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Thomas' Calculus 13th 英文
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微积分(Calculus)是高等数学中研究函数的微分(Differentiation)、积分(Integration)以及有关概念和应用的数学分支。它是数学的一个基础学科。内容主要包括极限、微分学、积分学及其应用。微分学包括求导数的运算,是一套关于变化率的理论。它使得函数、速度、加速度和曲线的斜率等均可用一套通用的符号进行讨论。积分学,包括求积分的运算,为定义和计算面积、体积等提供一套通用的方法。本书为有数学学习需求的用户准备
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Based on the original work by
George B. Thomas, Jr.
Massachusetts Institute of Technology
as revised by
Maurice D. Weir
Naval Postgraduate School
Joel Hass
University of California, Davis
with the assistance of
Christopher Heil
Georgia Institute of Technology
THOMAS’
CALCULUS
EARLY TRANSCENDENTALS
Thirteenth Edition
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Library of Congress Cataloging-in-Publication Data
Weir, Maurice D.
Thomas’ calculus : early transcendentals : based on the original work by George B. Thomas, Jr., Massachusetts
Institute of Technology.—Thirteenth edition / as revised by Maurice D. Weir, Naval Postgraduate School, Joel
Hass, University of California, Davis.
pages cm
ISBN 978-0-321-88407-7 (hardcover)
I. Hass, Joel.II. Thomas, George B. (George Brinton), Jr., 1914–2006. Calculus. Based on (Work):III.
Title. IV. Title: Calculus.
QA303.2.W45 2014
515–dc23 2013023096
Copyright © 2014, 2010, 2008 Pearson Education, Inc. All rights reserved. No part of this publication may be
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1 2 3 4 5 6 7 8 9 10—CRK—17 16 15 14 13
www.pearsonhighered.com
ISBN-10: 0-321-88407-8
ISBN-13: 978-0-321-88407-7

iii
Preface ix
1 Functions 1
1.1 Functions and Their Graphs 1
1.2 Combining Functions; Shifting and Scaling Graphs 14
1.3 Trigonometric Functions 21
1.4 Graphing with Software 29
1.5 Exponential Functions 36
1.6 Inverse Functions and Logarithms 41
Questions to Guide Your Review 54
Practice Exercises 54
Additional and Advanced Exercises 57
2 Limits and Continuity 59
2.1 Rates of Change and Tangents to Curves 59
2.2 Limit of a Function and Limit Laws 66
2.3 The Precise Definition of a Limit 77
2.4 One-Sided Limits 86
2.5 Continuity 93
2.6 Limits Involving Infinity; Asymptotes of Graphs 104
Questions to Guide Your Review 118
Practice Exercises 118
Additional and Advanced Exercises 120
3 Derivatives 123
3.1 Tangents and the Derivative at a Point 123
3.2 The Derivative as a Function 128
3.3 Differentiation Rules 136
3.4 The Derivative as a Rate of Change 146
3.5 Derivatives of Trigonometric Functions 156
3.6 The Chain Rule 163
3.7 Implicit Differentiation 171
3.8 Derivatives of Inverse Functions and Logarithms 177
3.9 Inverse Trigonometric Functions 187
3.10 Related Rates 193
3.11 Linearization and Differentials 202
Questions to Guide Your Review 214
Practice Exercises 215
Additional and Advanced Exercises 219
Contents

iv Contents
4 Applications of Derivatives 223
4.1 Extreme Values of Functions 223
4.2 The Mean Value Theorem 231
4.3 Monotonic Functions and the First Derivative Test 239
4.4 Concavity and Curve Sketching 244
4.5 Indeterminate Forms and L’Hôpital’s Rule 255
4.6 Applied Optimization 264
4.7 Newton’s Method 276
4.8 Antiderivatives 281
Questions to Guide Your Review 291
Practice Exercises 291
Additional and Advanced Exercises 295
5 Integrals 299
5.1 Area and Estimating with Finite Sums 299
5.2 Sigma Notation and Limits of Finite Sums 309
5.3 The Definite Integral 316
5.4 The Fundamental Theorem of Calculus 328
5.5 Indefinite Integrals and the Substitution Method 339
5.6 Definite Integral Substitutions and the Area Between Curves 347
Questions to Guide Your Review 357
Practice Exercises 357
Additional and Advanced Exercises 361
6 Applications of Definite Integrals 365
6.1 Volumes Using Cross-Sections 365
6.2 Volumes Using Cylindrical Shells 376
6.3 Arc Length 384
6.4 Areas of Surfaces of Revolution 390
6.5 Work and Fluid Forces 395
6.6 Moments and Centers of Mass 404
Questions to Guide Your Review 415
Practice Exercises 416
Additional and Advanced Exercises 417
7 Integrals and Transcendental Functions 420
7.1 The Logarithm Defined as an Integral 420
7.2 Exponential Change and Separable Differential Equations 430
7.3 Hyperbolic Functions 439
7.4 Relative Rates of Growth 448
Questions to Guide Your Review 453
Practice Exercises 453
Additional and Advanced Exercises 455

Contents v
8 Techniques of Integration 456
8.1 Using Basic Integration Formulas 456
8.2 Integration by Parts 461
8.3 Trigonometric Integrals 469
8.4 Trigonometric Substitutions 475
8.5 Integration of Rational Functions by Partial Fractions 480
8.6 Integral Tables and Computer Algebra Systems 489
8.7 Numerical Integration 494
8.8 Improper Integrals 504
8.9 Probability 515
Questions to Guide Your Review 528
Practice Exercises 529
Additional and Advanced Exercises 531
9 First-Order Differential Equations 536
9.1 Solutions, Slope Fields, and Euler’s Method 536
9.2 First-Order Linear Equations 544
9.3 Applications 550
9.4 Graphical Solutions of Autonomous Equations 556
9.5 Systems of Equations and Phase Planes 563
Questions to Guide Your Review 569
Practice Exercises 569
Additional and Advanced Exercises 570
10 Infinite Sequences and Series 572
10.1 Sequences 572
10.2 Infinite Series 584
10.3 The Integral Test 593
10.4 Comparison Tests 600
10.5 Absolute Convergence; The Ratio and Root Tests 604
10.6 Alternating Series and Conditional Convergence 610
10.7 Power Series 616
10.8 Taylor and Maclaurin Series 626
10.9 Convergence of Taylor Series 631
10.10 The Binomial Series and Applications of Taylor Series 638
Questions to Guide Your Review 647
Practice Exercises 648
Additional and Advanced Exercises 650
11 Parametric Equations and Polar Coordinates 653
11.1 Parametrizations of Plane Curves 653
11.2 Calculus with Parametric Curves 661
11.3 Polar Coordinates 671
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