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MATH 201 Section 3 Summary of Two Differentiation Rules Mathematics Worksheet

MATH 201

MATH

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Two pages summary about Section 3.2 The product/Quotient Rule, and Section 3.3 The Chain Rule.

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About the Cover Peter Blair Henry received his first lesson in international economics at the age of eight, when his family moved from the Caribbean island of Jamaica to affluent Wilmette, Illinois. Upon arrival in the United States, he wondered why people in his new home seemed to have so much more than people in Jamaica. The elusive answer to the question of why the average standard of living can be so different from one country to another still drives him today as a Professor of Economics in the Graduate School of Business at Stanford University. Peter began his academic career on the campus of the University of North Carolina at Chapel Hill, where he was a wide receiver on the varsity football team and a Phi Beta Kappa graduate in economics. With an intrinsic love of learning and a desire to make the world a better place, he knew that he wanted a career as an economist. He also knew that a firm foundation in mathematics would help him to answer the real-life questions that fueled his passion for economics—a passion that earned him a Rhodes Scholarship to Oxford University, where he received a B.A. in mathematics. PETER BLAIR HENRY International Economist This foundation in mathematics prepared Peter for graduate study at the Massachusetts Institute of Technology (MIT), where he received his Ph.D. in economics. While in graduate school, he served as a consultant to the Governors of the Bank of Jamaica and the Eastern Caribbean Central Bank (ECCB). His research at the ECCB helped provide the intellectual foundation for establishing the first stock market in the Eastern Caribbean Currency Area. His research and teaching at Stanford has been funded by the National Science Foundation’s Early Career Development Program (CAREER), which recognizes and supports the early career-development activities of those teacher-scholars who are most likely to become the academic leaders of the 21st century. Peter is also a member of the National Bureau of Economic Research (NBER), a nonpartisan economics think tank based in Cambridge, Massachusetts. Peter Blair Henry’s love of learning and his questioning nature have led him to his desired career as an international economist whose research positively impacts and addresses the tough decisions that face the world’s economies. It is his foundation in mathematics that enables him to grapple objectively with complex and emotionally charged issues of international economic policy reform, such as debt relief for developing countries and its effect on international stock markets. The images accompanying Peter Blair Henry’s portrait on the cover represent these vital issues faced by developing countries. Look for other featured applied researchers in the following forthcoming titles in the Tan applied mathematics series: CHRIS SHANNON Economics and Finance University of California, Berkeley MARK VAN DER LAAN Biostatistician University of California, Berkeley JONATHAN D. FARLEY Applied Mathematician California Institute of Technology NAVIN KHANEJA Applied Scientist Harvard University Now that you’ve bought the textbook . . . GET THE BEST GRADE IN THE SHORTEST TIME POSSIBLE! Visit www.iChapters.com to view over 10,000 print, digital, and audio study tools that allow you to: • Study in less time to get the grade you want . . . using online resources such as chapter pre- and post-tests and personalized study plans. • Prepare for tests anywhere, anytime . . . using chapter review audio files that are downloadable to your MP3 player. • Practice, review, and master course concepts . . . using printed guides and manuals that work hand-in-hand with each chapter of your textbook. Join the thousands of students who have benefited from www.iChapters.com. Just search by author, title, or ISBN, then filter the results by “Study Tools” and select the format best suited for you. www.iChapters.com. Your First Study Break EDITION 8 APPLIED CALCULUS FOR THE MANAGERIAL, LIFE, AND SOCIAL SCIENCES A BRIEF APPROACH S. T. TAN STONEHILL COLLEGE Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach, Eighth Edition S. T. Tan Senior Acquisitions Editor: Carolyn Crockett Development Editor: Danielle Derbenti Assistant Editor: Catie Ronquillo © 2009, 2006 Brooks/Cole, Cengage Learning ALL RIGHTS RESERVED. No part of this work covered by the copyright herein may be reproduced, transmitted, stored, or used in any form or by any means graphic, electronic, or mechanical, including but not limited to photocopying, recording, scanning, digitizing, taping, Web distribution, information networks, or information storage and retrieval systems, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without the prior written permission of the publisher. 