Designed for the freshman/sophomore Calculus I-II-III sequence, the eighth edition continues to evolve to fulfill the needs of a changing market by providing flexible solutions to teaching and learning needs of all kinds. The new edition retains the strengths of earlier editions such as Anton's trademark clarity of exposition, sound mathematics, excellent exercises and examples, and appropriate level. Anton also incorporates new ideas that have withstood the objective scrutiny of many skilled and thoughtful instructors and their students.
Howard Anton obtained his B.A. from Lehigh University, his M.A. from the University of Illinois, and his Ph.D. from the Polytechnic University of Brooklyn, all in mathematics. In the early 1960's he worked for Burroughs Corporation and Avco Corporation at Cape Canaveral, Florida, where he was involved with the manned space program. In 1968 he joined the Mathematics Department at Drexel University, where he taught full time until 1983. Since that time he has been an adjunct professor at Drexel and has devoted the majority of his time to textbook writing and activities for mathematical associations. Dr. Anton was president of the EPADEL Section of the Mathematical Association of America (MAA), Served on the board of Governors of that organization, and guided the creation of the Student Chapters of the MAA. He has published numerous research papers in functional analysis, approximation theory, and topology, as well as pedagogical papers. He is best known for his textbooks in mathematics, which are among the most widely used in the world. There are currently more than one hundred versions of his books, including translations into Spanish, Arabic, Portuguese, Italian, Indonesian, French, Japanese, Chinese, Hebrew, and German. For relaxation, Dr. Anton enjoys traveling and photography.
Irl C. Bivens, recipient of the George Polya Award and the Merten M. Hasse Prize for Expository Writing in Mathematics, received his A.B. from Pfeiffer College and his Ph.D. from the University of North Carolina at Chapel Hill, both in mathematics. Since 1982, he has taught at Davidson College, where he currently holds the position of professor of mathematics. A typical academic year sees him teaching courses in calculus, topology, and geometry. Dr. Bivens also enjoys mathematical history, and his annual History of Mathematics seminar is a perennial favorite with Davidson mathematics majors. He has published numerous articles on undergraduate mathematics, as well as research papers in his specialty, differential geometry. he is currently a member of the editorial board for the MAA Problem Book series and is a reviewer for Mathematical Reviews. When he is not pursuing mathematics, Professor Bivens enjoys juggling, swimming, walking, and spending time with his son Robert.
Stephen L. Davis received his B.A. from Lindenwood College and his Ph.D. from Rutgers University in mathematics. Having previously taught at Rutgers University and Ohio State University, Dr. Davis came to Davidson College in 1981, where he is currently a professor of mathematics. He regularly teaches calculus, linear algebra, abstract algebra, and computer science. A sabbatical in 1995-1996 took him to Swarthmore College as a visiting associate professor. Professor Davis has published numerous articles on calculus reform and testing, as well as research papers on finite group theory, his specialty. Professor Davis has held several offices in the Southeastern section of the MAA, including chair and secretary-treasurer. He is currently a faculty consultant for the Educational testing Service Advanced Placement Calculus Test, a board member of the North Carolina, Association of Advanced Placement Mathematics Teachers, and is actively involved in nurturing mathematically talented high school students through leadership in the Charlotte Mathematics Club. He was formerly North Carolina state director for the MAA. For relaxation, he plays basketball, juggles, and travels. Professor Davis and his wife Elisabeth have three children, Laura, Anne, and James, all former calculus Students.
