| 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 | (54) |
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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 | (16) |
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55 | (3) |
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Principles of Problem Solving |
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58 | (6) |
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Limits and Rates of Change |
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64 | (62) |
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The Tangent and Velocity Problems |
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65 | (5) |
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70 | (12) |
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Calculating Limits Using the Limit Laws |
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82 | (10) |
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The Precise Definition of a Limit |
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92 | (10) |
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102 | (10) |
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Tangents, Velocities, and Other Rates of Change |
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112 | (14) |
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121 | (3) |
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124 | (2) |
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126 | (96) |
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127 | (7) |
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Writing Project Early Methods for Finding Tangents |
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133 | (1) |
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The Derivative as a Function |
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134 | (11) |
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145 | (12) |
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Rates of Change in the Natural and Social Sciences |
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157 | (12) |
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Derivatives of Trigonometric Functions |
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169 | (6) |
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175 | (9) |
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184 | (6) |
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190 | (8) |
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Applied Project Where Should a Pilot Start Descent? |
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197 | (1) |
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Applied Project Building a Better Roller Coaster |
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198 | (1) |
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198 | (7) |
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Linear Approximations and Differentials |
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205 | (17) |
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Laboratory Project Taylor Polynomials |
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212 | (1) |
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213 | (5) |
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218 | (4) |
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Applications of Differentiation |
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222 | (92) |
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Maximum and Minimum Values |
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223 | (11) |
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Applied Project The Calculus of Rainbows |
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232 | (2) |
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234 | (6) |
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How Derivatives Affect the Shape of a Graph |
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240 | (9) |
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Limits at Infinity; Horizontal Asymptotes |
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249 | (14) |
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Summary of Curve Sketching |
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263 | (8) |
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Graphing with Calculus and Calculators |
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271 | (7) |
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278 | (11) |
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Applied Project The Shape of a Can |
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288 | (1) |
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Applications to Business and Economics |
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289 | (5) |
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294 | (6) |
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300 | (14) |
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308 | (3) |
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311 | (3) |
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314 | (60) |
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315 | (11) |
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326 | (14) |
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Discovery Project Area Functions |
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339 | (1) |
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The Fundamental Theorem of Calculus |
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340 | (10) |
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Indefinite Integrals and the Net Change Theorem |
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350 | (10) |
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Writing Project Newton, Leibniz, and the Invention of Calculus |
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359 | (1) |
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360 | (14) |
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368 | (4) |
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372 | (2) |
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Applications of Integration |
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374 | (38) |
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375 | (7) |
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382 | (11) |
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Volumes by Cylindrical Shells |
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393 | (5) |
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398 | (4) |
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Average Value of a Function |
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402 | (10) |
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406 | (2) |
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408 | (4) |
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Inverse Functions: Exponential, Logarithmic, and Inverse Trigonometric Functions |
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412 | (98) |
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413 | (8) |
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Instructors may cover either Sections 7.2--7.4 or Sections 7.2*--7.4*. See the Preface |
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Exponential Functions and Their Derivatives |
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421 | (13) |
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434 | (7) |
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Derivatives of Logarithmic Functions |
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441 | (10) |
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The Natural Logarithmic Function |
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451 | (9) |
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The Natural Exponential Function |
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460 | (7) |
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General Logarithmic and Exponential Functions |
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467 | (10) |
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Inverse Trigonometric Functions |
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477 | (9) |
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Applied Project Where to Sit at the Movies |
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485 | (1) |
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486 | (7) |
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Indeterminate Forms and L'Hospital's Rule |
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493 | (17) |
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Writing Project The Origins of L'Hospital's Rule |
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504 | (1) |
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504 | (4) |
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508 | (2) |
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Techniques of Integration |
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510 | (72) |
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511 | (7) |
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518 | (7) |
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Trigonometric Substitution |
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525 | (7) |
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Integration of Rational Functions by Partial Fractions |
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532 | (9) |
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541 | (6) |
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Integration Using Tables and Computer Algebra Systems |
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547 | (7) |
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Discovery Project Patterns in Integrals |
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553 | (1) |
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554 | (12) |
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566 | (16) |
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576 | (3) |
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579 | (3) |
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Further Applications of Integration |
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582 | (40) |
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583 | (7) |
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Discovery Project Arc Length Contest |
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590 | (1) |
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Area of a Surface of Revolution |
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590 | (7) |
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Discovery Project Rotating on a Slant |
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596 | (1) |
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Applications to Physics and Engineering |
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597 | (10) |
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Applications to Economics and Biology |
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607 | (4) |
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611 | (11) |
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618 | (2) |
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620 | (2) |
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622 | (64) |
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Modeling with Differential Equations |
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623 | (5) |
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Direction Fields and Euler's Method |
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628 | (9) |
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637 | (10) |
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Applied Project How Fast Does a Tank Drain? |
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645 | (1) |
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Applied Project Which Is Faster, Going Up or Coming Down? |
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646 | (1) |
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Exponential Growth and Decay |
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647 | (12) |
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Applied Project Calculus and Baseball |
