Calculus: An Integrated Approach to Functions and Their Rates of Change, Preliminary Edition

A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course. Cover......Page 1 Calculus: An Integrated Approach to Functions and Their Rates of Change (Preliminary Edition)......Page 4 0201709295......Page 5 Preface......Page 8 Contents......Page 14 1.1 Functions Are Everywhere......Page 22 * EXPLORATORY PROBLEMS FOR CHAPTER 1: Calibrating Bottles......Page 25 1.2 What Are Functions? Basic Vocabulary and Notation......Page 26 1.3 Representations of Functions......Page 36 2.1 Features of a Function: Positive/Negative, Increasing/Decreasing, Continuous/Discontinuous......Page 70 2.2 A Pocketful of Functions: Some Basic Examples......Page 82 2.3 Average Rates of Change......Page 94 * EXPLORATORY PROBLEMS FOR CHAPTER 2: Runners......Page 103 2.4 Reading a Graph to Get Information About a Function......Page 105 2.5 The Real Number System: An Excursion......Page 116 3.1 Combining Outputs: Addition, Subtraction, Multiplication, and Division of Functions......Page 122 3.2 Composition of Functions......Page 129 3.3 Decomposition of Functions......Page 140 * EXPLORATORY PROBLEMS FOR CHAPTER 3: Flipping, Shifting, Shrinking, and Stretching: Exercising Functions......Page 144 3.4 Altered Functions, Altered Graphs: Stretching, Shrinking, Shifting, and Flipping......Page 147 4.1 Making Predictions: An Intuitive Approach to Local Linearity......Page 160 4.2 Linear Functions......Page 164 4.3 Modeling and Interpreting the Slope......Page 174 * EXPLORATORY PROBLEM FOR CHAPTER 4: Thomas Wolfe's Royalties for The Story of a Novel......Page 179 4.4 Applications of Linear Models: Variations on a Theme......Page 180 5.1 Calculating the Slope of a Curve and Instantaneous Rate of Change......Page 190 5.2 The Derivative Function......Page 208 5.3 Qualitative Interpretation of the Derivative......Page 215 * EXPLORATORY PROBLEMS FOR CHAPTER 5: Running Again......Page 227 5.4 Interpreting the Derivative: Meaning and Notation......Page 229 6.1 A Prole of Quadratics from a Calculus Perspective......Page 238 6.2 Quadratics From A Noncalculus Perspective......Page 244 * EXPLORATORY PROBLEMS FOR CHAPTER 6: Tossing Around Quadratics......Page 247 6.3 Quadratics and Their Graphs......Page 252 6.4 The Free Fall of an Apple: A Quadratic Model......Page 258 7.1 Investigating Limits - Methods of Inquiry and a Definition......Page 266 7.2 Left- and Right-Handed Limits; Sometimes the Approach Is Critical......Page 279 7.3 A Streetwise Approach to Limits......Page 286 7.4 Continuity and the Intermediate and Extreme Value Theorems......Page 291 * EXPLORATORY PROBLEMS FOR CHAPTER 7: Pushing the Limit......Page 296 8.1 Local Linearity and the Derivative......Page 300 * EXPLORATORY PROBLEMS FOR CHAPTER 8: Circles and Spheres......Page 307 8.2 The First and Second Derivatives in Context: Modeling Using Derivatives......Page 309 8.3 Derivatives of Sums, Products, Quotients, and Power Functions......Page 311 9.1 Exponential Growth: Growth at a Rate Proportional to Amount......Page 324 9.2 Exponential: The Bare Bones......Page 330 9.3 Applications of the Exponential Function......Page 341 * EXPLORATORY PROBLEMS FOR CHAPTER 9: The Derivative of the Exponential Function......Page 349 9.4 The Derivative of an Exponential Function......Page 355 10.1 Analysis of Extrema......Page 362 10.2 Concavity and the Second Derivative......Page 377 10.3 Principles in Action......Page 382 * EXPLORATORY PROBLEMS FOR CHAPTER 10: Optimization......Page 386 11.1 A Portrait of Cubics from a Calculus Perspective......Page 394 11.2 Characterizing Polynomials......Page 400 11.3 Polynomial Functions and Their Graphs......Page 412 * EXPLORATORY PROBLEMS FOR CHAPTER 11: Functions and Their Graphs: Tinkering with Polynomials and Rational Functions......Page 425 11.4 Rational Functions and Their Graphs......Page 427 12.1 What Does It Mean for F and G to Be Inverse Functions?......Page 442 12.2 Finding the Inverse of a Function......Page 450 12.3 Interpreting the Meaning of Inverse Functions......Page 455 * EXPLORATORY PROBLEMS FOR CHAPTER 12: Thinking About the Derivatives of Inverse Functions......Page 458 13.1 The Logarithmic Function Dened......Page 460 13.2 The Properties of Logarithms......Page 465 13.3 Using Logarithms and Exponentiation to Solve Equations......Page 470 * EXPLORATORY PROBLEM FOR CHAPTER 13: Pollution Study......Page 479 13.4 Graphs of Logarithmic Functions: Theme and Variations......Page 483 14.1 The Derivative of Logarithmic Functions......Page 488 * EXPLORATORY PROBLEM FOR CHAPTER 14: The Derivative of the Natural Logarithm......Page 489 14.2 The Derivative of b^x Revisited......Page 494 14.3 Worked Examples Involving Differentiation......Page 