Rent 9780534343309 | Calculus: Concepts and Contexts

Table of Contents

A PREVIEW OF CALCULUS

(8)

1 FUNCTIONS AND MODELS

(84)

1.1 Four Ways to Represent a Function

(14)

... MORE

1.2 New Functions from Old Functions

(15)

1.3 Graphing Calculators and Computers

(7)

1.4 Parametric Curves

(8)

Laboratory Project: Families of Hypocycloids

(1)

1.5 Exponential Functions

(7)

1.6 Inverse Functions and Logarithms

(12)

1.7 Models and Curve Fitting

(9)

Review

(3)

Principles of Problem Solving

(7)

2 LIMITS AND DERIVATIVES

(96)

2.1 The Tangent and Velocity Problems

(5)

2.2 The Limit of a Function

101

(10)

2.3 Calculating Limits Using the Limit Laws

111

(9)

2.4 Continuity

120

(10)

2.5 Limits Involving Infinity

130

(12)

2.6 Tangents, Velocities, and Other Rates of Change

142

(9)

2.7 Derivatives

151

(7)

Writing Project: Early Methods for Finding Tangents

157

(1)

2.8 The Derivative as a Function

158

(13)

2.9 Linear Approximations

171

(4)

2.10 What Does f' Say about f?

175

(7)

Review

182

(4)

Focus on Problem Solving

186

(4)

3 DIFFERENTIATION RULES

190

(76)

3.1 Derivatives of Polynomials and Exponential Functions

192

(9)

3.2 The Product and Quotient Rules

201

(6)

3.3 Rates of Change in the Natural and Social Sciences

207

(13)

3.4 Derivatives of Trigonometric Functions

220

(7)

3.5 The Chain Rule

227

(12)

Laboratory Project: Bezier Curves

237

(1)

Applied Project: Where Should a Pilot Start Descent?

238

(1)

3.6 Implicit Differentiation

239

(8)

3.7 Derivatives of Logarithmic Functions

247

(7)

Discovery Project: Hyperbolic Functions

253

(1)

3.8 Linear Approximations and Differentials

254

(6)

Laboratory Project: Taylor Polynomials

259

(1)

Review

260

(3)

Focus on Problem Solving

263

(3)

4 APPLICATIONS OF DIFFERENTIATION

266

(82)

4.1 Related Rates

268

(6)

4.2 Maximum and Minimum Values

274

(9)

Applied Project: The Calculus of Rainbows

282

(1)

4.3 Derivatives and the Shapes of Curves

283

(11)

4.4 Graphing with Calculus and Calculators

294

(7)

4.5 Indeterminate Forms and L'Hospital's Rule

301

(9)

Writing Project: The Origins of L'Hospital's Rule

310

(1)

4.6 Optimization Problems

310

(12)

Applied Project: The Shape of a Can

321

(1)

4.7 Applications to Economics

322

(5)

4.8 Newton's Method

327

(5)

4.9 Antiderivatives

332

(8)

Review

340

(4)

Focus on Problem Solving

344

(4)

5 INTEGRALS

348

(98)

5.1 Areas and Distances

350

(11)

5.2 The Definite Integral

361

(11)

5.3 Evaluating Definite Integrals

372

(11)

Discovery Project: Area Functions

382

(1)

5.4 The Fundamental Theorem of Calculus

383

(10)

Writing Project: Newton, Leibniz, and the Invention of Calculus

392

(1)

5.5 The Substitution Rule

393

(9)

5.6 Integration by Parts

402

(6)

5.7 Integration Using Tables and Computer Algebra Systems

408

(8)

Discovery Project: Patterns in Integrals

414

(2)

5.8 Approximate Integration

416

(11)

5.9 Improper Integrals

427

(10)

Review

437

(5)

Focus on Problem Solving

442

(4)

6 APPLICATIONS OF INTEGRATION

446

(56)

6.1 More about Areas

448

(7)

6.2 Volumes

455

(9)

6.3 Arc Length

464

(5)

6.4 Average Value of a Function

469

(4)

Applied Project: Where to Sit at the Movies

472

(1)

6.5 Applications to Physics and Engineering

473

(11)

6.6 Applications to Economics and Biology

484

(5)

6.7 Probability

489

(7)

Review

496

(3)

Focus on Problem Solving

499

(3)

7 DIFFERENTIAL EQUATIONS

502

(56)

7.1 Modeling with Differential Equations

504

(5)

7.2 Direction Fields

509

(5)

7.3 Euler's Method

514

(4)

7.4 Separable Equations

518

(9)

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

526

(1)

7.5 Exponential Growth and Decay

527

(10)

Applied Project: Calculus and Baseball

536

(1)

7.6 The Logistic Equation

537

(9)

7.7 Predator-Prey Systems

546

(7)

