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About the Author
Deborah Hughes-Hallett is Adjunct Professor of Public Policy and Professor of Mathematics at the University of Arizona. She graduated from Cambridge University in England and has taught at Middle East Technical University in Ankara, Turkey. Her work is on strategies to improve the teaching of mathematics, and she is interested in promoting international cooperation between mathematicians. She has served on committees for the National Academy of Sciences and organized three international conferences on the teaching of mathematics. She is a fellow of the American Advancement of Science and the author or coauthor of seven books, which have been translated into several languages. Her work has been recognized by prizes from Harvard, Arizona, the Association for Women in Mathematics, and the Mathematical Association of America.
Table of ContentsChapter 1. A Library of Functions 1.1 Functions and Change Material from former 1.1 and 1.2 1.2 Exponential Functions Material from former 1.3 and 1.7 1.3 New Functions from Old Material from former 1.5 and 1.8 1.4 Logarithmic Functions Material from former 1.6 and 1.7 1.5 Trigonometric Functions Former 1.9 1.6 Powers, Polynomials, and Rational Functions Material from former 1.4 and 1.10 1.7 Introduction to Continuity Former 1.11 including Intermediate Value Theorem. The Binomial Theorem is now a section available on the web site. Chapter 2. Key Concept: The Derivative 2.1 How Do We Measure Speed? 2.2 Limits NEW section from former Focus on Theory section. 2.3 The Derivative at a Point 2.4 The Derivative Function 2.5 Interpretations of the Derivative 2.6 The Second Derivative 2.7 Continuity and Differentiability NEW section from former Focus on Theory section. Chapter 3. Short-Cuts to Differentiation 3.1 Powers and Polynomials 3.2 The Exponential Function 3.3 The Product and Quotient Rules 3.4 The Chain Rule 3.5 The Trigonometric Functions 3.6 Applications of the Chain Rule. 3.7 Implicit Functions 3.8 Parametric Equations NEW: Material on Motion and Parametric Curves and Differentiation based on Appendix F and G and 16.1 3.9 Linear Approximations and the Derivative NEW: Material on Estimating the Error in the Approximation and theory on Differentiability and Local Linearity included. 3.10 Using Local Linearity to Find Limits Includes L'Hopital's rule. NEW. Chapter 4. Using the Derivative 4.1 Using First and Second Derivatives 4.2 Families of Curves 4.3 Optimization 4.4 Applications toMarginality 4.5 More Optimization: Introduction to Modeling 4.6 Hyperbolic Functions 4. 7 Theorems about Continuous and Differentiable Functions NEW. Extreme Value Theorem, Local Extrema and Critical Points, Mean Value Theorem, Increasing Function Theorem, Constant Function Theorem, Racetrack Principle. Chapter 5. Key Concept: The Definite Integral 5.1 How Do We Measure Distance Traveled? 5.2 The Definite Integral Now includes general Riemann sum. 5.3 Interpretations of the Definite Integral Material about integrating rates of change is now in this section. 5.4 Theorems About Definite Integrals Chapter 6. Constructing Antiderivatives 6.1 Antiderivatives Graphically and Numerically 6.2 Constructing Antiderivatives Analytically 6.3 Differential Equations 6.4 Second Fundamental Theorem of Calculus 6.5 The Equations of Motion Former Focus on Modeling Section. Chapter 7. Integration 7.1 Integration by Substitution. 7.2 Integration by Parts 7.3 Tables of Integrals 7.4 Algebraic Identities and Trigonometric Substitutions. NEW section including partial factions and trigonometric substitutions involving completing the square. 7.5 Approximating Definite Integrals 7.6 Approximating Errors and Simpson's Rule 7.7 Improper Integrals 7.8 More on Improper Integrals Chapter 8. Using the Definite Integral 8.1 Areas and Volumes. More accessible introduction to setting up integrals focusing on basic concepts. 