497 15.1 An Interesting Limit......Page 508 15.2 Introducing Differential Equations......Page 518 * EXPLORATORY PROBLEMS FOR CHAPTER 15: Population Studies......Page 528 16.1 The Chain Rule......Page 534 16.2 The Derivative of x^n where n is any Real Number......Page 542 16.3 Using the Chain Rule......Page 544 * EXPLORATORY PROBLEMS FOR CHAPTER 16: Finding the Best Path......Page 549 17.1 Introductory Example......Page 556 17.2 Logarithmic Differentiation......Page 559 17.3 Implicit Differentiation......Page 562 17.4 Implicit Differentiation in Context: Related Rates of Change......Page 571 18.1 Geometric Sums......Page 580 18.2 Innite Geometric Series......Page 587 18.3 A More General Discussion of Innite Series......Page 593 18.4 Summation Notation......Page 596 18.5 Applications of Geometric Sums and Series......Page 600 19 Trigonometry: Introducing Periodic Functions......Page 614 19.1 The Sine and Cosine Functions: Denitions and Basic Properties......Page 615 19.2 Modifying the Graphs of Sine and Cosine......Page 624 19.3 The Function f(x) = tanx......Page 636 19.4 Angles and Arc Lengths......Page 640 20.1 Right-Triangle Trigonometry: The Denitions......Page 648 20.2 Triangles We Know and Love, and the Information They Give Us......Page 656 20.3 Inverse Trigonometric Functions......Page 666 20.4 Solving Trigonometric Equations......Page 672 20.5 Applying Trigonometry to a General Triangle: The Law of Cosines and the Law of Sines......Page 678 20.6 Trigonometric Identities......Page 688 20.7 A Brief Introduction to Vectors......Page 692 21.1 Investigating the Derivative of sinx Graphically, Numerically, and Using Physical Intuition......Page 704 21.2 Differentiating sinx and cosx......Page 709 21.3 Applications......Page 716 21.4 Derivatives of Inverse Trigonometric Functions......Page 724 21.5 Brief Trigonometry Summary......Page 728 22.1 Finding Net Change in Amount: Physical and Graphical Interplay......Page 732 22.2 The Denite Integral......Page 746 22.3 The Denite Integral: Qualitative Analysis and Signed Area......Page 752 22.4 Properties of the Denite Integral......Page 759 23.1 An Introduction to the Area Function \int_a^x f(t)......Page 764 23.2 Characteristics of the Area Function......Page 768 23.3 The Fundamental Theorem of Calculus......Page 778 24.1 Denite Integrals and the Fundamental Theorem......Page 782 24.2 The Average Value of a Function: An Application of the Denite Integral......Page 796 25.1 A List of Basic Antiderivatives......Page 804 25.2 Substitution: The Chain Rule in Reverse......Page 808 25.3 Substitution to Alter the Form of an Integral......Page 819 26.1 Approximating Sums: L_n, R_n, T_n, and M_n......Page 826 26.2 Simpson's Rule and Error Estimates......Page 841 27.1 Finding "Mass" When Density Varies......Page 848 27.2 Slicing to Find the Area Between Two Curves......Page 864 28.1 Computing Volumes......Page 874 28.2 Arc Length, Work, and Fluid Pressure: Additional Applications of the Denite Integral......Page 886 29.1 Integration by Parts - The Product Rule in Reverse......Page 898 29.2 Trigonometric Integrals and Trigonometric Substitution......Page 907 29.3 Integration Using Partial Fractions......Page 919 29.4 Improper Integrals......Page 924 30.1 Approximating a Function by a Polynomial......Page 940 30.2 Error Analysis and Taylor's Theorem......Page 955 30.3 Taylor Series......Page 962 30.4 Working with Series and Power Series......Page 973 30.5 Convergence Tests......Page 985 31.1 Introduction to Modeling with Differential Equations......Page 1004 31.2 Solutions to Differential Equations: An Introduction......Page 1012 31.3 Qualitative Analysis of Solutions to Autonomous Differential Equations......Page 1023 31.4 Solving Separable First Order Differential Equations......Page 1039 31.5 Systems of Differential Equations......Page 1045 31.6 Second Order Homogeneous Differential Equations with Constant Coefcients......Page 1066 A.1 Introduction to Algebra: Expressions and Equations......Page 1072 A.2 Working with Expressions......Page 1077 A.3 Solving Equations......Page 1091 APPENDIX B - Geometric Formulas......Page 1106 APPENDIX C - The Theoretical Basis of Applications of the Derivative......Page 1108 APPENDIX D - Proof by Induction......Page 1116 APPENDIX E - Conic Sections......Page 1120 E.1 Characterizing Conics from a Geometric Viewpoint......Page 1121 E.2 Dening Conics Algebraically......Page 1122 E.3 The Practical Importance of Conic Sections......Page 1127 F.1 Indeterminate Forms......Page 1132 APPENDIX G - Newton's Method: Using Derivatives to Approximate Roots......Page 1142 APPENDIX H - Proofs to Accompany Chapter 30, Series......Page 1148 Index......Page 1154 Robin J. Gottlieb. Includes Bibliographical References And Index.