Review

553

(3)

Focus on Problem Solving

556

(2)

8 INFINITE SEQUENCES AND SERIES

558

(86)

8.1 Sequences

560

(10)

Laboratory Project: Logistic Sequences

569

(1)

8.2 Series

570

(9)

8.3 The Integral and Comparison Tests: Estimating Sums

579

(10)

8.4 Other Convergence Tests

589

(8)

8.5 Power Series

597

(6)

8.6 Representations of Functions as Power Series

603

(5)

8.7 Taylor and Maclaurin Series

608

(11)

8.8 The Binomial Series

619

(5)

Writing Project: How Newton Discovered the Binomial Series

623

(1)

8.9 Applications of Taylor Polynomials

624

(9)

Applied Project: Radiation from the Stars

632

(1)

8.10 Using Series to Solve Differential Equations

633

(4)

Review

637

(3)

Focus on Problem Solving

640

(4)

9 VECTORS AND THE GEOMETRY OF SPACE

644

(58)

9.1 Three-Dimensional Coordinate Systems

646

(5)

9.2 Vectors

651

(9)

9.3 The Dot Product

660

(6)

9.4 The Cross Product

666

(9)

Discovery Project: The Geometry of a Tetrahedron

674

(1)

9.5 Equations of Lines and Planes

675

(9)

9.6 Functions and Surfaces

684

(8)

9.7 Cylindrical and Spherical Coordinates

692

(6)

Laboratory Project: Families of Surfaces

697

(1)

Review

698

(3)

Focus on Problem Solving

701

(1)

10 VECTOR FUNCTIONS

702

(44)

10.1 Vector Functions and Space Curves

704

(6)

10.2 Derivatives and Integrals of Vector Functions

710

(6)

10.3 Arc Length and Curvature

716

(8)

10.4 Motion in Space

724

(10)

Applied Project: Kepler's Laws

733

(1)

10.5 Parametric Surfaces

734

(6)

Review

740

(3)

Focus on Problem Solving

743

(3)

11 PARTIAL DERIVATIVES

746

(92)

11.1 Functions of Several Variables

748

(11)

11.2 Limits and Continuity

759

(7)

11.3 Partial Derivatives

766

(12)

11.4 Tangent Planes and Linear Approximations

778

(11)

11.5 The Chain Rule

789

(9)

11.6 Directional Derivatives and the Gradient Vector

798

(13)

11.7 Maximum and Minimum Values

811

(11)

Applied Project: Designing a Dumpster

820

(1)

Discovery Project: Quadratic Approximations and Critical Points

821

(1)

11.8 Lagrange Multipliers

822

(9)

Applied Project: Rocket Science

829

(1)

Applied Project: Hydro-Turbine Optimization

830

(1)

Review

831

(5)

Focus on Problem Solving

836

(2)

12 MULTIPLE INTEGRALS

838

(78)

12.1 Double Integrals over Rectangles

840

(9)

12.2 Iterated Integrals

849

(6)

12.3 Double Integrals over General Regions

855

(8)

12.4 Double Integrals in Polar Coordinates

863

(6)

12.5 Applications of Double Integrals

869

(10)

12.6 Surface Area

879

(5)

12.7 Triple Integrals

884

(10)

Discovery Project: Volumes of Hyperspheres

894

(1)

12.8 Triple Integrals in Cylindrical and Spherical Coordinates

894

(8)

Applied Project: Roller Derby

900

(1)

Discovery Project: The Intersection of Three Cylinders

901

(1)

12.9 Change of Variables in Multiple Integrals

902

(8)

Review

910

(4)

Focus on Problem Solving

914

(2)

13 VECTOR CALCULUS

916

13.1 Vector Fields

918

(6)

13.2 Line Integrals

924

(12)

13.3 The Fundamental Theorem for Line Integrals

936

(10)

13.4 Green's Theorem

946

(7)

13.5 Curl and Divergence

953

(8)

13.6 Surface Integrals

961

(12)

13.7 Stokes' Theorem

973

(6)

Writing Project: Three Men and Two Theorems

978

(1)

13.8 The Divergence Theorem

979

(7)

13.9 Summary

986

(1)

Review

987

(3)

Focus on Problem Solving

990

APPENDIXES

(128)

A Intervals, Inequalities, and Absolute Values

(5)

B Coordinate Geometry

(12)

C Trigonometry

A19

(13)

D Precise Definitions of Limits

A32

(9)

E A Few Proofs

A41

(5)

F Integration of Rational Functions by Partial Fractions

A46

(9)

G Polar Coordinates

A55

(16)

H Complex Numbers

A71

(9)

I Answers to Odd-Numbered Exercises

A80

(49)

INDEX

A129

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Calculus: Concepts and Contexts

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