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Project Manager, Editorial Production: Cheryll Linthicum Creative Director: Rob Hugel Library of Congress Control Number: 2007940269 Art Director: John Walker ISBN-13: 978-0-495-38754-1 Print Buyer: Judy Inouye ISBN-10: 0-495-38754-1 Permissions Editor: Bob Kauser Production Service: Martha Emry Text Designer: Diane Beasley Photo Researcher: Terri Wright Copy Editor: Betty Duncan Illustrator: Jade Myers, Matrix Productions Compositor: Graphic World Cover Designer: Irene Morris Cover Images: Peter Henry © Cengage Learning; Numbers, George Doyle/Getty Images; Foreign Currency © Imagemore Co., Ltd./Corbis; Woman Holding Money © Philippe Lissac/Godong/Corbis; Girl with Laptop, Symphonie/Getty Images Brooks/Cole 10 Davis Drive Belmont, CA 94002-3098 USA Cengage Learning is a leading provider of customized learning solutions with office locations around the globe, including Singapore, the United Kingdom, Australia, Mexico, Brazil, and Japan. Locate your local office at international.cengage.com/region. Cengage Learning products are represented in Canada by Nelson Education, Ltd. For your course and learning solutions, visit academic.cengage.com. Purchase any of our products at your local college store or at our preferred online store www.ichapters.com. Printed in Canada 1 2 3 4 5 6 7 12 11 10 09 08 CONTENTS Preface vi CHAPTER 1 Preliminaries 1.1 1.2 1.3 1.4 CHAPTER 2 1 Precalculus Review I 3 Precalculus Review II 15 The Cartesian Coordinate System 25 Straight Lines 33 Chapter 1 Summary of Principal Formulas and Terms 46 Chapter 1 Concept Review Questions 46 Chapter 1 Review Exercises 47 Chapter 1 Before Moving On 48 Functions, Limits, and the Derivative 2.1 2.2 2.3 49 Functions and Their Graphs 50 Using Technology: Graphing a Function 63 The Algebra of Functions 67 Functions and Mathematical Models 75 PORTFOLIO: Deb Farace 82 2.4 2.5 2.6 CHAPTER 3 Using Technology: Finding the Points of Intersection of Two Graphs and Modeling 92 Limits 97 Using Technology: Finding the Limit of a Function 115 One-Sided Limits and Continuity 117 Using Technology: Finding the Points of Discontinuity of a Function 131 The Derivative 133 Using Technology: Graphing a Function and Its Tangent Line 150 Chapter 2 Summary of Principal Formulas and Terms 152 Chapter 2 Concept Review Questions 152 Chapter 2 Review Exercises 153 Chapter 2 Before Moving On 156 Differentiation 3.1 3.2 3.3 3.4 3.5 157 Basic Rules of Differentiation 158 Using Technology: Finding the Rate of Change of a Function 169 The Product and Quotient Rules 171 Using Technology: The Product and Quotient Rules 180 The Chain Rule 182 Using Technology: Finding the Derivative of a Composite Function 193 Marginal Functions in Economics 194 Higher-Order Derivatives 208 Using Technology: Finding the Second Derivative of a Function at a Given Point 214 iv CONTENTS 3.6 3.7 CHAPTER 4 Applications of the Derivative 4.1 4.2 4.3 4.4 4.5 CHAPTER 5 Implicit Differentiation and Related Rates 215 Differentials 227 Using Technology: Finding the Differential of a Function 236 Chapter 3 Summary of Principal Formulas and Terms 237 Chapter 3 Concept Review Questions 238 Chapter 3 Review Exercises 239 Chapter 3 Before Moving On 242 243 Applications of the First Derivative 244 Using Technology: Using the First Derivative to Analyze a Function 261 Applications of the Second Derivative 264 Using Technology: Finding the Inflection Points of a Function 282 Curve Sketching 283 Using Technology: Analyzing the Properties of a Function 295 Optimization I 298 Using Technology: Finding the Absolute Extrema of a Function 311 Optimization II 312 Chapter 4 Summary of Principal Terms 323 Chapter 4 Concept Review Questions 324 Chapter 4 Review Exercises 324 Chapter 4 Before Moving On 327 Exponential and Logarithmic Functions 5.1 5.2 5.3 329 Exponential Functions 330 Using Technology 336 Logarithmic Functions 338 Compound Interest 346 PORTFOLIO: Richard Mizak 349 5.4 5.5 5.6 CHAPTER 6 Using Technology: Finding the Accumulated Amount of an Investment, the Effective Rate of Interest, and the Present Value of an Investment 359 Differentiation of Exponential Functions 360 Using Technology 370 Differentiation of Logarithmic Functions 372 Exponential Functions as Mathematical Models 380 Using Technology: Analyzing Mathematical Models 390 Chapter 5 Summary of Principal Formulas and Terms 393 Chapter 5 Concept Review Questions 393 Chapter 5 Review Exercises 394 Chapter 5 Before Moving On 396 Integration 6.1 6.2 6.3 6.4 397 Antiderivatives and the Rules of Integration 398 Integration by Substitution 411 Area and the Definite Integral 421 The Fundamental Theorem of Calculus 430 CONTENTS 6.5 6.6 6.7 CHAPTER 7 Evaluating Definite Integrals 441 Using Technology: Evaluating Definite Integrals for Piecewise-Defined Functions 451 Area between Two Curves 453 Using Technology: Finding the Area between Two Curves 463 Applications of the Definite Integral to Business and Economics 464 Using Technology: Business and Economic Applications 476 Chapter 6 Summary of Principal Formulas and Terms 477 Chapter 6 Concept Review Questions 478 Chapter 6 Review Exercises 479 Chapter 6 Before