I. FUNCTIONS: AN INTRODUCTION.
1. Functions Are Lurking Everywhere. Functions Are Everywhere.
EXPLORATORY PROBLEMS FOR CHAPTER 1: Calibrating Bottles.
What Are Functions? Basic Vocabulary and Notation.
Representations of Functions.
2. Characterizing Functions and Introducing Rates of Change. Features of a Function: Positive/Negative, Increasing/Decreasing, Continuous/Discontinuous.
A Pocketful of Functions: Some Basic Examples.
Average Rates of Change.
EXPLORATORY PROBLEMS FOR CHAPTER 2: Runners.
Reading a Graph to Get Information About a Function.
The Real Number System: An Excursion.
3. Functions Working Together. Combining Outputs: Addition, Subtraction, Multiplication, and Division of Functions.
Composition of Functions.
Decomposition of Functions.
EXPLORATORY PROBLEMS FOR CHAPTER 3: Flipping, Shifting, Shrinking, and Stretching: Exercising Functions.
Altered Functions, Altered Graphs: Stretching, Shrinking, Shifting, and Flipping.
II. RATES OF CHANGE: AN INTRODUCTION TO THE DERIVATIVE.
4. Linearity and Local Linearity. Making Predictions: An Intuitive Approach to Local Linearity.
Linear Functions.
Modeling and Interpreting the Slope.
EXPLORATORY PROBLEMS FOR CHAPTER 4: Thomas Wolfe's Royalties for
The Story of a Novel. Applications of Linear Models: Variations on a Theme.
5. The Derivative Function. Calculating the Slope of a Curve and Instantaneous Rates of Change.
The Derivative Function.
Qualitative Interpretation of the Derivative.
EXPLORATORY PROBLEMS FOR CHAPTER 5: Running Again.
Interpreting the Derivative: Meaning and Notation.
6. The Quadratics: A Profile of a Prominent Family of Functions. A Profile of Quadratics from a Calculus Perspective.
Quadratics from a Noncalculus Perspective.
EXPLORATORY PROBLEMS FOR CHAPTER 6: Tossing Around Quadratics.
Quadratics and Their Graphs.
The Free Fall of an Apple: A Quadratic Model.
7. The Theoretical Backbone: Limits and Continuity. Investigating Limits-Methods of Inquiry and a Definition.
Left- and Right-Handed Limits; Sometimes the Approach is Critical.
A Streetwise Approach to Limits.
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. Local Linearity and the Derivative.
EXPLORATORY PROBLEMS FOR CHAPTER 8: Circles and Spheres.
The First and Second Derivatives in Context: Modeling Using Derivatives.
Derivatives of Sums, Products, Quotients, and Power Functions.
III. EXPONENTIAL, POLYNOMIAL, AND RATIONAL FUNCTIONS - WITH APPLICATIONS.
9. Exponential Functions. Exponential Growth: Growth at a Rate Proportional to Amount.
Exponential: The Bare Bones.
Applications of the Exponential Function.
EXPLORATORY PROBLEMS FOR CHAPTER 9: The Derivative of the Exponential Function.
The Derivative of an Exponential Function.
10. Optimization. Analysis of Extrema.
Concavity and the Second Derivative.
Principles in Action.
EXPLORATORY PROBLEMS FOR CHAPTER 10: Optimization.
11. A Portrait of Polynomials and Rational Functions. A Portrait of Cubics from a Calculus Perspective.
Characterizing Polynomials.
Polynomial Functions and Their Graphs.
EXPLORATORY PROBLEMS FOR CHAPTER 11: Functions and Their Graphs: Tinkering with Polynomials and Rational Functions.
Rational Functions and Their Graphs.
IV. INVERSE FUNCTIONS: CAN WHAT IS DONE BE UNDONE?
12. Inverse Functions. What Does It Mean for
f and
g to Be Inverse Functions?
Finding the Inverse of a Function.
Interpreting the Meaning of Inverse Functions.
EXPLORATORY PROBLEMS FOR CHAPTER 12: Thinking About the Derivatives of Inverse Functions.
13. Logarithmic Functions. The Logarithmic Function Defined.
The Properties of Logarithms.
Using Logarithms and Exponentiation to Solve Equations.
EXPLORATORY PROBLEMS FOR CHAPTER 13: Pollution Study.
Graphs of the Logarithmic Function: Theme and Variations.
14. Differentiating Logarithmic and Exponential Functions. The Derivative of Logarithmic Functions.
EXPLORATORY PROBLEMS FOR CHAPTER 14: The Derivative of the Natural Logarithm.
The Derivative of
bx Revisited.