chapter one FUNCTIONS 1
1.1 Functions 1
1.2 Graphing Functions Using Calculators and Computer Algebra Systems16
1.3 New Functions from Old 27
1.4 Families of Functions40
1.5 Inverse Functions; Inverse Trigonometric Functions 51
1.6 Mathematical Models 59
1.7 Parametric Equations 69
chapter two LIMITS AND CONTINUITY 84
2.1 Limits (An Intuitive Approach) 84
2.2 Computing Limits 96
2.3 Limits at Infinity; End Behavior of a Function 105
2.4 Limits (Discussed More Rigorously) 116
2.5 Continuity 125
2.6 Continuity of Trigonometric and Inverse Functions 137
chapter three THE DERIVATIVE 146
3.1 Tangent Lines, Velocity, and General Rates of Change 146
3.2 The Derivative Function 159
3.3 Techniques of Differentiation 171
3.4 The Product and Quotient Rules 179
3.5 Derivatives of Trigonometric Functions 185
3.6 The Chain Rule 190
3.7 Implicit Differentiation 198
3.8 Related Rates 206
3.9 Local Linear Approximation; Differentials 213
chapter four THE DERIVATIVE IN GRAPHING AND APPLICATIONS 225
4.1 Analysis of Functions I: Increase, Decrease, and Concavity 225
4.2 Analysis of Functions II: Relative Extrema; Graphing Polynomials 234
4.3 More on Curve Sketching: Rational Functions; Curves with Cusps and Vertical Tangent Lines; Using Technology 245
4.4 Absolute Maxima and Minima 254
4.5 Applied Maximum and Minimum Problems 262
4.6 Newton’s Method 276
4.7 Rolle’s Theorem; Mean-Value Theorem 281
4.8 Rectilinear Motion 289
chapter five INTEGRATION 302
5.1 An Overview of the Area Problem 302
5.2 The Indefinite Integral 308
5.3 Integration by Substitution 318
5.4 The Definition of Area as a Limit; Sigma Notation 324
5.5 The Definite Integral 337
5.6 The Fundamental Theorem of Calculus 347
5.7 Rectilinear Motion Revisited Using Integration 361
5.8 Evaluating Definite Integrals by Substitution 370
chapter six APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING 380
6.1 Area Between Two Curves 380
6.2 Volumes by Slicing; Disks and Washers 388
6.3 Volumes by Cylindrical Shells 397
6.4 Length of a Plane Curve 403
6.5 Area of a Surface of Revolution 409
6.6 Average Value of a Function and its Applications 414
6.7 Work 419
6.8 Fluid Pressure and Force 427
chapter seven EXPONENTIAL, LOGARITHMIC, AND INVERSE TRIGONOMETRIC FUNCTIONS 435
7.1 Exponential and Logarithmic Functions 435
7.2 Derivatives and Integralsd Involving Logarithmic Functions 447
7.3 Derivatives of Inverse Functions; Derivatives and Integrals Involving Exponential Functions 453
7.4 Graphs and Applications Involving Logarithmic and Exponential Functions 460
7.5 L’Hôpital’s Rule; Indeterminate Forms 467
7.6 Logarithmic Functions from the Integral Point of View 476
7.7 Derivatives and Integrals Involving Inverse Trigonometric Functions 488
7.8 Hyperbolic Functions and Hanging Cables 498
chapter eight PRINCIPLES OF INTEGRAL EVALUATION 514
8.1 An Overview of Integration Methods 514
8.2 Integration by Parts 517
8.3 Trigonometric Integrals 526
8.4 Trigonometric Substitutions 534
8.5 Integrating Rational Functions by Partial Fractions 5441
8.6 Using Computer Algebra Systems and Tables of Integrals 549
8.7 Numerical Integration; Simpson’s Rule 560
8.8 Improper Integrals 573
chapter 9 MATHEMATICAL MODELING WITH DIFFERENTIAL EQUATIONS 586
9.1 First-Order Differential Equations and Applications 586
9.2 Slope Fields; Euler’s Method 600
9.3 Modeling with First-Order Differential Equations 607
9.4 Second-Order Linear Homogeneous Differential Equations; The Vibrating Spring 616
chapter ten INFINITE SERIES 628
10.1 Sequences 628
10.2 Monotone Sequences 639
10.3 Infinite Series 647
10.4 Convergence Tests 656
10.5 The Comparison, Ratio, and Root Tests 663
10.6 Alternating Series; Conditional Convergence 670
10.7 Maclaurin and Taylor Polynomials 679
10.8 Maclaurin and Taylor Series; PowerSeries 689
10.9 Convergence of Taylor Series 698
10.10 Differentiating and Integrating Power Series; Modeling with Taylor Series 708
chapter eleven ANALYTIC GEOMETRY IN CALCULUS 721
11.1 Polar Coordinates 721
11.2 Tangent Lines and Arc Length for Parametric and Polar Curves 735
11.3 Area in Polar Coordinates 744
11.4 Conic Sections in Calculus 750
11.5 Rotation of Axes; Second-Degree Equations 769
11.6 Conic Sections in Polar Coordinates 775
Horizon Module: Comet Collision 787
appendix a TRIGONOMETRY REVIEW A1
appendix b SOLVING POLYNOMIAL EQUATIONS A15
appendix c SELECTED PROOFS A22
ANSWERS A33
PHOTOCREDITS C1
INDEX I-1
web appendix d REAL NUMBERS,INTERVALS, AND INEQUALITIES W1
web appendix e ABSOLUTE VALUE W11
web appendix f COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS W16
web appendix g DISTANCE, CIRCLES, AND QUADRATIC FUNCTIONS W32
web appendix h THE DISCRIMINANT W41