.3 Monotonic Functions and the First Derivative Test 229 4.4 Concavity and Curve Sketching 243 4.5 Indeterminate Forms and L’Hôpital’s Rule 267 4.6 Applied Optimization 276 4.7 Newton's Method 291 4.8 Antiderivatives 296 Practice Exercises 306 Additional and Advanced Exercises 327 5 Integrals 335 5.1 Area and Estimating with Finite Sums 335 5.2 Sigma Notation and Limits of Finite Sums 340 5.3 The Definite Integral 346 5.4 The Fundamental Theorem of Calculus 361 5.5 Indefinite Integrals and the Substitution Method 372 5.6 Definite Integral Substitutions and the Area Between Curves 380 Practice Exercises 399 Additional and Advanced Exercises 415 6 Applications of Definite Integrals 421 6.1 Volumes Using Cross-Sections 421 6.2 Volumes Using Cylindrical Shells 433 6.3 Arc Length 445 6.4 Areas of Surfaces of Revolution 454 6.5 Work and Fluid Forces 459 6.6 Moments and Centers of Mass 466 Practice Exercises 479 Additional and Advanced Exercises 489 7 Integrals and Transcendental Functions 495 7.1 The Logarithm Defined as an Integral 495 7.2 Exponential Change and Separable Differential Equations 502 7.3 Hyperbolic Functions 507 Practice Exercises 516 Additional and Advanced Exercises 520 v 8 Techniques of Integration 523 8.1 Integration by Parts 523 8.2 Trigonometric Integrals 537 8.3 Trigonometric Substitutions 546 8.4 Integration of Rational Functions by Partial Fractions 556 8.5 Integral Tables and Computer Algebra Systems 566 8.6 Numerical Integration 578 8.7 Improper Integrals 588 Practice Exercises 601 Additional and Advanced Exercises 615 9 Infinite Sequences and Series 623 9.1 Sequences 623 9.2 Infinite Series 635 9.3 The Integral Test 644 9.4 Comparison Tests 652 9.5 Absolute Convergence; The Ratio and Root Tests 663 9.6 Alternating Series and Conditional Convergence 669 9.7 Power Series 679 9.8 Taylor and Maclaurin Series 691 9.9 Convergence of Taylor Series 697 9.10 The Binomial Series and Applications of Taylor Series 705 Practice Exercises 714 Additional and Advanced Exercises 725 10 Parametric Equations and Polar Coordinates 731 10.1 Parametrizations of Plane Curves 731 10.2 Calculus with Parametric Curves 737 10.3 Polar Coordinates 747 10.4 Graphing Polar Coordinate Equations 753 10.5 Areas and Lengths in Polar Coordinates 761 Practice Exercises 767 Additional and Advanced Exercises 774 vi 11 Vectors and the Geometry of Space 777 11.1 Three-Dimensional Coordinate Systems 777 11.2 Vectors 782 11.3 The Dot Product 788 11.4 The Cross Product 794 11.5 Lines and Planes in Space 801 11.6 Cylinders and Quadric Surfaces 809 Practice Exercises 815 Additional and Advanced Exercises 823 12 Vector-Valued Functions and Motion in Space 829 12.1 Curves in Space and Their Tangents 829 12.2 Integrals of Vector Functions; Projectile Motion 836 12.3 Arc Length in Space 842 12.4 Curvature and Normal Vectors of a Curve 846 12.5 Tangential and Normal Components of Acceleration 853 12.6 Velocity and Acceleration in Polar Coordinates 857 Practice Exercises 859 Additional and Advanced Exercises 866 13 Partial Derivatives 869 13.1 Functions of Several Variables 869 13.2 Limits and Continuity in Higher Dimensions 879 13.3 Partial Derivatives 887 13.4 The Chain Rule 896 13.5 Directional Derivatives and Gradient Vectors 906 13.6 Tangent Planes and Differentials 912 13.7 Extreme Values and Saddle Points 921 13.8 Lagrange Multipliers 937 Practice Exercises 949 Additional and Advanced Exercises 965 vii 14 Multiple Integrals 971 14.1 Double and Iterated Integrals over Rectangles 971 14.2 Double Integrals over General Regions 974 14.3 Area by Double Integration 988 14.4 Double Integrals in Polar Form 993 14.5 Triple Integrals in Rectangular Coordinates 999 14.6 Applications 1005 14.7 Triple Integrals in Cylindrical and Spherical Coordinates 1011 14.8 Substitutions in Multiple Integrals 1024 Practice Exercises 1031 Additional and Advanced Exercises 1039 15 Integrals and Vector Fields 1045 15.1 Line Integrals 1045 15.