| Preface |
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xiv | |
| To the Student |
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xxvi | |
| A Preview of Calculus |
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2 | (8) |
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10 | (76) |
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Four Ways to Represent a Function |
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11 | (14) |
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Mathematical Models: A Catalog of Essential Functions |
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25 | (13) |
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New Functions from Old Functions |
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38 | (10) |
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Graphing Calculators and Computers |
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48 | (7) |
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55 | (8) |
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Inverse Functions and Logarithms |
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63 | (23) |
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77 | (3) |
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Principles of Problem Solving |
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80 | (6) |
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86 | (96) |
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The Tangent and Velocity Problems |
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87 | (5) |
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92 | (12) |
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Calculating Limits Using the Limit Laws |
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104 | (10) |
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The Precise Definition of a Limit |
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114 | (10) |
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124 | (11) |
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Limits at Infinity; Horizontal Asymptotes |
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135 | (14) |
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Tangents, Velocities, and Other Rates of Change |
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149 | (9) |
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158 | (7) |
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Writing Project Early Methods for Finding Tangents |
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164 | (1) |
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The Derivative as a Function |
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165 | (17) |
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176 | (4) |
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180 | (2) |
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182 | (96) |
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Derivatives of Polynomials and Exponential Functions |
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183 | (9) |
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The Product and Quotient Rules |
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192 | (7) |
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Rates of Change in the Natural and Social Sciences |
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199 | (12) |
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Derivatives of Trigonometric Functions |
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211 | (6) |
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217 | (10) |
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227 | (9) |
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236 | (8) |
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Applied Project Where Should a Pilot Start Descent? |
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243 | (1) |
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Applied Project Building a Better Roller Coaster |
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243 | (1) |
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Derivatives of Logarithmic Functions |
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244 | (6) |
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250 | (6) |
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256 | (6) |
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Linear Approximations and Differentials |
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262 | (16) |
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Laboratory Project Taylor Polynomials |
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269 | (1) |
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270 | (4) |
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274 | (4) |
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Applications of Differentiation |
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278 | (90) |
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Maximum and Minimum Values |
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279 | (11) |
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Applied Project The Calculus of Rainbows |
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288 | (2) |
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290 | (6) |
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How Derivatives Affect the Shape of a Graph |
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296 | (11) |
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Indeterminate Forms and L'Hospital's Rule |
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307 | (9) |
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Writing Project The Origins of L'Hospital's Rule |
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315 | (1) |
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Summary of Curve Sketching |
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316 | (8) |
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Graphing with Calculus and Calculators |
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324 | (7) |
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331 | (11) |
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Applied Project The Shape of a Can |
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341 | (1) |
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Applications to Business and Economics |
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342 | (5) |
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347 | (6) |
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353 | (15) |
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361 | (4) |
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365 | (3) |
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368 | (68) |
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369 | (11) |
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380 | (14) |
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Discovery Project Area Functions |
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393 | (1) |
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The Fundamental Theorem of Calculus |
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394 | (11) |
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Indefinite Integrals and the Net Change Theorem |
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405 | (9) |
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Writing Project Newton, Leibniz, and the Invention of Calculus |
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413 | (1) |
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414 | (8) |
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The Logarithm Defined as an Integral |
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422 | (14) |
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430 | (4) |
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434 | (2) |
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Applications of Integration |
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436 | (38) |
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437 | (7) |
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444 | (11) |
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Volumes by Cylindrical Shells |
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455 | (5) |
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460 | (4) |
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Average Value of a Function |
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464 | (10) |
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Applied Project Where to Sit at the Movies |
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468 | (1) |
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468 | (2) |
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470 | (4) |
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Techniques of Integration |
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474 | (72) |
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475 | (7) |
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482 | (7) |
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Trigonometric Substitution |
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489 | (7) |
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Integration of Rational Functions by Partial Fractions |
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496 | (9) |
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505 | (6) |
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Integration Using Tables and Computer Algebra Systems |
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511 | (7) |
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Discovery Project Patterns in Integrals |
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517 | (1) |
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518 | (12) |
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530 | (16) |
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540 | (3) |
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543 | (3) |
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Further Applications of Integration |
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546 | (40) |
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547 | (7) |
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Discovery Project Arc Length Contest |
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554 | (1) |
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Area of a Surface of Revolution |
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554 | (7) |
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Discovery Project Rotating on a Slant |
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560 | (1) |
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Applications to Physics and Engineering |
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561 | (10) |
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Applications to Economics and Biology |
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571 | (4) |
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575 | (11) |
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582 | (2) |
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584 | (2) |
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586 | (64) |
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Modeling with Differential Equations |
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587 | (5) |
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Direction Fields and Euler's Method |
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592 | (9) |
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601 | (10) |
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Applied Project How Fast Does a Tank Drain? |
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609 | (1) |
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Applied Project Which Is Faster, Going Up or Coming Down? |
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610 | (1) |
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Exponential Growth and Decay |
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611 | (12) |
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Applied Project Calculus and Baseball |
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622 | (1) |
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623 | (9) |
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632 | (6) |
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638 | (12) |
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644 | (4) |