Moving On 482 Additional Topics in Integration 7.1 7.2 7.3 7.4 7.5 483 Integration by Parts 484 Integration Using Tables of Integrals 491 Numerical Integration 497 Improper Integrals 511 Applications of Calculus to Probability 520 PORTFOLIO: Gary Li 526 Chapter 7 Summary of Principal Formulas and Terms 531 Chapter 7 Concept Review Questions 532 Chapter 7 Review Exercises 532 Chapter 7 Before Moving On 534 CHAPTER 8 Calculus of Several Variables 8.1 8.2 8.3 535 Functions of Several Variables 536 Partial Derivatives 545 Using Technology: Finding Partial Derivatives at a Given Point 557 Maxima and Minima of Functions of Several Variables 558 PORTFOLIO: Kirk Hoiberg 560 8.4 8.5 8.6 APPENDIX The Method of Least Squares 568 Using Technology: Finding an Equation of a Least-Squares Line 577 Constrained Maxima and Minima and the Method of Lagrange Multipliers 579 Double Integrals 589 Chapter 8 Summary of Principal Terms 603 Chapter 8 Concept Review Questions 603 Chapter 8 Review Exercises 604 Chapter 8 Before Moving On 606 Inverse Functions 607 Answers to Odd-Numbered Exercises Index 651 613 v PREFACE M ath is an integral part of our daily life. Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach, Eighth Edition, attempts to illustrate this point with its applied approach to mathematics. This text is appropriate for use in a onesemester or a two-quarter introductory calculus course for students in the managerial, life, and social sciences. My objective for this Eighth Edition is twofold: (1) To write an applied text that motivates students and (2) to make the book a useful teaching tool for instructors. I hope that with the present edition I have come one step closer to realizing my goal. THE APPROACH Level of Presentation My approach is intuitive, and I state the results informally. However, I have taken special care to ensure that this approach does not compromise the mathematical content and accuracy. Problem-Solving Approach A problem-solving approach is stressed throughout the book. Numerous examples and applications illustrate each new concept and result. Special emphasis is placed on helping students formulate, solve, and interpret the results of the problems involving applications. Because students often have difficulty setting up and solving word problems, extra care has been taken to help students master these skills: ■ ■ ■ Very early on in the text, students are given practice in setting up word problems (see Section 2.3). Guidelines are given to help formulate and solve related-rates problems in Section 3.6. In Chapter 4, optimization problems are covered in two sections. First, the techniques of calculus are used to solve problems in which the function to be optimized is given (Section 4.4); second, in Section 4.5, optimization problems that require the additional step of formulating the problem are solved. Intuitive Introduction to Concepts Mathematical concepts are introduced with concrete, real-life examples wherever appropriate. An illustrative list of some of the topics introduced in this manner follows: ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Limits: The Motion of a Maglev The algebra of functions: The U.S. Budget Deficit The chain rule: The Population of Americans Aged 55 Years and Older Differentials: Calculating Mortgage Payments Increasing and decreasing functions: The Fuel Economy of a Car Concavity: U.S. and World Population Growth Inflection points: The Point of Diminishing Returns Curve sketching: The Dow Jones Industrial Average on “Black Monday” Exponential functions: Income Distribution of American Families Area between two curves: Petroleum Saved with Conservation Measures Approximating definite integrals: The Cardiac Output of a Heart PREFACE vii Connections One example (the maglev) is used as a common thread throughout the development of calculus—from limits through integration. The goal here is to show students the connections between the concepts presented—limits, continuity, rates of change, the derivative, the definite integral, and so on. Motivation Illustrating the practical value of mathematics in applied areas is an important objective of my approach. Many of the applications are based on mathematical models (functions) that I have constructed using data drawn from various sources, including current newspapers, magazines, and the Internet. Sources are given in the text for these applied problems. Modeling I believe that one of the important skills that a student should acquire is the ability to translate a real problem into a mathematical model that can provide insight into the problem. In Section 2.3, the modeling process is discussed, and students are asked to use models (functions) constructed from real-life data to answer questions. Students get hands-on experience constructing these models in the Using Technology sections. NEW TO THIS EDITION Algebra Review Gives Students a Plan of Action A Diagnostic Test now precedes the precalculus review. Each question is referenced by the section and example in the text where the relevant topic can be reviewed. Students can now use this test to diagnose their weaknesses and review the material on an as needed basis. Algebra Review Where Students Need It Most Well-placed algebra review notes, keyed to the review chapter, appear where students need them most throughout the text. These are indicated by the (x ) icon. See this feature in action on pages 105 and 537. 