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 1051 15.3 Path Independence, Conservative Fields, and Potential Functions 1063 15.4 Green's Theorem in the Plane 1069 15.5 Surfaces and Area 1077 15.6 Surface Integrals 1087 15.7 Stokes' Theorem 1098 15.8 The Divergence Theorem and a Unified Theory 1105 Practice Exercises 1112 Additional and Advanced Exercises 1122 16 First-Order Differential Equations 1127 16.1 Solutions, Slope Fields, and Euler's Method 1127 16.2 First-Order Linear Equations 1137 16.3 Applications 1141 16.4 Graphical Solutions of Autonomous Equations 1145 16.5 Systems of Equations and Phase Planes 1152 Practice Exercises 1158 Additional and Advanced Exercises 1166 viii 17 Second-Order Differential Equations 1169 17.1 Second-Order Linear Equations 1169 17.2 Nonhomogeneous Linear Equations 1174 17.3 Applications 1182 17.4 Euler Equations 1186 17.5 Power-Series Solutions 1189 B Appendix 1197 B.1 Relative Rates of Growth 1197 B.2 Probability 1202 B.3 Conics in Polar Coordinates 1210 B.4 Taylor’s Formula for Two Variables 1220 B.5 Partial Derivatives with Constrained Variables 1223 1 CHAPTER 1 FUNCTIONS 1.1 FUNCTIONS AND THEIR GRAPHS 1. domain ( , ); range [1, )      2. domain [0, ); range ( , 1]     3. domain [ 2, ); y    in range and y  5 10 x  0 y   can be any positive real number range   [0, ).  4. domain ( , 0] [3, ); y     in range and 2 3 0 y x x y     can be any positive real number  range [0, ).   5. domain ( , 3) (3, ); y     in range and 4 3 , t y   now if 4 3 3 3 0 0, t t t        or if 3 t   4 3 3 0 0 t t y       can be any nonzero real number range ( , 0) (0, ).      6. domain ( , 4) ( 4, 4) (4, ); y        in range and 2 2 16 , t y   now if 2 2 2 16 4 16 0 0, t t t         or if 2 2 2 2 16 16 4 4 16 16 0 t t t             , or if 2 2 2 16 4 16 0 0 t t t y         can be any nonzero real number 1 8 range ( , ] (0, ).       7. (a) Not the graph of a function of x since it fails the vertical line test. (b) Is the graph of a function of x since any vertical line intersects the graph at most once. 8. (a) Not the graph of a function of x since it fails the vertical line test. (b) Not the graph of a function of x since it fails the vertical line test. 9. base   2 2 2 3 2 2 ; (height) height ; x x x x      area is 1 2 ( ) a x  (base)(height)   2 3 3 1 2 2 4 ( ) ; x x x   perimeter is ( ) 3 . p x x x x x     10. 2 2 2 2 side length ; d s s s d s       and area is 2 2 1 2 a s a d    11. Let D  diagonal length of a face of the cube and   the length of an edge. Then 2 2 2 D d    and 2 2 2 2 3 2 3 . d D d         The surface area is 2 2 2 6 3 6 2 d d    and the volume is   2 3 3/2 3 3 3 3 . d d    12. The coordinates of P are   , x x so the slope of the line joining P to the origin is 1 ( 0). x x x m x    Thus,     2 1 1 , , . m m x x  13. 2 2 2 2 2 2 5 5 5 25 1 1 1 2 4 2 4 4 4 16 2 4 5 ; ( 0) ( 0) ( ) x y y x L x y x x x x x                   2 2 20 20 25 20 20 25 2 5 5 25 4 4 16 16 4 x x x x x x          14. 2 2 2 2 2 2 2 2 2 3 3 ; ( 4) ( 0) ( 3 4) ( 1) y x y x L x y y y y y                 4 2 2 4 2 2 1 1 y y y y y        2 Chapter 1 Functions 15. The domain is ( , ).   16. The domain is ( , ).   