INSTRUCTOR’S SOLUTIONS MANUAL THOMAS’ CALCULUS TWELFTH EDITION WILLIAM ARDIS Collin County Community College

Functions 1 1.1 Functions and Their Graphs 1 1.2 Combining Functions; Shifting and Scaling Graphs 8 1.3 Trigonometric Functions 19 1.4 Graphing with Calculators and Computers 26 Practice Exercises 30 Additional and Advanced Exercises 38 2 Limits and Continuity 43 2.1 Rates of Change and Tangents to Curves 43 2.2 Limit of a Function and Limit Laws 46 2.3 The Precise Definition of a Limit 55 2.4 One-Sided Limits 63 2.5 Continuity 67 2.6 Limits Involving Infinity; Asymptotes of Graphs 73 Practice Exercises 82 Additional and Advanced Exercises 86 3 Differentiation 93 3.1 Tangents and the Derivative at a Point 93 3.2 The Derivative as a Function 99 3.3 Differentiation Rules 109 3.4 The Derivative as a Rate of Change 114 3.5 Derivatives of Trigonometric Functions 120 3.6 The Chain Rule 127 3.7 Implicit Differentiation 135 3.8 Related Rates 142 3.9 Linearizations and Differentials 146 Practice Exercises 151 Additional and Advanced Exercises 162 4 Applications of Derivatives 167 4.1 Extreme Values of Functions 167 4.2 The Mean Value Theorem 179 4.3 Monotonic Functions and the First Derivative Test 188 4.4 Concavity and Curve Sketching 196 4.5 Applied Optimization 216 4.6 Newton's Method 229 4.7 Antiderivatives 233 Practice Exercises 239 Additional and Advanced Exercises 251 5 Integration 257 5.1 Area and Estimating with Finite Sums 257 5.2 Sigma Notation and Limits of Finite Sums 262 5.3 The Definite Integral 268 5.4 The Fundamental Theorem of Calculus 282 5.5 Indefinite Integrals and the Substitution Rule 290 5.6 Substitution and Area Between Curves 296 Practice Exercises 310 Additional and Advanced Exercises 3206 Applications of Definite Integrals 327 6.1 Volumes Using Cross-Sections 327 6.2 Volumes Using Cylindrical Shells 337 6.3 Arc Lengths 347 6.4 Areas of Surfaces of Revolution 353 6.5 Work and Fluid Forces 358 6.6 Moments and Centers of Mass 365 Practice Exercises 376 Additional and Advanced Exercises 384 7 Transcendental Functions 389 7.1 Inverse Functions and Their Derivatives 389 7.2 Natural Logarithms 396 7.3 Exponential Functions 403 7.4 Exponential Change and Separable Differential Equations 414 7.5 Indeterminate Forms and L'Hopital's Rule 418 ^ 7.6 Inverse Trigonometric Functions 425 7.7 Hyperbolic Functions 436 7.8 Relative Rates of Growth 443 Practice Exercises 447 Additional and Advanced Exercises 458 8 Techniques of Integration 461 8.1 Integration by Parts 461 8.2 Trigonometric Integrals 471 8.3 Trigonometric Substitutions 478 8.4 Integration of Rational Functions by Partial Fractions 484 8.5 Integral Tables and Computer Algebra Systems 491 8.6 Numerical Integration 502 8.7 Improper Integrals 510 Practice Exercises 518 Additional and Advanced Exercises 528 9 First-Order Differential Equations 537 9.1 Solutions, Slope Fields and Euler's Method 537 9.2 First-Order Linear Equations 543 9.3 Applications 546 9.4 Graphical Solutions of Autonomous Equations 549 9.5 Systems of Equations and Phase Planes 557 Practice Exercises 562 Additional and Advanced Exercises 567 10 Infinite Sequences and Series 569 10.1 Sequences 569 10.2 Infinite Series 577 10.3 The Integral Test 583 10.4 Comparison Tests 590 10.5 The Ratio and Root Tests 597 10.6 Alternating Series, Absolute and Conditional Convergence 602 10.7 Power Series 608 10.8 Taylor and Maclaurin Series 617 10.9 Convergence of Taylor Series 621 10.10 The