Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition ~ ABOUT CONCEPT

by: Robin J. Gottlieb

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.

Table of Contents

Preface

Contents

PART I - Functions: An Introduction

1 Functions Are Lurking Everywhere
- 1.1 Functions Are Everywhere
- * EXPLORATORY PROBLEMS FOR CHAPTER 1: Calibrating Bottles
- 1.2 What Are Functions? Basic Vocabulary and Notation
- 1.3 Representations of Functions
2 Characterizing Functions and Introducing Rates of Change
- 2.1 Features of a Function: Positive/Negative, Increasing/Decreasing, Continuous/Discontinuous
- 2.2 A Pocketful of Functions: Some Basic Examples
- 2.3 Average Rates of Change
- * EXPLORATORY PROBLEMS FOR CHAPTER 2: Runners
- 2.4 Reading a Graph to Get Information About a Function
- 2.5 The Real Number System: An Excursion
3 Functions Working Together
- 3.1 Combining Outputs: Addition, Subtraction, Multiplication, and Division of Functions
- 3.2 Composition of Functions
- 3.3 Decomposition of Functions
- * EXPLORATORY PROBLEMS FOR CHAPTER 3: Flipping, Shifting, Shrinking, and Stretching: Exercising Functions
- 3.4 Altered Functions, Altered Graphs: Stretching, Shrinking, Shifting, and Flipping

PART II - Rates of Change: An Introduction to the Derivative

4 Linearity and Local Linearity
- 4.1 Making Predictions: An Intuitive Approach to Local Linearity
- 4.2 Linear Functions
- 4.3 Modeling and Interpreting the Slope
- * EXPLORATORY PROBLEM FOR CHAPTER 4: Thomas Wolfe's Royalties for The Story of a Novel
- 4.4 Applications of Linear Models: Variations on a Theme
5 The Derivative Function
- 5.1 Calculating the Slope of a Curve and Instantaneous Rate of Change
- 5.2 The Derivative Function
- 5.3 Qualitative Interpretation of the Derivative
- * EXPLORATORY PROBLEMS FOR CHAPTER 5: Running Again
- 5.4 Interpreting the Derivative: Meaning and Notation
5 The Quadratics: A ProÞle of a Prominent Family of Functions
- 6.1 A Prole of Quadratics from a Calculus Perspective
- 6.2 Quadratics From A Noncalculus Perspective
- * EXPLORATORY PROBLEMS FOR CHAPTER 6: Tossing Around Quadratics
- 6.3 Quadratics and Their Graphs
- 6.4 The Free Fall of an Apple: A Quadratic Model
7 The Theoretical Backbone: Limits and Continuity
- 7.1 Investigating Limits - Methods of Inquiry and a Definition
- 7.2 Left- and Right-Handed Limits; Sometimes the Approach Is Critical
- 7.3 A Streetwise Approach to Limits
- 7.4 Continuity and the Intermediate and Extreme Value Theorems
- * EXPLORATORY PROBLEMS FOR CHAPTER 7: Pushing the Limit
8 Fruits of Our Labor: Derivatives and Local Linearity Revisited
- 8.1 Local Linearity and the Derivative
- * EXPLORATORY PROBLEMS FOR CHAPTER 8: Circles and Spheres
- 8.2 The First and Second Derivatives in Context: Modeling Using Derivatives
- 8.3 Derivatives of Sums, Products, Quotients, and Power Functions

PART III - Exponential, Polynomial, and Rational Functions with Applications

9 Exponential Functions
- 9.1 Exponential Growth: Growth at a Rate Proportional to Amount
- 9.2 Exponential: The Bare Bones
- 9.3 Applications of the Exponential Function
- * EXPLORATORY PROBLEMS FOR CHAPTER 9: The Derivative of the Exponential Function
- 9.4 The Derivative of an Exponential Function
10 Optimization
- 10.1 Analysis of Extrema
- 10.2 Concavity and the Second Derivative
- 10.3 Principles in Action
- * EXPLORATORY PROBLEMS FOR CHAPTER 10: Optimization
11 A Portrait of Polynomials and Rational Functions
- 11.1 A Portrait of Cubics from a Calculus Perspective
- 11.2 Characterizing Polynomials
- 11.3 Polynomial Functions and Their Graphs
- * EXPLORATORY PROBLEMS FOR CHAPTER 11: Functions and Their Graphs: Tinkering with Polynomials and Rational Functions
- 11.4 Rational Functions and Their Graphs

PART IV - Inverse Functions: A Case Study of Exponential and Logarithmic Functions

12 Inverse Functions: Can What Is Done Be Undone?
- 12.1 What Does It Mean for F and G to Be Inverse Functions?
- 12.2 Finding the Inverse of a Function
- 12.3 Interpreting the Meaning of Inverse Functions
- * EXPLORATORY PROBLEMS FOR CHAPTER 12: Thinking About the Derivatives of Inverse Functions
13 Logarithmic Functions
- 13.1 The Logarithmic Function Dened
- 13.2 The Properties of Logarithms
- 13.3 Using Logarithms and Exponentiation to Solve Equations
- * EXPLORATORY PROBLEM FOR CHAPTER 13: Pollution Study
- 13.4 Graphs of Logarithmic Functions: Theme and Variations
14 Differentiating Logarithmic and Exponential Functions
- 14.1 The Derivative of Logarithmic Functions
- * EXPLORATORY PROBLEM FOR CHAPTER 14: The Derivative of the Natural Logarithm
- 14.2 The Derivative of b^x Revisited
- 14.3 Worked Examples Involving Differentiation
15 Take It to the Limit
- 15.1 An Interesting Limit
- 15.2 Introducing Differential Equations
- * EXPLORATORY PROBLEMS FOR CHAPTER 15: Population Studies

