Calculus with Differential Equations

by Varberg, Dale; Purcell, Edwin J.; Rigdon, Steve E.

• ISBN13: 9780132306331
• ISBN10: 0132306336
• Edition: 9th
• Format: Hardcover
• Copyright: 2006-04-10
• Publisher: Pearson

Summary

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

Preface	p. ix
Preliminaries	p. 1
Real Numbers, Estimation, and Logic	p. 1
Inequalities and Absolute Values	p. 8
The Rectangular Coordinate System	p. 16
Graphs of Equations	p. 24
Functions and Their Graphs	p. 29
Operations on Functions	p. 35
Trigonometric Functions	p. 41
Chapter Review	p. 51
Review and Preview Problems	p. 54
Limits	p. 55
Introduction to Limits	p. 55
Rigorous Study of Limits	p. 61
Limit Theorems	p. 68
Limits Involving Trigonometric Functions	p. 73
Limits at Infinity; Infinite Limits	p. 77
Continuity of Functions	p. 82
Chapter Review	p. 90
Review and Preview Problems	p. 92
The Derivative	p. 93
Two Problems with One Theme	p. 93
The Derivative	p. 100
Rules for Finding Derivatives	p. 107
Derivatives of Trigonometric Functions	p. 114
The Chain Rule	p. 118
Higher-Order Derivatives	p. 125
Implicit Differentiation	p. 130
Related Rates	p. 135
Differentials and Approximations	p. 142
Chapter Review	p. 147
Review and Preview Problems	p. 150
Applications of the Derivative	p. 151
Maxima and Minima	p. 151
Monotonicity and Concavity	p. 155
Local Extrema and Extrema on Open Intervals	p. 162
Practical Problems	p. 167
Graphing Functions Using Calculus	p. 178
The Mean Value Theorem for Derivatives	p. 185
Solving Equations Numerically	p. 190
Antiderivatives	p. 197
Introduction to Differential Equations	p. 203
Chapter Review	p. 209
Review and Preview Problems	p. 214
The Definite Integral	p. 215
Introduction to Area	p. 215
The Definite Integral	p. 224
The First Fundamental Theorem of Calculus	p. 232
The Second Fundamental Theorem of Calculus and the Method of Substitution	p. 243
The Mean Value Theorem for Integrals and the Use of Symmetry	p. 253
Numerical Integration	p. 260
Chapter Review	p. 270
Review and Preview Problems	p. 274
Applications of the Integral	p. 275
The Area of a Plane Region	p. 275
Volumes of Solids: Slabs, Disks, Washers	p. 281
Volumes of Solids of Revolution: Shells	p. 288
Length of a Plane Curve	p. 294
Work and Fluid Force	p. 301
Moments and Center of Mass	p. 308
Probability and Random Variables	p. 316
Chapter Review	p. 322
Review and Preview Problems	p. 324
Transcendental Functions	p. 325
The Natural Logarithm Function	p. 325
Inverse Functions and Their Derivatives	p. 331
The Natural Exponential Function	p. 337
General Exponential and Logarithmic Functions	p. 342
Exponential Growth and Decay	p. 347
First-Order Linear Differential Equations	p. 355
Approximations for Differential Equations	p. 359
The Inverse Trigonometric Functions and Their Derivatives	p. 365
The Hyperbolic Functions and Their Inverses	p. 374
Chapter Review	p. 380
Review and Preview Problems	p. 382
Techniques of Integration	p. 383
Basic Integration Rules	p. 383
Integration by Parts	p. 387
Some Trigonometric Integrals	p. 393
Rationalizing Substitutions	p. 399
Integration of Rational Functions Using Partial Fractions	p. 404
Strategies for Integration	p. 411
Chapter Review	p. 419
Review and Preview Problems	p. 422
Indeterminate Forms and Improper Integrals	p. 423
Indeterminate Forms of Type 0/0	p. 423
Other Indeterminate Forms	p. 428
Improper Integrals: Infinite Limits of Integration	p. 433
Improper Integrals: Infinite Integrands	p. 442
Chapter Review	p. 446
Review and Preview Problems	p. 448
Infinite Series	p. 449
Infinite Sequences	p. 449
Infinite Series	p. 455
Positive Series: The Integral Test	p. 463
Positive Series: Other Tests	p. 468
Alternating Series, Absolute Convergence, and Conditional Convergence	p. 474
Power Series	p. 479
Operations on Power Series	p. 484
Taylor and Maclaurin Series	p. 489
The Taylor Approximation to a Function	p. 497
Chapter Review	p. 504
Review and Preview Problems	p. 508
Conics and Polar Coordinates	p. 509
The Parabola	p. 509
Ellipses and Hyperbolas	p. 513
Translation and Rotation of Axes	p. 523
Parametric Representation of Curves in the Plane	p. 530
The Polar Coordinate System	p. 537
Graphs of Polar Equations	p. 542
Calculus in Polar Coordinates	p. 547
Chapter Review	p. 552
Review and Preview Problems	p. 554
Geometry in Space and Vectors	p. 555
Cartesian Coordinates in Three-Space	p. 555
Vectors	p. 560
The Dot Product	p. 566
The Cross Product	p. 574
Vector-Valued Functions and Curvilinear Motion	p. 579
Lines and Tangent Lines in Three-Space	p. 589
Curvature and Components of Acceleration	p. 593
Surfaces in Three-Space	p. 603
Cylindrical and Spherical Coordinates	p. 609
Chapter Review	p. 613
Review and Preview Problems	p. 616
Derivatives for Functions of Two or More Variables	p. 617
Functions of Two or More Variables	p. 617
Partial Derivatives	p. 624
Limits and Continuity	p. 629
Differentiability	p. 635
Directional Derivatives and Gradients	p. 641
The Chain Rule	p. 647
Tangent Planes and Approximations	p. 652
Maxima and Minima	p. 657
The Method of Lagrange Multipliers	p. 666
Chapter Review	p. 672
Review and Preview Problems	p. 674
Multiple Integrals	p. 675
Double Integrals over Rectangles	p. 675
Iterated Integrals	p. 680
Double Integrals over Nonrectangular Regions	p. 684
Double Integrals in Polar Coordinates	p. 691
Applications of Double Integrals	p. 696
Surface Area	p. 700
Triple Integrals in Cartesian Coordinates	p. 706
Triple Integrals in Cylindrical and Spherical Coordinates	p. 713
Change of Variables in Multiple Integrals	p. 718
Chapter Review	p. 728
Review and Preview Problems	p. 730
Vector Calculus	p. 731
Vector Fields	p. 731
Line Integrals	p. 735
Independence of Path	p. 742
Green's Theorem in the Plane	p. 749
Surface Integrals	p. 755
Gauss's Divergence Theorem	p. 764
Stokes's Theorem	p. 770
Chapter Review	p. 773
Differential Equations	p. 775
Linear Homogeneous Equations	p. 775
Nonhomogeneous Equations	p. 779
Applications of Second-Order Equations	p. 783
Chapter Review	p. 788
Appendix	p. A-1
Mathematical Induction	p. A-1
Proofs of Several Theorems	p. A-3
Answers to Odd-Numbered Problems	p. A-7
Index	p. I-1
Photo Credits	P-1
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