A PREVIEW OF CALCULUS

1 Functions and Models

1.1 Four Ways to Represent a Function

1.2 Mathematical Models:A Catalog of Essential Functions

1.3 New Functions from Old Functions

1.4 Graphing Calculators and Computers

1.5 Exponential Functions

1.6 Inverse Functions and Logarithms

Review

Principles of Problem Solving

2 Limits and Derivatives

2.1 The Tangent and Velocity Problems

2.2 The Limit of a Function

2.3 Calculating Limits Using the Limit Laws

2.4 The Precise Definition of a Limit

2.5 Continuity

2.6 Limits at Infinity;Horizontal Asymptotes

2.7 Derivatives and Rates of Change

Writing Project Early Methods for Finding Tangents

2.8 The Derivative as a Function

Review

Problems Plus

3 Differentiation Rules

3.1 Derivatives of Polynomials and Exponential Functions

Applied Project Building a Better Roller Coaster

3.2 The Product and Quotient Rules

3.3 Derivatives of Trigonometric Functions

3.4 The Chain Rule

Applied Project Where Should a Pilot Start Descent?

3.5 Implicit Differentiation

Laboratory Project Families of Implicit Curves

3.6 Derivatives of Logarithmic Functions

3.7 Rates of Change in the Natural and Social Sciences

3.8 Exponential Growth and Decay

3.9 Related Rates

3.10 Linear Approximations and Differentials

Laboratory Project Taylor Polynomials

3.11 Hyperbolic Functions

Review

Problems Plus

4 Applications of Differentiation

4.1 Maximum and Minimum Values

Applied Project The Calculus of Rainbows

4.2 The Mean Value Theorem

4.3 How Derivatives Affect the Shape of a Graph

4.4 Indeterminate Forms and l'Hospital's Rule

Writing Project The Origins of I'Hospital's Rule

4.5 Summary of Curve Sketching

4.6 Graphing with Calculus and Calculators

4.7 Optimization Problems

Applied Project The Shape of a Can

4.8 Newton's Method

4.9 Antiderivatives

Review

Problems Plus

5 Integrals

5.1 Areas and Distances

5.2 The Definite Integral

Discovery Project Area Functions

5.3 The Fundamental Theorem of Calculus

5.4 Indefinite Integrals and the Net Change Theorem

Writing Project Newton,Leibniz,and the Invention of Calculus

5.5 The Substitution Rule

Review

Problems Plus

6 Applications of Integration

6.1 Areas Between Curves

Applied Project The Gini Index

6.2 Volumes

6.3 Volumes by Cylindrical Shells

6.4 Work

6.5 Average Value of a Function

Applied Project Calculus and Baseball

Applied Project Where to Sit at the Movies

Review

Problems Plus

7 Techniques of Integration

7.1 Integration by Parts

7.2 Trigonometric Integrals

7.3 Trigonometric Substitution

7.4 Integration of Rational Functions by Partial Fractions

7.5 Strategy for Integration

7.6 Integration Using Tables and Computer Algebra Systems

Discovery Project Patterns in Integrals

7.7 Approximate Integration

7.8 Improper Integrals

Review

Problems Plus

8 Further Applications of Integration

8.1 Arc Length

Discovery Project Arc Length Contest

8.2 Area of a Surface of Revolution

Discovery Project Rotating on a Slant

8.3 Applications to Physics and Engineering

Discovery Project Complementary Coffee Cups

8.4 Applications to Economics and Biology

8.5 Probability

Review

Problems Plus

9 Differential Equations

9.1 Modeling with Differential Equations

9.2 Direction Fields and Euler's Method

9.3 Separable Equations

Applied Project How Fast Does a Tank Drain?

Applied Project Which Is Faster,Going Up or Coming Down?

9.4 Models for Population Growth

9.5 Linear Equations

9.6 Predator-Prey Systems

Review

Problems Plus

10 Parametric Equations and Polar Coordinates

10.1 Curves Defined by Parametric Equations

Laboratory Project Running Circles around Circles

10.2 Calculus with Parametric Curves

Laboratory Project Bézier Curves

10.3 Polar Coordinates

Laboratory Project Families of Polar Curves

10.4 Areas and Lengths in Polar Coordinates

10.5 Conic Sections

10.6 Conic Sections in Polar Coordinates

Review

Problems Plus