Calculus Single and Multivariable

The Calculus Consortium’s focus on the “Rule of Four” (viewing problems graphically, numerically, symbolically, and verbally) has become an integral part of teaching calculus in a way that promotes critical thinking to reveal solutions to mathematical problems. Their approach reinforces the conceptual understanding necessary to reduce complicated problems to simple procedures without losing sight of the practical value of mathematics. In this edition, the authors continue their focus on introducing different perspectives for students with an increased emphasis on active learning in a ‘flipped’ classroom.

The 8^th edition of Calculus: Single and Multivariable features a variety of problems with applications from the physical sciences, health, biology, engineering, and economics, allowing for engagement across multiple majors. The Consortium brings Calculus to (real) life with current, relevant examples and a focus on active learning. 

Deborah Hughes Hallett is Professor of Mathematics at the University of Arizona and Adjunct Professor of Public Policy at the Harvard Kennedy School. With Andrew M. Gleason at Harvard, she organized the Calculus Consortium based at Harvard, which brought together faculty from a wide variety of schools to work on undergraduate curricular issues. She is regularly consulted on the design of curricula and pedagogy for undergraduate mathematics at the national and international level and she is an author of several college level mathematics texts. In 1998 and 2002 and 2006, she was co-chair of the International Conference on the Teaching of Mathematics in Greece and Turkey, attended by several hundred faculty from about 50 countries. She has designed courses in Brunei, Colombia and Niger. She was awarded the Louise Hay Prize and elected a fellow of the American Association for the Advancement of Science for contributions to mathematics education. Her work has been recognized by prizes from Harvard, the University of Arizona, and as national winner MAA Award for Distinguished Teaching. Deb was also recently awarded with the 2022 AMS Award for Impact on the Teaching and Learning of Mathematics. This award is given annually to a mathematician (or group of mathematicians) who has made significant contributions of lasting value to mathematics education.

1 FOUNDATION FOR CALCULUS: FUNCTIONS AND LIMITS 1

1.1 FUNCTIONS AND CHANGE 2

1.2 EXPONENTIAL FUNCTIONS 13

1.3 NEW FUNCTIONS FROM OLD 23

1.4 LOGARITHMIC FUNCTIONS 32

1.5 TRIGONOMETRIC FUNCTIONS 39

1.6 POWERS, POLYNOMIALS, AND RATIONAL FUNCTIONS 49

1.7 INTRODUCTION TO LIMITS AND CONTINUITY 58

1.8 EXTENDING THE IDEA OF A LIMIT 67

1.9 FURTHER LIMIT CALCULATIONS USING ALGEBRA 75

1.10 OPTIONAL PREVIEW OF THE FORMAL DEFINITION OF A LIMIT ONLINE

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2 KEY CONCEPT: THE DERIVATIVE 83

2.1 HOW DO WE MEASURE SPEED? 84

2.2 THE DERIVATIVE AT A POINT 91

2.3 THE DERIVATIVE FUNCTION 99

2.4 INTERPRETATIONS OF THE DERIVATIVE 108

2.5 THE SECOND DERIVATIVE 115

2.6 DIFFERENTIABILITY 123

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3 SHORT-CUTS TO DIFFERENTIATION 129

3.1 POWERS AND POLYNOMIALS 130

3.2 THE EXPONENTIAL FUNCTION 140

3.3 THE PRODUCT AND QUOTIENT RULES 144

3.4 THE CHAIN RULE 151

3.5 THE TRIGONOMETRIC FUNCTIONS 158

3.6 THE CHAIN RULE AND INVERSE FUNCTIONS 164

3.7 IMPLICIT FUNCTIONS 171

3.8 HYPERBOLIC FUNCTIONS 174

3.9 LINEAR APPROXIMATION AND THE DERIVATIVE 178

3.10 THEOREMS ABOUT DIFFERENTIABLE FUNCTIONS 186

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4 USING THE DERIVATIVE 191

4.1 USING FIRST AND SECOND DERIVATIVES 192

4.2 OPTIMIZATION 203

4.3 OPTIMIZATION AND MODELING 212

4.4 FAMILIES OF FUNCTIONS AND MODELING 224

4.5 APPLICATIONS TO MARGINALITY 233

4.6 RATES AND RELATED RATES 243

4.7 L’HOPITAL’S RULE, GROWTH, AND DOMINANCE 252

4.8 PARAMETRIC EQUATIONS 259

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5 KEY CONCEPT: THE DEFINITE INTEGRAL 271