Worked Examples Involving Differentiation.
15. Take It to the Limit. An Interesting Limit.
Introducing Differential Equations.
EXPLORATORY PROBLEMS FOR CHAPTER 15: Population Studies.
V. ADDING SOPHISTICATION TO YOUR DIFFERENTIATION.
16. Taking the Derivative of Composite Functions. The Chain Rule.
The Derivative of
xn Where
n is any Real Number.
Using the Chain Rule.
EXPLORATORY PROBLEMS FOR CHAPTER 16: Finding the Best Path.
17. Implicit Differentiation and its Applications. Introductory Example.
Logarithmic Differentiation.
Implicit Differentiation.
Implicit Differentiation in Context: Related Rates of Change.
VI. AN EXCURSION INTO GEOMETRIC SERIES.
18. Geometric Sums, Geometric Series. Geometric Sums.
Infinite Geometric Series.
A More General Discussion of Infinite Series.
Summation Notation.
Applications of Geometric Sums and Series.
VII. TRIGONOMETRIC FUNCTIONS.
19. Trigonometry: Introducing Periodic Functions. The Sine and Cosine Functions: Definitions and Basic Properties.
Modifying the Graphs of Sine and Cosine.
The Function
f (x) = tan
x.
Angles and Arc Lengths.
20. Trigonometry-Circles and Triangles. Right Triangle Trigonometry: The Definitions.
Triangles We Know and Love, and the Information They Give Us.
Inverse Trigonometric Functions.
Solving Trigonometric Equations.
Applying Trigonometry to a General Triangle: The Law of Cosines and the Law of Sines.
Trigonometric Identities.
A Brief Introduction to Vectors.
21. Differentiation of Trigonometric Functions. Investigating the Derivative of sin
x Graphically, Numerically, and Using Physical Intuition.
Differentiating sin
x and cos
x.
Applications.
Derivatives of Inverse Trigonometric Functions.
Brief Trigonometry Summary.
VIII. INTEGRATION: AN INTRODUCTION.
22. Net Change in Amount and Area: Introducing the Definite Integral. Finding Net Change in Amount: Physical and Graphical Interplay.
The Definite Integral.
The Definite Integral: Qualitative Analysis and Signed Area.
Properties of the Definite Integral.
23. The Area Function and Its Characteristics. An Introduction to the Area Function
f (t) dt.
Characteristics of the Area Function.
The Fundamental Theorem of Calculus.
24. The Fundamental Theorem of Calculus. Definite Integrals and the Fundamental Theorem.
The Average Value of a Function: An Application of the Definite Integral.
IX. APPLICATIONS AND COMPUTATION OF THE INTEGRAL.
25. Finding Antiderivatives-An Introduction to Indefinite Integration. A List of Basic Antiderivatives.
Substitution: The Chain Rule in Reverse.
Substitution to Alter the Form of an Integral.
26. Numerical Methods of Approximating Definite Integrals. Approximating Sums:
Ln, Rn, Tn, and
Mn.
Simpson's Rule and Error Estimates.
27. Applying the Definite Integral: Slice and Conquer. Finding Mass When Density Varies.
Slicing to Find the Area Between Two Curves.
28. More Applications of Integration. Computing Volumes.
Arc Length, Work, and Fluid Pressure: Additional Applications of the Definite Integral.
29. Computing Integrals. Integration by Parts-The Product Rule in Reverse.
Trigonometric Integrals and Trigonometric Substitution.
Integration Using Partial Fractions.
Improper Integrals.
X. SERIES.
30. Series. Approximating a Function by a Polynomial.
Error Analysis and Taylor's Theorem.
Taylor Series.
Working with Series and Power Series.
Convergence Tests.
31. Differential Equations. Introduction to Modeling with Differential Equations.
Solutions to Differential Equations: An Introduction.
Qualitative Analysis of Solutions to Autonomous Differential Equations.
Solving Separable First Order Differential Equations.
Systems of Differential Equations.
Second Order Homogeneous Differential Equations with Constant Coefficients.
APPENDICES.
A. Algebra. Introduction to Algebra: Expressions and Equations.
Working with Expressions.
Solving Equations.
B. Geometric Formulas. C. The Theoretical Basis of Applications of the Derivative. D. Proof by Induction. E. Conic Sections. Characterizing Conics from a Geometric Viewpoint.
Defining Conics Algebraically.
The Practical Importance of Conic Sections.
F. L'Hopital's Rule: Using Relative Rates of Change to Evaluate Limits. G. Newton's Method: Using Derivatives to Approximate Roots. H. Proofs to Accompany Chapter 30, Series. Index. 