Binomial Series and Applications of Taylor Series 627 Practice Exercises 634 Additional and Advanced Exercises 642TABLE OF CONTENTS 10 Infinite Sequences and Series 569 10.1 Sequences 569 10.2 Infinite Series 577 10.3 The Integral Test 583 10.4 Comparison Tests 590 10.5 The Ratio and Root Tests 597 10.6 Alternating Series, Absolute and Conditional Convergence 602 10.7 Power Series 608 10.8 Taylor and Maclaurin Series 617 10.9 Convergence of Taylor Series 621 10.10 The Binomial Series and Applications of Taylor Series 627 Practice Exercises 634 Additional and Advanced Exercises 642 11 Parametric Equations and Polar Coordinates 647 11.1 Parametrizations of Plane Curves 647 11.2 Calculus with Parametric Curves 654 11.3 Polar Coordinates 662 11.4 Graphing in Polar Coordinates 667 11.5 Areas and Lengths in Polar Coordinates 674 11.6 Conic Sections 679 11.7 Conics in Polar Coordinates 689 Practice Exercises 699 Additional and Advanced Exercises 709 12 Vectors and the Geometry of Space 715 12.1 Three-Dimensional Coordinate Systems 715 12.2 Vectors 718 12.3 The Dot Product 723 12.4 The Cross Product 728 12.5 Lines and Planes in Space 734 12.6 Cylinders and Quadric Surfaces 741 Practice Exercises 746 Additional Exercises 754 13 Vector-Valued Functions and Motion in Space 759 13.1 Curves in Space and Their Tangents 759 13.2 Integrals of Vector Functions; Projectile Motion 764 13.3 Arc Length in Space 770 13.4 Curvature and Normal Vectors of a Curve 773 13.5 Tangential and Normal Components of Acceleration 778 13.6 Velocity and Acceleration in Polar Coordinates 784 Practice Exercises 785 Additional Exercises 791 Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.14 Partial Derivatives 795 14.1 Functions of Several Variables 795 14.2 Limits and Continuity in Higher Dimensions 804 14.3 Partial Derivatives 810 14.4 The Chain Rule 816 14.5 Directional Derivatives and Gradient Vectors 824 14.6 Tangent Planes and Differentials 829 14.7 Extreme Values and Saddle Points 836 14.8 Lagrange Multipliers 849 14.9 Taylor's Formula for Two Variables 857 14.10 Partial Derivatives with Constrained Variables 859 Practice Exercises 862 Additional Exercises 876 15 Multiple Integrals 881 15.1 Double and Iterated Integrals over Rectangles 881 15.2 Double Integrals over General Regions 882 15.3 Area by Double Integration 896 15.4 Double Integrals in Polar Form 900 15.5 Triple Integrals in Rectangular Coordinates 904 15.6 Moments and Centers of Mass 909 15.7 Triple Integrals in Cylindrical and Spherical Coordinates 914 15.8 Substitutions in Multiple Integrals 922 Practice Exercises 927 Additional Exercises 933 16 Integration in Vector Fields 939 16.1 Line Integrals 939 16.2 Vector Fields and Line Integrals; Work, Circulation, and Flux 944 16.3 Path Independence, Potential Functions, and Conservative Fields 952 16.4 Green's Theorem in the Plane 957 16.5 Surfaces and Area 963 16.6 Surface Integrals 972 16.7 Stokes's Theorem 980 16.8 The Divergence Theorem and a Unified Theory 984 Practice Exercises 989 Additional Exercises 997 Copyright © 2010 Pearson Education Inc. Publishing as Addison-Wesley.CHAPTER 1 FUNCTIONS 1.1 FUNCTIONS AND THEIR GRAPHS 1. domain ( ); range [1 ) 2. domain [0 ); range ( 1] œ
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È È 4 2 2 4 2 Copyright © 2010 Pearson Education, Inc. Publishing as Addison-Wesley.2 Chapter 1 Functions 15. The domain is . 16. The domain is . a b a b
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