5.1 HOW DO WE MEASURE DISTANCE TRAVELED? 272

5.2 THE DEFINITE INTEGRAL 283

5.3 THE FUNDAMENTAL THEOREM AND INTERPRETATIONS 292

5.4 THEOREMS ABOUT DEFINITE INTEGRALS 302

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6 CONSTRUCTING ANTIDERIVATIVES 315

6.1 ANTIDERIVATIVES GRAPHICALLY AND NUMERICALLY 316

6.2 CONSTRUCTING ANTIDERIVATIVES ANALYTICALLY 322

6.3 DIFFERENTIAL EQUATIONS AND MOTION 329

6.4 SECOND FUNDAMENTAL THEOREM OF CALCULUS 335

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7 INTEGRATION 341

7.1 INTEGRATION BY SUBSTITUTION 342

7.2 INTEGRATION BY PARTS 353

7.3 TABLES OF INTEGRALS 360

7.4 ALGEBRAIC IDENTITIES AND TRIGONOMETRIC SUBSTITUTIONS 366

7.5 NUMERICAL METHODS FOR DEFINITE INTEGRALS 376

7.6 IMPROPER INTEGRALS 385

7.7 COMPARISON OF IMPROPER INTEGRALS 394

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8 USING THE DEFINITE INTEGRAL 401

8.1 AREAS AND VOLUMES 402

8.2 APPLICATIONS TO GEOMETRY 410

8.3 AREA AND ARC LENGTH IN POLAR COORDINATES 420

8.4 DENSITY AND CENTER OF MASS 429

8.5 APPLICATIONS TO PHYSICS 439

8.6 APPLICATIONS TO ECONOMICS 450

8.7 DISTRIBUTION FUNCTIONS 457

8.8 PROBABILITY, MEAN, AND MEDIAN 464

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9 SEQUENCES AND SERIES 473

9.1 SEQUENCES 474

9.2 GEOMETRIC SERIES 480

9.3 CONVERGENCE OF SERIES 488

9.4 TESTS FOR CONVERGENCE 494

9.5 POWER SERIES AND INTERVAL OF CONVERGENCE 504

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10 APPROXIMATING FUNCTIONS USING SERIES 513

10.1 TAYLOR POLYNOMIALS 514

10.2 TAYLOR SERIES 523

10.3 FINDING AND USING TAYLOR SERIES 530

10.4 THE ERROR IN TAYLOR POLYNOMIAL APPROXIMATIONS 539

10.5 FOURIER SERIES 546

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11 DIFFERENTIAL EQUATIONS 561

11.1 WHAT IS A DIFFERENTIAL EQUATION? 562

11.2 SLOPE FIELDS 567

11.3 EULER’S METHOD 575

11.4 SEPARATION OF VARIABLES 580

11.5 GROWTH AND DECAY 586

11.6 APPLICATIONS AND MODELING 597

11.7 THE LOGISTIC MODEL 606

11.8 SYSTEMS OF DIFFERENTIAL EQUATIONS 616

11.9 ANALYZING THE PHASE PLANE 626

11.10 SECOND-ORDER DIFFERENTIAL EQUATIONS: OSCILLATIONS 632

11.11 LINEAR SECOND-ORDER DIFFERENTIAL EQUATIONS 640

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12 FUNCTIONS OF SEVERAL VARIABLES 651

