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Solution Manual for Mathematics for Elementary Teachers with Activities, 5th Edition

Solution Manual for Mathematics for Elementary Teachers with Activities, 5th Edition provides expert solutions to textbook questions, making complex problems easier to solve.

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    S OLUTIONS M ANUAL D UANE K OUBA University of California, Davis T HOMAS ’ C ALCULUS E ARLY T RANSCENDENTALS F OURTEENTH E DITION Based on the original work by George B. Thomas, Jr Massachusetts Institute of Technology as revised by Joel Hass University of California, Davis Christopher Heil Georgia Institute of Technology Maurice D. Weir Naval Postgraduate School

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    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 Practice Exercises 45 Additional and Advanced Exercises 55 2 Limits and Continuity 61 2.1 Rates of Change and Tangents to Curves 61 2.2 Limit of a Function and Limit Laws 65 2.3 The Precise Definition of a Limit 75 2.4 One-Sided Limits 83 2.5 Continuity 88 2.6 Limits Involving Infinity; Asymptotes of Graphs 94 Practice Exercises 105 Additional and Advanced Exercises 111 3 Derivatives 119 3.1 Tangents and the Derivative at a Point 119 3.2 The Derivative as a Function 125 3.3 Differentiation Rules 136 3.4 The Derivative as a Rate of Change 142 3.5 Derivatives of Trigonometric Functions 148 3.6 The Chain Rule 157 3.7 Implicit Differentiation 168 3.8 Derivatives of Inverse Functions and Logarithms 176 3.9 Inverse Trigonometric Functions 186 3.10 Related Rates 193 3.11 Linearization and Differentials 198 Practice Exercises 206 Additional and Advanced Exercises 220

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    iv 4 Applications of Derivatives 227 4.1 Extreme Values of Functions 227 4.2 The Mean Value Theorem 238 4.3 Monotonic Functions and the First Derivative Test 245 4.4 Concavity and Curve Sketching 259 4.5 Indeterminate Forms and L’Hôpital’s Rule 287 4.6 Applied Optimization 296 4.7 Newton's Method 311 4.8 Antiderivatives 316 Practice Exercises 326 Additional and Advanced Exercises 348 5 Integrals 355 5.1 Area and Estimating with Finite Sums 355 5.2 Sigma Notation and Limits of Finite Sums 360 5.3 The Definite Integral 366 5.4 The Fundamental Theorem of Calculus 381 5.5 Indefinite Integrals and the Substitution Method 391 5.6 Definite Integral Substitutions and the Area Between Curves 398 Practice Exercises 418 Additional and Advanced Exercises 434 6 Applications of Definite Integrals 443 6.1 Volumes Using Cross-Sections 443 6.2 Volumes Using Cylindrical Shells 455 6.3 Arc Length 467 6.4 Areas of Surfaces of Revolution 476 6.5 Work and Fluid Forces 482 6.6 Moments and Centers of Mass 493 Practice Exercises 507 Additional and Advanced Exercises 518 7 Integrals and Transcendental Functions 523 7.1 The Logarithm Defined as an Integral 523 7.2 Exponential Change and Separable Differential Equations 531 7.3 Hyperbolic Functions 537 7.4 Relative Rates of Growth 545 Practice Exercises 550 Additional and Advanced Exercises 556