17. The domain is ( , ).   18. The domain is ( , 0].  19. The domain is ( , 0) (0, ).    20. The domain is ( , 0) (0, ).    21. The domain is ( , 5) ( 5, 3] [3, 5) (5, )         22. The range is [5, )  . 23. Neither graph passes the vertical line test (a) (b) Section 1.1 Functions and Their Graphs 3 24. Neither graph passes the vertical line test (a) (b) 1 1 1 or or 1 1 x y y x x y x y y x                                   25. 0 1 2 0 1 0 x y 26. 0 1 2 1 0 0 x y 27. 2 2 4 , 1 ( ) 2 , 1 x x F x x x x           28. 1 , 0 ( ) , 0 x x G x x x         29. (a) Line through (0, 0) and (1, 1): ; y x  Line through (1, 1) and (2, 0): 2 y x    , 0 1 ( ) 2, 1 2 x x f x x x           (b) 2, 0 1 0, 1 2 ( ) 2, 2 3 0, 3 4 x x f x x x                 30. (a) Line through (0, 2) and (2, 0): 2 y x    Line through (2, 1) and (5, 0): 0 1 1 1 5 2 3 3 , m        so 5 1 1 3 3 3 ( 2) 1 y x x        5 1 3 3 2, 0 2 ( ) , 2 5 x x f x x x               4 Chapter 1 Functions (b) Line through ( 1, 0)  and (0, 3):  3 0 0 ( 1) 3, m        so 3 3 y x    Line through (0, 3) and 1 3 4 2 0 2 (2, 1) : 2, m          so 2 3 y x    3 3, 1 0 ( ) 2 3, 0 2 x x f x x x              31. (a) Line through ( 1, 1)  and (0, 0): y x   Line through (0, 1) and (1, 1): 1 y  Line through (1, 1) and (3, 0): 0 1 1 1 3 1 2 2 , m        so 3 1 1 2 2 2 ( 1) 1 y x x        3 1 2 2 1 0 ( ) 1 0 1 1 3 x x f x x x x                   (b) Line through ( 2, 1)   and (0, 0): 1 2 y x  Line through (0, 2) and (1, 0): 2 2 y x    Line through (1, 1)  and (3, 1):  1 y   32. (a) Line through   2 , 0 T and ( T , 1): 1 0 2 ( /2) , T T T m     so   2 2 2 0 1 T T T y x x      2 2 2 0, 0 ( ) 1, T T T x f x x x T            (b) 2 2 3 2 3 2 , 0 , ( ) , , 2 T T T T A x A x T f x A T x A x T                     33. (a) 0 for [0, 1) x x       (b) 0 for ( 1, 0] x x        34. x x          only when x is an integer. 35. For any real number , 1, x n x n    where n is an integer. Now: 1 ( 1) . n x n n x n           By definition: and . x n x n x n                     So x x            for all real x . 36. To find f ( x ) you delete the decimal or fractional portion of x , leaving only the integer part. 1 2 2 0 ( ) 2 2 0 1 1 1 3 x x f x x x x                   Section 1.1 Functions and Their Graphs 5 37. Symmetric about the origin Dec: x     Inc: nowhere 38. Symmetric about the y -axis Dec: 0 x    Inc: 0 x    39. Symmetric about the origin Dec: nowhere Inc: 0 0 x x       40. Symmetric about the y -axis Dec: 0 x    Inc: 0 x    41. Symmetric about the y -axis Dec: 0 x    Inc: 0 x    42. No symmetry Dec: 0 x    Inc: nowhere 6 Chapter 1 Functions 43. Symmetric about the origin Dec: nowhere Inc: x     44. No symmetry Dec: 0 x    Inc: nowhere 45. No symmetry Dec: 0 x    Inc: nowhere 46. Symmetric about the y -axis Dec: 0 x    Inc: 0 x    47. Since a horizontal line not through the origin is symmetric with respect to the y -axis, but not with respect to the origin, the function is even. 48.   5 5 5 5 5 1 1 1 ( ) ( ) and ( ) ( ) ( ). x x x f x x f x x f x              Thus the function is odd. 49. Since 2 2 ( ) 1 ( ) 1 ( ). f x x x f x        The function is even. 50. Since 2 2 [ ( ) ] [ ( ) ( ) ] f x x x f x x x         and 2 2 [ ( ) ] [ ( ) ( ) ] f x x x f x x x         the function is neither even nor odd. 51. Since 3 3 3 ( ) , ( ) ( ) ( ). g x x x g x x x x x g x            So the function is odd. 52. 4 2 4 2 ( ) 3 1 ( ) 3( ) 1 ( ), g x x x x x g x           thus the function is even. 53. 2 2 1 1 1 ( ) 1 ( ) ( ). x x g x g x        Thus the function is even. 54. 