Calculus (8^th Ed.) Single and Multivariable

Calculus: Single and Multivariable, 8th Edition teaches calculus in a way that promotes critical thinking to reveal solutions to mathematical problems while highlighting the practical value of mathematics. From the Calculus Consortium based at Harvard University, this leading text reinforces the conceptual understanding students require to reduce complicated problems to simple procedures. In this new edition, the authors retain their emphasis on the Rule of Four?viewing problems graphically, numerically, symbolically, and verbally?with a special focus on introducing different perspectives for students with different learning styles.

The ideal textbook for promoting active learning in a 'flipped' classroom, Calculus engages students across multiple majors by providing a variety of problems with applications from the physical sciences, economics, health, biology, engineering, and economics. Throughout the text, the Consortium brings calculus to life with current and relevant examples and numerous opportunities to master key mathematical concepts and skills. The eighth edition includes new graphing questions and visualizations powered by GeoGebra?enabling complex, multi-part questions that reinforce the Rule of Four and strengthen student comprehension.

1 Foundation For Calculus: Functions and Limits 1

1.1 Functions and Change 2

1.2 Exponential Functions 14

1.3 New Functions From Old 26

1.4 Logarithmic Functions 34

1.5 Trigonometric Functions 42

1.6 Powers, Polynomials, and Rational Functions 53

1.7 Introduction To Limits and Continuity 62

1.8 Extending The Idea of A Limit 71

1.9 Further Limit Calculations Using Algebra 80

1.10 Preview of The Formal Definition of A Limit Online

2 Key Concept: The Derivative 87

2.1 How Do We Measure Speed? 88

2.2 The Derivative At A Point 96

2.3 The Derivative Function 105

2.4 Interpretations of The Derivative 113

2.5 The Second Derivative 121

2.6 Differentiability 130

3 Short-Cuts To Differentiation 135

3.1 Powers and Polynomials 136

3.2 The Exponential Function 146

3.3 The Product and Quotient Rules 151

3.4 The Chain Rule 158

3.5 The Trigonometric Functions 165

3.6 The Chain Rule and Inverse Functions 171

3.7 Implicit Functions 178

3.8 Hyperbolic Functions 181

3.9 Linear Approximation and The Derivative 185

3.10 Theorems About Differentiable Functions 193

4 Using The Derivative 199

4.1 Using First and Second Derivatives 200

4.2 Optimization 211

4.3 Optimization and Modeling 220

4.4 Families of Functions and Modeling 234

4.5 Applications To Marginality 244

4.6 Rates and Related Rates 253

4.7 L’hopital’s Rule, Growth, and Dominance 264

4.8 Parametric Equations 271

5 Key Concept: The Definite Integral 285

5.1 How Do We Measure Distance Traveled? 286

5.2 The Definite Integral 298

5.3 The Fundamental Theorem and Interpretations 308

5.4 Theorems About Definite Integrals 319

6 Constructing Antiderivatives 333

6.1 Antiderivatives Graphically and Numerically 334

6.2 Constructing Antiderivatives Analytically 341

6.3 Differential Equations and Motion 348

6.4 Second Fundamental Theorem of Calculus 355

7 Integration 361

7.1 Integration By Substitution 362

7.2 Integration By Parts 373

7.3 Tables of Integrals 380

7.4 Algebraic Identities and Trigonometric Substitutions 386

7.5 Numerical Methods For Definite Integrals 398

7.6 Improper Integrals 408

7.7 Comparison of Improper Integrals 417

8 Using The Definite Integral 425

8.1 Areas and Volumes 426

8.2 Applications To Geometry 436

8.3 Area and Arc Length In Polar Coordinates 447

8.4 Density and Center of Mass 456

8.5 Applications To Physics 467

8.6 Applications To Economics 478

8.7 Distribution Functions 489

8.8 Probability, Mean, and Median 497

9 Sequences and Series 507

9.1 Sequences 508

9.2 Geometric Series 514

9.3 Convergence of Series 522

9.4 Tests For Convergence 529

9.5 Power Series and Interval of Convergence 539

10 Approximating Functions Using Series 549

10.1 Taylor Polynomials 550

10.2 Taylor Series 560

10.3 Finding and Using Taylor Series 567

10.4 The Error In Taylor Polynomial Approximations 577

10.5 Fourier Series 584

11 Differential Equations 599

11.1 What is a Differential Equation? 600

11.2 Slope Fields 605

11.3 Euler’s Method 614

11.4 Separation of Variables 619

11.5 Growth and Decay 625

11.6 Applications and Modeling 637

11.7 The Logistic Model 647

11.8 Systems of Differential Equations 657

11.9 Analyzing The Phase Plane 667

11.10 Second-Order Differential Equations: Oscillations 674

11.11 Linear Second-Order Differential Equations 682

12 Functions of Several Variables 693

12.1 Functions of Two Variables 694

12.2 Graphs and Surfaces 702

12.3 Contour Diagrams 711

12.4 Linear Functions 725

12.5 Functions of Three Variables 732

12.6 Limits and Continuity 739

13 A Fundamental Tool: Vectors 745

13.1 Displacement Vectors 746

13.2 Vectors In General 755

13.3 The Dot Product 763

13.4 The Cross Product 774

14 Differentiating Functions of Several Variables 785

14.1 The Partial Derivative 786

14.2 Computing Partial Derivatives Algebraically 795

14.3 Local Linearity and The Differential 800

14.4 Gradients and Directional Derivatives In The Plane 809

14.5 Gradients and Directional Derivatives In Space 819

14.6 The Chain Rule 827

14.7 Second-Order Partial Derivatives 838

14.8 Differentiability 847

15 Optimization: Local and Global Extrema 855

15.1 Critical Points: Local Extrema and Saddle Points 856

15.2 Optimization 866

15.3 Constrained Optimization: Lagrange Multipliers 876

16 Integrating Functions of Several Variables 889

16.1 The Definite Integral of A Function of Two Variables 890

16.2 Iterated Integrals 898

16.3 Triple Integrals 908

16.4 Double Integrals In Polar Coordinates 916

16.5 Integrals In Cylindrical and Spherical Coordinates 921

16.6 Applications of Integration To Probability 931

17 Parameterization and Vector Fields 937

17.1 Parameterized Curves 938

17.2 Motion, Velocity, and Acceleration 948

17.3 Vector Fields 958

17.4 The Flow of A Vector Field 966

18 Line Integrals 973

18.1 The Idea of A Line Integral 974

18.2 Computing Line Integrals Over Parameterized Curves 984

18.3 Gradient Fields and Path-Independent Fields 992

18.4 Path-Dependent Vector Fields and Green’s Theorem 1003

19 Flux Integrals and Divergence 1017

19.1 The Idea of A Flux Integral 1018

19.2 Flux Integrals For Graphs, Cylinders, and Spheres 1029

19.3 The Divergence of A Vector Field 1039

19.4 The Divergence Theorem 1048

20 The Curl and Stokes’ Theorem 1055

20.1 The Curl of A Vector Field 1056

20.2 Stokes’ Theorem 1064

20.3 The Three Fundamental Theorems 1071

21 Parameters, Coordinates, and Integrals 1077

21.1 Coordinates and Parameterized Surfaces 1078

21.2 Change of Coordinates In A Multiple Integral 1089

21.3 Flux Integrals Over Parameterized Surfaces 1094

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 1099

Answers To Odd Numbered Problems 1117

Index 1177

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.