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648 | (2) |
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Parametric Equations and Polar Coordinates |
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650 | (50) |
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Curves Defined by Parametric Equations |
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651 | (9) |
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Laboratory Project Running Circles around Circles |
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659 | (1) |
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Calculus with Parametric Curves |
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660 | (9) |
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Laboratory Project Bezier Curves |
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669 | (1) |
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669 | (10) |
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Areas and Lengths in Polar Coordinates |
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679 | (5) |
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684 | (8) |
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Conic Sections in Polar Coordinates |
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692 | (8) |
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696 | (3) |
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699 | (1) |
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Infinite Sequences and Series |
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700 | (92) |
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701 | (12) |
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Laboratory Project Logistic Sequences |
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713 | (1) |
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713 | (10) |
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The Integral Test and Estimates of Sums |
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723 | (7) |
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730 | (5) |
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735 | (5) |
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Absolute Convergence and the Ratio and Root Tests |
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740 | (7) |
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Strategy for Testing Series |
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747 | (2) |
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749 | (5) |
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Representations of Functions as Power Series |
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754 | (6) |
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Taylor and Maclaurin Series |
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760 | (12) |
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Laboratory Project An Elusive Limit |
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772 | (1) |
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772 | (4) |
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Writing Project How Newton Discovered the Binomial Series |
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776 | (1) |
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Applications of Taylor Polynomials |
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776 | (16) |
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Applied Project Radiation from the Stars |
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785 | (1) |
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786 | (3) |
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789 | (3) |
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Vectors and the Geometry of Space |
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792 | (56) |
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Three-Dimensional Coordinate Systems |
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793 | (5) |
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798 | (9) |
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807 | (7) |
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814 | (8) |
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Discovery Project The Geometry of a Tetrahedron |
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822 | (1) |
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Equations of Lines and Planes |
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822 | (10) |
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Laboratory Project Putting 3D in Perspective |
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832 | (1) |
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Cylinders and Quadric Surfaces |
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832 | (7) |
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Cylindrical and Spherical Coordinates |
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839 | (9) |
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Laboratory Project Families of Surfaces |
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844 | (1) |
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844 | (3) |
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847 | (1) |
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848 | (38) |
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Vector Functions and Space Curves |
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849 | (7) |
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Derivatives and Integrals of Vector Functions |
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856 | (6) |
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862 | (8) |
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Motion in Space: Velocity and Acceleration |
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870 | (16) |
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Applied Project Kepler's Laws |
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880 | (1) |
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881 | (3) |
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884 | (2) |
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886 | (94) |
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Functions of Several Variables |
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887 | (15) |
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902 | (7) |
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909 | (14) |
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Tangent Planes and Linear Approximations |
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923 | (8) |
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931 | (9) |
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Directional Derivatives and the Gradient Vector |
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940 | (13) |
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Maximum and Minimum Values |
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953 | (12) |
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Applied Project Designing a Dumpster |
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963 | (1) |
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Discovery Project Quadratic Approximations and Critical Points |
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964 | (1) |
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965 | (15) |
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Applied Project Rocket Science |
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972 | (1) |
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Applied Project Hydro-Turbine Optimization |
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973 | (1) |
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974 | (4) |
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978 | (2) |
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980 | (74) |
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Double Integrals over Rectangles |
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981 | (8) |
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989 | (6) |
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Double Integrals over General Regions |
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995 | (8) |
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Double Integrals in Polar Coordinates |
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1003 | (6) |
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Applications of Double Integrals |
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1009 | (10) |
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1019 | (4) |
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1023 | (10) |
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Discovery Project Volumes of Hyperspheres |
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1032 | (1) |
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Triple Integrals in Cylindrical and Spherical Coordinates |
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1033 | (8) |
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Applied Project Roller Derby |
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1039 | (1) |
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Discovery Project The Intersection of Three Cylinders |
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1040 | (1) |
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Change of Variables in Multiple Integrals |
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1041 | (13) |
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1049 | (3) |
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1052 | (2) |
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1054 | (86) |
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1055 | (7) |
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1062 | (12) |
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The Fundamental Theorem for Line Integrals |
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1074 | (9) |
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1083 | (7) |
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1090 | (8) |
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Parametric Surfaces and Their Areas |
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1098 | (11) |
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1109 | (12) |
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1121 | (6) |
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Writing Project Three Men and Two Theorems |
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1126 | (1) |
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1127 | (7) |
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1134 | (6) |
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1135 | (3) |
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1138 | (2) |
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Second-Order Differential Equations |
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1140 | |
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Second-Order Linear Equations |
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1141 | (6) |
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Nonhomogeneous Linear Equations |
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1147 | (8) |
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Applications of Second-Order Differential Equations |
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1155 | (8) |
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1163 | |
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1167 | |
| Appendixes |
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1 | (124) |
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A Numbers, Inequalities, and Absolute Values |
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2 | (8) |
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B Coordinate Geometry and Lines |
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10 | (6) |
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C Graphs of Second-Degree Equations |
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16 | (8) |
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24 | (10) |
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34 | (5) |
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39 | (10) |
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49 | (8) |
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H Answers to Odd-Numbered Exercises |
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57 | (68) |
| Index |
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125 | |