Calculus with Differential Equations

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  • Edition: 9th
  • Format: Hardcover
  • Copyright: 2006-04-10
  • Publisher: Pearson

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This the shortest mainstream calculus book available. The authors make effective use of computing technology, graphics, and applications, and provide at least two technology projects per chapter. This popular book is correct without being excessively rigorous, up-to-date without being faddish.Maintains a strong geometric and conceptual focus. Emphasizes explanation rather than detailed proofs. Presents definitions consistently throughout to maintain a clear conceptual framework. Provides hundreds of new problems, including problems on approximations, functions defined by tables, and conceptual questions.Ideal for readers preparing for the AP Calculus exam or who want to brush up on their calculus with a no-nonsense, concisely written book.

Table of Contents

Prefacep. ix
Preliminariesp. 1
Real Numbers, Estimation, and Logicp. 1
Inequalities and Absolute Valuesp. 8
The Rectangular Coordinate Systemp. 16
Graphs of Equationsp. 24
Functions and Their Graphsp. 29
Operations on Functionsp. 35
Trigonometric Functionsp. 41
Chapter Reviewp. 51
Review and Preview Problemsp. 54
Limitsp. 55
Introduction to Limitsp. 55
Rigorous Study of Limitsp. 61
Limit Theoremsp. 68
Limits Involving Trigonometric Functionsp. 73
Limits at Infinity; Infinite Limitsp. 77
Continuity of Functionsp. 82
Chapter Reviewp. 90
Review and Preview Problemsp. 92
The Derivativep. 93
Two Problems with One Themep. 93
The Derivativep. 100
Rules for Finding Derivativesp. 107
Derivatives of Trigonometric Functionsp. 114
The Chain Rulep. 118
Higher-Order Derivativesp. 125
Implicit Differentiationp. 130
Related Ratesp. 135
Differentials and Approximationsp. 142
Chapter Reviewp. 147
Review and Preview Problemsp. 150
Applications of the Derivativep. 151
Maxima and Minimap. 151
Monotonicity and Concavityp. 155
Local Extrema and Extrema on Open Intervalsp. 162
Practical Problemsp. 167
Graphing Functions Using Calculusp. 178
The Mean Value Theorem for Derivativesp. 185
Solving Equations Numericallyp. 190
Antiderivativesp. 197
Introduction to Differential Equationsp. 203
Chapter Reviewp. 209
Review and Preview Problemsp. 214
The Definite Integralp. 215
Introduction to Areap. 215
The Definite Integralp. 224
The First Fundamental Theorem of Calculusp. 232
The Second Fundamental Theorem of Calculus and the Method of Substitutionp. 243
The Mean Value Theorem for Integrals and the Use of Symmetryp. 253
Numerical Integrationp. 260
Chapter Reviewp. 270
Review and Preview Problemsp. 274
Applications of the Integralp. 275
The Area of a Plane Regionp. 275
Volumes of Solids: Slabs, Disks, Washersp. 281
Volumes of Solids of Revolution: Shellsp. 288
Length of a Plane Curvep. 294
Work and Fluid Forcep. 301
Moments and Center of Massp. 308
Probability and Random Variablesp. 316
Chapter Reviewp. 322
Review and Preview Problemsp. 324
Transcendental Functionsp. 325
The Natural Logarithm Functionp. 325
Inverse Functions and Their Derivativesp. 331
The Natural Exponential Functionp. 337
General Exponential and Logarithmic Functionsp. 342
Exponential Growth and Decayp. 347
First-Order Linear Differential Equationsp. 355
Approximations for Differential Equationsp. 359
The Inverse Trigonometric Functions and Their Derivativesp. 365
The Hyperbolic Functions and Their Inversesp. 374
Chapter Reviewp. 380
Review and Preview Problemsp. 382
Techniques of Integrationp. 383
Basic Integration Rulesp. 383
Integration by Partsp. 387
Some Trigonometric Integralsp. 393
Rationalizing Substitutionsp. 399
Integration of Rational Functions Using Partial Fractionsp. 404
Strategies for Integrationp. 411
Chapter Reviewp. 419
Review and Preview Problemsp. 422
Indeterminate Forms and Improper Integralsp. 423
Indeterminate Forms of Type 0/0p. 423
Other Indeterminate Formsp. 428
Improper Integrals: Infinite Limits of Integrationp. 433
Improper Integrals: Infinite Integrandsp. 442
Chapter Reviewp. 446
Review and Preview Problemsp. 448
Infinite Seriesp. 449
Infinite Sequencesp. 449
Infinite Seriesp. 455
Positive Series: The Integral Testp. 463
Positive Series: Other Testsp. 468
Alternating Series, Absolute Convergence, and Conditional Convergencep. 474
Power Seriesp. 479
Operations on Power Seriesp. 484
Taylor and Maclaurin Seriesp. 489
The Taylor Approximation to a Functionp. 497
Chapter Reviewp. 504
Review and Preview Problemsp. 508
Conics and Polar Coordinatesp. 509
The Parabolap. 509
Ellipses and Hyperbolasp. 513
Translation and Rotation of Axesp. 523
Parametric Representation of Curves in the Planep. 530
The Polar Coordinate Systemp. 537
Graphs of Polar Equationsp. 542
Calculus in Polar Coordinatesp. 547
Chapter Reviewp. 552
Review and Preview Problemsp. 554
Geometry in Space and Vectorsp. 555
Cartesian Coordinates in Three-Spacep. 555
Vectorsp. 560
The Dot Productp. 566
The Cross Productp. 574
Vector-Valued Functions and Curvilinear Motionp. 579
Lines and Tangent Lines in Three-Spacep. 589
Curvature and Components of Accelerationp. 593
Surfaces in Three-Spacep. 603
Cylindrical and Spherical Coordinatesp. 609
Chapter Reviewp. 613
Review and Preview Problemsp. 616
Derivatives for Functions of Two or More Variablesp. 617
Functions of Two or More Variablesp. 617
Partial Derivativesp. 624
Limits and Continuityp. 629
Differentiabilityp. 635
Directional Derivatives and Gradientsp. 641
The Chain Rulep. 647
Tangent Planes and Approximationsp. 652
Maxima and Minimap. 657
The Method of Lagrange Multipliersp. 666
Chapter Reviewp. 672
Review and Preview Problemsp. 674
Multiple Integralsp. 675
Double Integrals over Rectanglesp. 675
Iterated Integralsp. 680
Double Integrals over Nonrectangular Regionsp. 684
Double Integrals in Polar Coordinatesp. 691
Applications of Double Integralsp. 696
Surface Areap. 700
Triple Integrals in Cartesian Coordinatesp. 706
Triple Integrals in Cylindrical and Spherical Coordinatesp. 713
Change of Variables in Multiple Integralsp. 718
Chapter Reviewp. 728
Review and Preview Problemsp. 730
Vector Calculusp. 731
Vector Fieldsp. 731
Line Integralsp. 735
Independence of Pathp. 742
Green's Theorem in the Planep. 749
Surface Integralsp. 755
Gauss's Divergence Theoremp. 764
Stokes's Theoremp. 770
Chapter Reviewp. 773
Differential Equationsp. 775
Linear Homogeneous Equationsp. 775
Nonhomogeneous Equationsp. 779
Applications of Second-Order Equationsp. 783
Chapter Reviewp. 788
Appendixp. A-1
Mathematical Inductionp. A-1
Proofs of Several Theoremsp. A-3
Answers to Odd-Numbered Problemsp. A-7
Indexp. I-1
Photo CreditsP-1
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