2 viii PREFACE Motivating Real-World Applications More than 140 new applications have been added to the Applied Examples and Exercises. Among these applications are global warming, depletion of Social Security trust fund assets, driving costs for a 2007 medium-sized sedan, hedge fund investments, mobile instant messaging accounts, hiring lobbyists, Web conferencing, the autistic brain, the revenue of Polo Ralph Lauren, U.S. health-care IT spending, and consumption of bottled water. Modeling with Data Modeling with Data exercises are now found in many of the Using Technology sections throughout the text. Students can actually see how some of the functions found in the exercises are constructed. (See Internet users in China, Exercise 44, page 335, and the corresponding exercise where the model is derived in Exercise 14, page 337.) Making Connections with Technology Many Using Technology sections have been updated. A new example—TV mobile phones—has been added to Using Technology 4.3. A new Using Technology section has been added to Section 5.3 (“Compound Interest”). Using Technology 5.6 includes a new example in which an exponential model is constructed—Internet gaming sales—using the logistic function of a graphing utility. Additional graphing calculator screens have been added in some sections. PREFACE Variety of Problem Types Additional rote questions, true or false questions, and concept questions have been added throughout the text to enhance the exercise sets. (See, for example, the graphical questions added to Concept Questions 2.1, page 57.) Action-Oriented Study Tabs Convenient color-coded study tabs, similar to Post it® flags, make it easy for students to tab pages that they want to return to later, whether it be for additional review, exam preparation, online exploration, or identifying a topic to be discussed with the instructor. ix x PREFACE Specific Content Changes ■ ■ ■ ■ The precalculus review in Sections 1.1 and 1.2 has been reorganized. Operations with algebraic expressions and factoring are now covered in Section 1.1, and inequalities and absolute value are covered in Section 1.2. Two examples illustrating how nonlinear inequalities are solved have been added. Section 2.3 on functions and mathematical models has been reorganized, and new models have been introduced. Here, students are now asked to use a model describing global warming to predict the amount of carbon dioxide (CO2) that will be present in the atmosphere in 2010 and a model describing the assets of the Social Security trust fund to determine when those assets are expected to be depleted. The chain rule in Section 3.3 is now introduced with an application—the population of Americans aged 55 years and older. A How-To Technology Index has been added for easy reference. TRUSTED FEATURES In addition to the new features, we have retained many of the following hallmarks that have made this series so usable and well-received in past editions: ■ ■ ■ ■ Section exercises to help students understand and apply concepts Optional technology sections to explore mathematical ideas and solve problems End-of-chapter review sections to assess understanding and problem-solving skills Features to motivate further exploration Self-Check Exercises Offering students immediate feedback on key concepts, these exercises begin each end of section exercise set. Fully worked-out solutions can be found at the end of each exercise section. Concept Questions Designed to test students’ understanding of the basic concepts discussed in the section, these questions encourage students to explain learned concepts in their own words. Exercises Each exercise section contains an ample set of problems of a routine computational nature followed by an extensive set of ap ...
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Final Answer

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Summary of two differentiation rules
This report summarizes two differentiation rules from sections 3.2 and 3.3 of the textbook.
The two rules are
1) The product/quotient rule
2) The chain rule

The product rule
The product rule states that the derivative of the product of two differentiable functions is given
by
u '( x) = f ( x) g '( x) + f '( x) g ( x)

where
f ( x ) = the first function, whose derivative is f ' ( x ) ,
g ( x ) = the second function, whose derivative is g ' ( x ) ,
u ( x ) = f ( x ) g ( x ) , the product of the two functions.
Example
Problem:
Find the derivative of the function ( 3x3 − 2 x 2 + 5 x − 1) x + x

(

)

Solution:
Let f ( x ) = 3x3 − 2 x 2 + 5 x − 1, g ( x ) = x + x , so that u ( x ) = f ( x ) g ( x ) .
f ' ( x ) = 9x2 − 4x + 5
g '( x) = 1+

1 −1/2
1
x = 1+
2
2 x

Therefore

(

1 

2
u '...

abotsi55 (1152)
Duke University

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