A Preview of Calculus

1 Functions and Models

1.1Four Ways to Represent a Function

1.2Mathematical Models：A Catalog of Essential Functions

1.3New Functions from Old Functions

1.4Graphing Calculators and Computers

1.5Exponential Functions

1.6Inverse Functions and Logarithms

Review

Principles of Problem Solving

2 Limits and Derivatives

2.1The Tangent and Velocity Problems

2.2The Limit of a Function

2.3Calculating Limits Using the Limit Laws

2.4The Precise Definition of a Limit

2.5Continuity

2.6Limits at Infinity；Horizontal Asymptotes

2.7Tangents，Velocities，and Other Rates of Change

2.8Derivatives

Writing Project。Early Methods for Finding Tangents

2.9The Derivative as a Function

Review

Problems Plus

3 Differentiation Aules

3.1Derivatives of Polynomials and Exponential Functions

3.2The Product and Quotient Rules

3.3Rates of Change in the Natural and Social Sciences

3.4Derivatives of Trigonometric Functions

3.5The Chain Rule

3.6Implicit Differentiation

3.7Higher Derivatives

Applied Project。Where Should a Pilot Start Descent?

Applied Project。Building a Better Roller Goaster

3.8Derivatives of Logarithmic Functions

3.9Hyperbolic Functions

3.10Related Rates

3.11Linear Approximations and Differentials

Laboratory Project。Taylor Polynomials

Review

Problems Plus

4 Applications of Differentiation

4.1Maximum and Minimum Values

Applied Project。The Calculus of Rainbows

4.2The Mean Value Theorem

4.3How Derivatives Affect the Shape of a Graph

4.4Indeterminate Forms and L’Hospital’s Rule

Writing Project。The Origins of L’Hospital’s Rule

4.5Summary of Curve Sketching

4.6Graphing with Calculus and Calculators

4.7 Optimization Problems

Applied Project。The Shape of a Can

4.8 Applications to Business and Economics

4.9 Newton’s Method

4.10Antiderivatives

Review

Problems PIus

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 Galculus

5.5 The Substitution Rule

5.6 The Logarithm Defined as an Integral

Review

Problems Plus

6 Applications of lntegration

6.1 Areas between Curves

6.2 Volumes

6.3 Volumes by Cylindrical Shells

6.4 Work

6.5 Average Value of a Function

Applied Project。Where to Sit at the Movies

Review

Problems Plus

7 Techniques of Integration

7.1Integration by Parts

7.2Trigonometric Integrals

7.3Trigonometric Substitution

7.4Integration of Rational Functions by Partial Fractions

7.5Strategy for Integration

7.6Integration Using Tables and Computer Algebra Systems

Discovery Project 。 Patterns in Integrals

7.7Approximate Integration

7.8Improper Integrals

Review

Problems Plus

8 Further Applications of lntegration

8.1Arc Length

Discovery Project。Arc Length Contest

8.2Area of a Surface of Revolution

Discovery Project 。 Rotating on a 51ant

8.3Applications to Physics and Engineering

8.4Applications to Economics and Biology

8.5Probability

Review

Problems Plus

9 Differential Equations

9.1Modeling with Differential Equations

9.2Direction Fields and Euler’s Method

9.3Separable Equations

Applied Project。How Fast Does a Tank Drain?

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

9.4Exponential Growth and Decay

Applied Project。Calculus and Baseball

9.5The Logistic Equation

9.6Linear Equations

9.7Predator-Prey Systems

Review

Problems Plus

10 Parametric Equations and Poiar Coordinates

10.1Curves Defined by Parametric Equations

Laboratory Project。Running Circles around Circles

10.2Calculus with Parametric Curves

Laboratory Project。5ezier Curves

10.3Polar Coordinates

10.4Areas and Lengths in Polar Coordinates

10.5Conic Sections

10.6Conic Sections in Polar Coordinates

Review

Problems Plus

11 Infinite Sequences and Series

11.1Sequences

Laboratory Project。Logistic Sequences

11.2Series

11.3The Integral Test and Estimates of Sums

11.4The Comparison Tests

11.5Alternating Series

11.6Absolute Convergence and the Ratio and Root Tests

11.7Strategy for Testing Series

11.8Power Series

11.9Representations of Functions as Power Series

11.10Taylor and Maclaurin Series

Laboratory Project。An Elusive Limit

11.11The Binomial Series

Writing Project。How Newton Discovered the Binomial 5eriee

11.12Applications of Taylor Polynomials

Applied Project。Radiation from the Stars

Review

Problems Plue

12 Vectors and the Geometrq of Space

12.1Three-Dimensional Coordinate Systems

12.2Vectors

12.3The Dot Product

12.4The Cross Product

Discovery Project。The Geometry of a Tetrahedron

12.5Equations of Lines and Planes

Laboratory Project。Putting 3D in Perspective

12.6Cylinders and Quadric Surfaces

12.7Cylindrical and Spherical Coordinates

Laboratory Project。Families of Surfaces

Review

Problems Plus

13 Vector Functions

13.1Vector Functions and Space Curves

13.2Derivatives and Integrals of Vector Functions

13.3Arc Length and Curvature

13.4Motion in Space：Velocity and Acceleration

Applied Project。Kepler’s Laws

Review

Problems Plus

14 Partial DeriVatiVeS

14.1Functions of Several Variables

14.2Limits and Continuity

14.3Partial Derivatives

14.4Tangent Planes and Linear Approximations

14.5The Chain Rule

14.6Directional Derivatives and the Gradient Vector

14.7Maximum and Minimum Values

Applied Project。Designing a Dumpster

Discovery Project。Quadratic Approximations and Critical Points

14.8Lagrange Multipliers

Applied ProjectRocket Science

Applied ProjectHydro-Turbine Optimization

Review

Problems Plus

15 Multiple Integrals

15.1Double Integrals over Rectangles

15.2Iterated Integrals

15.3Double Integrals over General Regions

15.4Double Integrals in Polar Coordinates

15.5Applications of Double Integrals

15.6Surface Area

15.7Triple Integrals

Discovery Project。Volumes of Hyperspheres

15.8Triple Integrals in Cylindrical and Spherical Coordinates

Applied Project。Roller Derby

Discovery Project。The Intersection of Three Cylinders

15.9Change of Variables in Multiple Integrals

Review

Problems Plus

16VeCtor CalCUlUS

16.1Vector Fields

16.2Line Integrals

16.3The Fundamental Theorem for Line Integrals

16.4Green’s Theorem

16.5Curl and Divergence

16.6Parametric Surfaces and Their Areas

16.7Surface Integrals

16.8Stokes’Theorem

Writing Project。Three Men and Two Theorems

16.9The Divergence Theorem

16.10Summary

Review

Problems Plus

17 Second-Drder Differential Equations

17.1Second-Order Linear Equations

17.2Nonhomogeneous Linear Equations

17.3Applications of Second-Order Differential Equations

17.4Series Solutions

Review

AppendixesAl

ANumbers，Inequalities，and Absolute ValuesA

BCoordinate Geometry and LinesA

CGraphs of Second-Degree EquationsA

DTrigonometryA

ESigma NotationA

FProofs of TheoremsA

GComplex NumbersA

HAnswers to Odd-Numbered ExercisesA

IndexA