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    v 8 Techniques of Integration 559 8.1 Using Basic Integration Formulas 559 8.2 Integration by Parts 572 8.3 Trigonometric Integrals 586 8.4 Trigonometric Substitutions 595 8.5 Integration of Rational Functions by Partial Fractions 604 8.6 Integral Tables and Computer Algebra Systems 615 8.7 Numerical Integration 626 8.8 Improper Integrals 637 8.9 Probability 649 Practice Exercises 658 Additional and Advanced Exercises 672 9 First-Order Differential Equations 681 9.1 Solutions, Slope Fields, and Euler's Method 681 9.2 First-Order Linear Equations 690 9.3 Applications 694 9.4 Graphical Solutions of Autonomous Equations 699 9.5 Systems of Equations and Phase Planes 706 Practice Exercises 712 Additional and Advanced Exercises 720 10 Infinite Sequences and Series 723 10.1 Sequences 723 10.2 Infinite Series 735 10.3 The Integral Test 743 10.4 Comparison Tests 752 10.5 Absolute Convergence; The Ratio and Root Tests 762 10.6 Alternating Series and Conditional Convergence 768 10.7 Power Series 778 10.8 Taylor and Maclaurin Series 791 10.9 Convergence of Taylor Series 797 10.10 The Binomial Series and Applications of Taylor Series 805 Practice Exercises 814 Additional and Advanced Exercises 825

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    vi 11 Parametric Equations and Polar Coordinates 831 11.1 Parametrizations of Plane Curves 831 11.2 Calculus with Parametric Curves 840 11.3 Polar Coordinates 850 11.4 Graphing Polar Coordinate Equations 855 11.5 Areas and Lengths in Polar Coordinates 863 11.6 Conic Sections 869 11.7 Conics in Polar Coordinates 880 Practice Exercises 890 Additional and Advanced Exercises 901 12 Vectors and the Geometry of Space 907 12.1 Three-Dimensional Coordinate Systems 907 12.2 Vectors 912 12.3 The Dot Product 918 12.4 The Cross Product 923 12.5 Lines and Planes in Space 930 12.6 Cylinders and Quadric Surfaces 939 Practice Exercises 944 Additional and Advanced Exercises 952 13 Vector-Valued Functions and Motion in Space 959 13.1 Curves in Space and Their Tangents 959 13.2 Integrals of Vector Functions; Projectile Motion 966 13.3 Arc Length in Space 975 13.4 Curvature and Normal Vectors of a Curve 979 13.5 Tangential and Normal Components of Acceleration 987 13.6 Velocity and Acceleration in Polar Coordinates 993 Practice Exercises 996 Additional and Advanced Exercises 1003

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    vii 14 Partial Derivatives 1007 14.1 Functions of Several Variables 1007 14.2 Limits and Continuity in Higher Dimensions 1017 14.3 Partial Derivatives 1025 14.4 The Chain Rule 1034 14.5 Directional Derivatives and Gradient Vectors 1044 14.6 Tangent Planes and Differentials 1050 14.7 Extreme Values and Saddle Points 1059 14.8 Lagrange Multipliers 1075 14.9 Taylor's Formula for Two Variables 1087 14.10 Partial Derivatives with Constrained Variables 1090 Practice Exercises 1093 Additional and Advanced Exercises 1111 15 Multiple Integrals 1117 15.1 Double and Iterated Integrals over Rectangles 1117 15.2 Double Integrals over General Regions 1120 15.3 Area by Double Integration 1134 15.4 Double Integrals in Polar Form 1139 15.5 Triple Integrals in Rectangular Coordinates 1145 15.6 Moments and Centers of Mass 1151 15.7 Triple Integrals in Cylindrical and Spherical Coordinates 1158 15.8 Substitutions in Multiple Integrals 1172 Practice Exercises 1179 Additional and Advanced Exercises 1186 16 Integrals and Vector Fields 1193 16.1 Line Integrals 1193 16.2 Vector Fields and Line Integrals: Work, Circulation, and Flux 1199 16.3 Path Independence, Conservative Fields, and Potential Functions 1211 16.4 Green's Theorem in the Plane 1217 16.5 Surfaces and Area 1225 16.6 Surface Integrals 1235 16.7 Stokes' Theorem 1246 16.8 The Divergence Theorem and a Unified Theory 1253 Practice Exercises 1260 Additional and Advanced Exercises 1270

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    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       

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    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)

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    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              

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    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                  

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    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

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    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.

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    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.

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    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 .

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