12.1 FUNCTIONS OF TWO VARIABLES 652

12.2 GRAPHS AND SURFACES 660

12.3 CONTOUR DIAGRAMS 668

12.4 LINEAR FUNCTIONS 682

12.5 FUNCTIONS OF THREE VARIABLES 689

12.6 LIMITS AND CONTINUITY 695

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13 A FUNDAMENTAL TOOL: VECTORS 701

13.1 DISPLACEMENT VECTORS 702

13.2 VECTORS IN GENERAL 710

13.3 THE DOT PRODUCT 718

13.4 THE CROSS PRODUCT 728

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14 DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES 739

14.1 THE PARTIAL DERIVATIVE 740

14.2 COMPUTING PARTIAL DERIVATIVES ALGEBRAICALLY 748

14.3 LOCAL LINEARITY AND THE DIFFERENTIAL 753

14.4 GRADIENTS AND DIRECTIONAL DERIVATIVES IN THE PLANE 762

14.5 GRADIENTS AND DIRECTIONAL DERIVATIVES IN SPACE 772

14.6 THE CHAIN RULE 780

14.7 SECOND-ORDER PARTIAL DERIVATIVES 790

14.8 DIFFERENTIABILITY 799

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15 OPTIMIZATION: LOCAL AND GLOBAL EXTREMA 805

15.1 CRITICAL POINTS: LOCAL EXTREMA AND SADDLE POINTS 806

15.2 OPTIMIZATION 815

15.3 CONSTRAINED OPTIMIZATION: LAGRANGE MULTIPLIERS 825

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16 INTEGRATING FUNCTIONS OF SEVERAL VARIABLES 839

16.1 THE DEFINITE INTEGRAL OF A FUNCTION OF TWO VARIABLES 840

16.2 ITERATED INTEGRALS 847

16.3 TRIPLE INTEGRALS 857

16.4 DOUBLE INTEGRALS IN POLAR COORDINATES 864

16.5 INTEGRALS IN CYLINDRICAL AND SPHERICAL COORDINATES 869

16.6 APPLICATIONS OF INTEGRATION TO PROBABILITY 878

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17 PARAMETERIZATION AND VECTOR FIELDS 885

17.1 PARAMETERIZED CURVES 886

17.2 MOTION, VELOCITY, AND ACCELERATION 896

17.3 VECTOR FIELDS 905

17.4 THE FLOW OF A VECTOR FIELD 913

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18 LINE INTEGRALS 921

18.1 THE IDEA OF A LINE INTEGRAL 922

18.2 COMPUTING LINE INTEGRALS OVER PARAMETERIZED CURVES 931

18.3 GRADIENT FIELDS AND PATH-INDEPENDENT FIELDS 939

18.4 PATH-DEPENDENT VECTOR FIELDS AND GREEN’S THEOREM 949

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19 FLUX INTEGRALS AND DIVERGENCE 961

19.1 THE IDEA OF A FLUX INTEGRAL 962

19.2 FLUX INTEGRALS FOR GRAPHS, CYLINDERS, AND SPHERES 973

19.3 THE DIVERGENCE OF A VECTOR FIELD 982

19.4 THE DIVERGENCE THEOREM 991

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20 THE CURL AND STOKES’ THEOREM 999

20.1 THE CURL OF A VECTOR FIELD 1000

20.2 STOKES’ THEOREM 1008

20.3 THE THREE FUNDAMENTAL THEOREMS 1015

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21 PARAMETERS, COORDINATES, AND INTEGRALS 1021

21.1 COORDINATES AND PARAMETERIZED SURFACES 1022

21.2 CHANGE OF COORDINATES IN A MULTIPLE INTEGRAL 1033

21.3 FLUX INTEGRALS OVER PARAMETERIZED SURFACES 1038

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APPENDICES Online

A ROOTS, ACCURACY, AND BOUNDS ONLINE

B COMPLEX NUMBERS ONLINE

C NEWTON’S METHOD ONLINE

D VECTORS IN THE PLANE ONLINE

E DETERMINANTS ONLINE

READY REFERENCE 1043

ANSWERS TO ODD-NUMBERED PROBLEMS 1061

INDEX 1131