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658 | (1) |
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659 | (9) |
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668 | (6) |
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674 | (12) |
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680 | (4) |
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684 | (2) |
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Parametric Equations and Polar Coordinates |
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686 | (50) |
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Curves Defined by Parametric Equations |
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687 | (9) |
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Laboratory Project Running Circles around Circles |
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695 | (1) |
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Calculus with Parametric Curves |
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696 | (9) |
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Laboratory Project Bezier Curves |
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705 | (1) |
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705 | (10) |
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Areas and Lengths in Polar Coordinates |
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715 | (5) |
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720 | (8) |
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Conic Sections in Polar Coordinates |
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728 | (8) |
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732 | (3) |
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735 | (1) |
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Infinite Sequences and Series |
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736 | (92) |
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737 | (12) |
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Laboratory Project Logistic Sequences |
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749 | (1) |
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749 | (10) |
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The Integral Test and Estimates of Sums |
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759 | (7) |
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766 | (5) |
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771 | (5) |
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Absolute Convergence and the Ratio and Root Tests |
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776 | (7) |
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Strategy for Testing Series |
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783 | (2) |
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785 | (5) |
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Representations of Functions as Power Series |
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790 | (6) |
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Taylor and Maclaurin Series |
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796 | (12) |
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Laboratory Project An Elusive Limit |
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808 | (1) |
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808 | (4) |
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Writing Project How Newton Discovered the Binomial Series |
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812 | (1) |
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Applications of Taylor Polynomials |
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812 | (16) |
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Applied Project Radiation from the Stars |
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821 | (1) |
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822 | (3) |
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825 | (3) |
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Vectors and the Geometry of Space |
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828 | (56) |
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Three-Dimensional Coordinate Systems |
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829 | (5) |
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834 | (9) |
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843 | (7) |
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850 | (8) |
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Discovery Project The Geometry of a Tetrahedron |
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858 | (1) |
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Equations of Lines and Planes |
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858 | (10) |
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Laboratory Project Putting 3D in Perspective |
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868 | (1) |
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Cylinders and Quadric Surfaces |
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868 | (7) |
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Cylindrical and Spherical Coordinates |
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875 | (9) |
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Laboratory Project Families of Surfaces |
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880 | (1) |
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880 | (3) |
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883 | (1) |
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884 | (38) |
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Vector Functions and Space Curves |
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885 | (7) |
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Derivatives and Integrals of Vector Functions |
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892 | (6) |
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898 | (8) |
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Motion in Space: Velocity and Acceleration |
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906 | (16) |
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Applied Project Kepler's Laws |
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916 | (1) |
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917 | (3) |
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920 | (2) |
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922 | (94) |
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Functions of Several Variables |
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923 | (15) |
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938 | (7) |
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945 | (14) |
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Tangent Planes and Linear Approximations |
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959 | (8) |
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967 | (9) |
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Directional Derivatives and the Gradient Vector |
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976 | (13) |
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Maximum and Minimum Values |
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989 | (12) |
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Applied Project Designing a Dumpster |
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999 | (1) |
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Discovery Project Quadratic Approximations and Critical Points |
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1000 | (1) |
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1001 | (15) |
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Applied Project Rocket Science |
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1008 | (1) |
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Applied Project Hydro-Turbine Optimization |
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1009 | (1) |
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1010 | (4) |
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1014 | (2) |
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1016 | (74) |
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Double Integrals over Rectangles |
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1017 | (8) |
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1025 | (6) |
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Double Integrals over General Regions |
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1031 | (8) |
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Double Integrals in Polar Coordinates |
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1039 | (6) |
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Applications of Double Integrals |
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1045 | (10) |
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1055 | (4) |
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1059 | (10) |
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Discovery Project Volumes of Hyperspheres |
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1068 | (1) |
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Triple Integrals in Cylindrical and Spherical Coordinates |
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1069 | (8) |
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Applied Project Roller Derby |
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1075 | (1) |
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Discovery Project The Intersection of Three Cylinders |
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1076 | (1) |
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Change of Variables in Multiple Integrals |
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1077 | (13) |
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1085 | (3) |
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1088 | (2) |
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1090 | (86) |
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1091 | (7) |
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1098 | (12) |
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The Fundamental Theorem for Line Integrals |
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1110 | (9) |
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1119 | (7) |
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1126 | (8) |
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Parametric Surfaces and Their Areas |
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1134 | (11) |
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1145 | (12) |
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1157 | (6) |
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Writing Project Three Men and Two Theorems |
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1162 | (1) |
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1163 | (7) |
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1170 | (6) |
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1171 | (3) |
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1174 | (2) |
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Second-Order Differential Equations |
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1176 | |
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Second-Order Linear Equations |
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1177 | (6) |
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Nonhomogeneous Linear Equations |
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1183 | (8) |
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Applications of Second-Order Differential Equations |
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1191 | (8) |
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1199 | |
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1203 | |
| 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 | (9) |
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48 | (8) |
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H Answers to Odd-Numbered Exercises |
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56 | (69) |
| Index |
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125 | |
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