2 2 1 1 ( ) ; ( ) ( ). x x x x g x g x g x         So the function is odd. 55. 1 1 1 1 1 1 ( ) ; ( ) ; ( ) . t t t h t h t h t          Since ( ) ( ) and ( ) ( ), h t h t h t h t       the function is neither even nor odd. Section 1.1 Functions and Their Graphs 7 56. Since 3 3 | | |( ) |, ( ) ( ) t t h t h t     and the function is even. 57. ( ) 2 1, ( ) 2 1. h t t h t t       So ( ) ( ). ( ) 2 1, h t h t h t t        so ( ) ( ). h t h t    The function is neither even nor odd. 58. ( ) 2| | 1 and ( ) 2| | 1 2| | 1. h t t h t t t         So ( ) ( ) h t h t   and the function is even. 59. ( ) sin 2 ; ( ) sin 2 ( ). g x x g x x g x       So the function is odd. 60. 2 2 ( ) sin ; ( ) sin ( ). g x x g x x g x     So the function is even. 61. ( ) cos3 ; ( ) cos3 ( ). g x x g x x g x     So the function is even. 62. ( ) 1 cos ; ( ) 1 cos ( ). g x x g x x g x       So the function is even. 63. 1 1 1 3 3 3 25 (75) ; 60 180 s kt k k s t t t           64. 2 2 2 2 12960 (18) 40 40 ; 40(10) 4000 joules K c v c c K v K          65. 24 24 12 4 5 6 24 ; 10 k k s s s r k r s           66. 3 14700 14700 24500 1000 39 14.7 14700 ; 23.4 628.2 in k k V V V P k P V            67. 3 2 ( ) (14 2 )(22 2 ) 4 72 308 ; 0 7. V f x x x x x x x x          68. (a) Let h  height of the triangle. Since the triangle is isosceles,     2 2 2 2 2. AB AB AB     So,   2 2 2 1 2 1 h h B      is at (0, 1)  slope of 1 A B    The equation of AB is ( ) 1; [0, 1]. y f x x x      (b) 2 ( ) 2 2 ( 1 [0, 1]. ) 2 2 ; A x xy x x x x x         69. (a) Graph h because it is an even function and rises less rapidly than does Graph g . (b) Graph f because it is an odd function. (c) Graph g because it is an even function and rises more rapidly than does Graph h . 70. (a) Graph f because it is linear. (b) Graph g because it contains (0, 1). (c) Graph h because it is a nonlinear odd function. 8 Chapter 1 Functions 71. (a) From the graph, 4 2 1 ( 2, 0) (4, ) x x x        (b) 4 4 2 2 1 1 0 x x x x       2 2 8 ( 4)( 2) 4 2 2 2 0: 1 0 0 0 x x x x x x x x x             4 x   since x is positive; 2 2 8 ( 4)( 2) 4 2 2 2 0: 1 0 0 0 x x x x x x x x x             2 x    since x is negative; sign of ( 4)( 2) x x   Solution interval: ( 2, 0) (4, )    72. (a) From the graph, 3 2 1 1 ( , 5) ( 1, 1) x x x          (b) Case 1: x   3( 1) 3 2 1 1 1 2 x x x x        3 3 2 2 5. x x x        Thus, ( , 5) x    solves the inequality. Case 1 1: x    3 2 1 1 x x     3( 1) 1 2 x x    3 3 2 2 5 x x x        which is true if 1. x   Thus, ( 1, 1) x   solves the inequality. Case 1 : x  3 2 1 1 3 x x x      3 2 2 x    5 x   which is never true if 1 , x  so no solution here. In conclusion, ( , 5) ( 1, 1). x      73. A curve symmetric about the x -axis will not pass the vertical line test because the points ( x , y ) and ( , ) x y  lie on the same vertical line. The graph of the function ( ) 0 y f x   is the x -axis, a horizontal line for which there is a single y -value, 0, for any x . 74. price 40 5 , x   quantity  300 25 x   ( ) R x  (40 5 )(300 25 ) x x   75. 2 2 2 2 2 2 ; h h x x h x      cost  5(2 ) 10 x h     2 2 ( ) 10 10 h C h h      5 2 2 h  76. (a) Note that 2 mi 10,560 ft,  so there are 2 2 800 x  feet of river cable at $180 per foot and (10,560 ) x  feet of land cable at $100 per foot. The cost is 2 2 ( ) 180 800 C x x    100(10,560 - x ). (b) (0) $1, 200, 000 (500) $1,175,812 (1000) $1,186,512 (1500) $1, 212, 000 (2000) $1, 243, 732 (2500) $1, 278, 479 (3000) $1,314,870 C C C C C C C        Values beyond this are all larger. It would appear that the least expensive location is less than 2000 feet from the point P .