8.2 Applications to Geometry Section simplified and made easier to use. 8.3 Density and Center of Mass Material on Center of Mass expanded. 8.4 Applications to Physics Section simplified and made easier to use. 8.5 Applications to Economics 8.6 Distribution Functions 8.7 Probability and More on Distributions NOTE: Chapter 9 and 10 replace Chapter 9 in the 2nd edition. The material has been expanded and extensively reorganized and rewritten. All sections have new problems. Material is clearly divided between series and convergence (Chapter 9) and approximations of functions (Chapter 10) for users who wish to emphasize one or the other. Chapter 9. Series 9.1 Geometric Series Former 9.4. 9.2 Convergence of Sequences and Series NEW section from former Focus on Theory Section including new material on integral test added. 9.3 Tests for Convergence From Focus on Theory section with substantial new material on integral test, ratio test, and alternating series added. 9.4 Power Series Material from former 9.2 with substantial new material on intervals and radius of convergence added. Chapter 10. Approximating Functions 10.1 Taylor Polynomials First part of former 9.1. 10.2 Taylor Series Second part of former 9.1 and 9.2. 10.3 Finding and Using Series Former 9.3 10.4 The Error in Taylor Polynomial Approximations Former Focus on Theory section, substantially rewritten. 10.5 Fourier Series Chapter 11. Differential Equations 11.1 What Is a Differential Equation? 11.2 Slope Fields 11.3 Euler's Method 11.4 Separation of Variables 11.5 Growth and Decay 11.6 Applications and Modeling 11.7 Models of Population Growth 11.8 Systems of Differential Equations 11.9 Analyzing the Phase Plane 11.10 Second-Order Differential Equations: Oscillations 11.11 Linear Second-Order Differential Equations Chapter 12. Functions of Several Variables 12.1 Functions of Two Variables Former 11.1 and 11.2 12.2 Graphs of Functions of Two Variables 12.3 Contour Diagrams 12.4 Linear Functions 12.5 Functions of More than Two Variables 12.6 Limits and Continuity. NEW section from former Focus on Theory section Chapter 13. A Fundamental Tool: Vectors 13.1 Displacement Vectors 13.2 Vectors in General 13.3 The Dot Product 13.4 The Cross Product Chapter 14. Differentiating Functions of Many Variables 14.1 The Partial Derivative 14.2 Computing Partial Derivatives Algebraically 14.3 Local Linearity and the Differential 14.4 Gradients and Directional Derivatives in the Plane 14.5 Gradients and Directional Derivatives in Space 14.6 The Chain Rule 14.7 Second Order Partial Derivatives 14.8 Differentiability and Error Bounds Chapter 15. Optimization: Local and Global Extrema 15.1 Local Extrema 15.2 Global Extrema:Unconstrained Optimization 15.3 Constrained Optimization: Lagrange Multipliers Chapter 16. Integrating Functions of Many Variables 16.1 The Definite Integral of a Function of Two Variables 16.2 Iterated Integrals 16.3 Triple Integrals 16.4 Double Integrals in Polar Coordinates 16.5 Integrals in Cylindrical and Spherical Coordinates 16.6 Applications of Integrationto Probability 16.7 Change of Variables in a Multiple Integral NEW section from former Focus on Theory section. Chapter 17. Parameterized Curves and Vector Fields 17.1 Parameterized Curves 17.2 Motion, Velocity, and Acceleration 17.3 Vector Fields 17.4 The Flow of a Vector Field Chapter 18. Line Integrals 18.1 The Idea of a Line Integral 18.2 Computing Line Integrals Over Parameterized Curves 18.3 Gradient Fields and Path-Independent Fields 18.4 Path-Independent Vector Fields and Green's Theorem 18.5 Proof of Green's Theorem. NEW section from former Focus on Theory section Chapter 19. Flux Integrals 19.1 The Idea of a Flux Integral 19.2 Flux Integrals for Graphs, Cylinders, and Spheres Chapter 20. Calculus of Vector Fields 20.1 The Divergence of a Vector Field 20.2 The Divergence Theorem 20.3 The Curl of a Vector Field 20.4 Stokes' Theorem 20.5 The Three Fundamental Theorems. NEW section from former Focus on Theory section Appendices Calculus Easy Reference - now includes page numbers