PART V - Adding Sophistication to Your Differentiation

16 Taking the Derivative of Composite Functions
- 16.1 The Chain Rule
- 16.2 The Derivative of x^n where n is any Real Number
- 16.3 Using the Chain Rule
- * EXPLORATORY PROBLEMS FOR CHAPTER 16: Finding the Best Path
17 Implicit Differentiation and its Applications
- 17.1 Introductory Example
- 17.2 Logarithmic Differentiation
- 17.3 Implicit Differentiation
- 17.4 Implicit Differentiation in Context: Related Rates of Change

PART VI - An Excursion into Geometric Series

18 Geometric Sums, Geometric Series
- 18.1 Geometric Sums
- 18.2 Innite Geometric Series
- 18.3 A More General Discussion of Innite Series
- 18.4 Summation Notation
- 18.5 Applications of Geometric Sums and Series

PART VII - Trigonometric Functions

19 Trigonometry: Introducing Periodic Functions
- 19.1 The Sine and Cosine Functions: Denitions and Basic Properties
- 19.2 Modifying the Graphs of Sine and Cosine
- 19.3 The Function f(x) = tanx
- 19.4 Angles and Arc Lengths
20 Trigonometry - Circles and Triangles
- 20.1 Right-Triangle Trigonometry: The Denitions
- 20.2 Triangles We Know and Love, and the Information They Give Us
- 20.3 Inverse Trigonometric Functions
- 20.4 Solving Trigonometric Equations
- 20.5 Applying Trigonometry to a General Triangle: The Law of Cosines and the Law of Sines
- 20.6 Trigonometric Identities
- 20.7 A Brief Introduction to Vectors
21 Differentiation of Trigonometric Functions
- 21.1 Investigating the Derivative of sinx Graphically, Numerically, and Using Physical Intuition
- 21.2 Differentiating sinx and cosx
- 21.3 Applications
- 21.4 Derivatives of Inverse Trigonometric Functions
- 21.5 Brief Trigonometry Summary

PART VIII - Integration: An Introduction

22 Net Change in Amount and Area: Introducing the Definite Integral
- 22.1 Finding Net Change in Amount: Physical and Graphical Interplay
- 22.2 The Denite Integral
- 22.3 The Denite Integral: Qualitative Analysis and Signed Area
- 22.4 Properties of the Denite Integral
23 The Area Function and Its Characteristics
- 23.1 An Introduction to the Area Function \int_a^x f(t)
- 23.2 Characteristics of the Area Function
- 23.3 The Fundamental Theorem of Calculus
24 The Fundamental Theorem of Calculus
- 24.1 Denite Integrals and the Fundamental Theorem
- 24.2 The Average Value of a Function: An Application of the Denite Integral

PART IX - Applications and Computation of the Integral

25 Finding Antiderivatives - An Introduction to Indefinite Integration
- 25.1 A List of Basic Antiderivatives
- 25.2 Substitution: The Chain Rule in Reverse
- 25.3 Substitution to Alter the Form of an Integral
26 Numerical Methods of Approximating Definite Integrals
- 26.1 Approximating Sums: L_n, R_n, T_n, and M_n
- 26.2 Simpson's Rule and Error Estimates
27 Applying the DeÞnite Integral: Slice and Conquer
- 27.1 Finding "Mass" When Density Varies
- 27.2 Slicing to Find the Area Between Two Curves
28 More Applications of Integration
- 28.1 Computing Volumes
- 28.2 Arc Length, Work, and Fluid Pressure: Additional Applications of the Denite Integral
29 Computing Integrals
- 29.1 Integration by Parts - The Product Rule in Reverse
- 29.2 Trigonometric Integrals and Trigonometric Substitution
- 29.3 Integration Using Partial Fractions
- 29.4 Improper Integrals

PART X - Series

30 Series
- 30.1 Approximating a Function by a Polynomial
- 30.2 Error Analysis and Taylor's Theorem
- 30.3 Taylor Series
- 30.4 Working with Series and Power Series
- 30.5 Convergence Tests
31 Differential Equations
- 31.1 Introduction to Modeling with Differential Equations
- 31.2 Solutions to Differential Equations: An Introduction
- 31.3 Qualitative Analysis of Solutions to Autonomous Differential Equations
- 31.4 Solving Separable First Order Differential Equations
- 31.5 Systems of Differential Equations
- 31.6 Second Order Homogeneous Differential Equations with Constant Coefcients

APPENDIX A - Algebra

A.1 Introduction to Algebra: Expressions and Equations
A.2 Working with Expressions
A.3 Solving Equations

APPENDIX B - Geometric Formulas

APPENDIX C - The Theoretical Basis of Applications of the Derivative

APPENDIX D - Proof by Induction

APPENDIX E - Conic Sections

E.1 Characterizing Conics from a Geometric Viewpoint
E.2 Dening Conics Algebraically
E.3 The Practical Importance of Conic Sections

APPENDIX F - L' Hopital's Rule: Using Relative Rates of Change to Evaluate Limits

ABOUT CONCEPT

Sabtu, 13 Agustus 2011

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

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