Brief Calculus with Applications

by Armstrong, Bill; Davis, Don

• ISBN13: 9780130655912
• ISBN10: 0130655910
• Edition: 2nd
• Format: Hardcover
• Copyright: 2003-01-01
• Publisher: Pearson College Div

Summary

This book, modern in its writing style as well as in its applications, contains numerous exercisesboth skill oriented and applications, real data problems, and a problem solving method.The book features exercises based on data form the World Wide Web, technology options for those who wish to use a graphing calculator, review boxes, strategic checkpoints, interactive activities, section summaries and projects, and chapter openers and reviews.For anyone who wants to see and understand how mathematics are used in everyday life.

Table of Contents

Preface	vii
Functions, Models, and Average Rate of Change	1	(128)
The Coordinate System and Functions	2	(11)
Introduction to Problem Solving	13	(9)
Linear Functions and Average Rate of Change	22	(19)
Quadratic Functions and Average Rate of Change on an Interval	41	(11)
Operations on Functions	52	(18)
Rational, Radical, and Power Functions	70	(15)
Exponential Functions	85	(14)
Logarithmic Functions	99	(11)
Modeling Data with Functions (Optional Section---Requires Graphing Calculator)	110	(19)
Chapter Review Exercises	121	(7)
Chapter 1 Project	128	(1)
Limits, Instantaneous Rate of Change, and the Dervative	129	(96)
Limits	130	(13)
Limits and Asymptotes	143	(15)
Problem Solving: Rates of Change	158	(14)
The Derivative	172	(12)
Derivatives of Constants, Powers, and Sums	184	(13)
Derivatives of Products and Quotients	197	(11)
Continuity and Nondifferentiability	208	(17)
Chapter Review Exercises	218	(5)
Chapter 2 Project	223	(2)
Applications of the Derivative	225	(38)
The Differential and Linear Approximations	226	(9)
Marginal Analysis	235	(12)
Measuring Rates and Errors	247	(16)
Chapter Review Exercises	257	(4)
Chapter 3 Project	261	(2)
Additional Differentiation Techniques	263	(56)
The Chain Rule	264	(9)
Derivatives of Logarithmic Functions	273	(8)
Derivatives of Exponential Functions	281	(10)
Implicit Differentiation and Related Rates	291	(13)
Elasticity of Demand	304	(15)
Chapter Review Exercises	314	(3)
Chapter 4 Project	317	(2)
Further Applications of the Derivative	319	(80)
First Derivatives and Graphs	320	(17)
Second Derivatives and Graphs	337	(15)
Graphical Analysis and Curve Sketching	352	(13)
Optimizing Functions on a Closed Interval	365	(14)
The Second Derivative Test and Optimization	379	(20)
Chapter Review Exercises	392	(6)
Chapter 5 Project	398	(1)
Integral Calculus	399	(104)
The Indefinite Integral	400	(13)
Area and the Definite Integral	413	(12)
Fundamental Theorem of Calculus	425	(8)
Problem Solving: Integral Calculus and Total Accumulation	433	(13)
Integration by u-Substitution	446	(11)
Integrals That Yield Logarithmic and Exponential Functions	457	(13)
Differential Equations: Separation of Variables	470	(10)
Differential Equations: Growth and Decay	480	(23)
Chapter Review Exercises	496	(5)
Chapter 6 Project	501	(2)
Applications of Integral Calculus	503	(92)
Average Value of a Function and the Definite Integral in Finance	504	(14)
Area between Curves and Applications	518	(14)
Economic Applications of Area between Two Curves	532	(15)
Integration by Parts	547	(13)
Numerical Integration	560	(14)
Improper Integrals	574	(21)
Chapter Review Exercises	589	(5)
Chapter 7 Project	594	(1)
Calculus of Several Variables	595	(87)
Functions of Several Independent Variables	596	(12)
Level Curves, Contour Maps, and Cross-sectional Analysis	608	(14)
Partial Derivatives and Second-order Partial Derivatives	622	(15)
Maxima and Minima	637	(11)
Lagrange Multipliers	648	(13)
Double Integrals	661	(21)
Chapter Review Exercises	675	(5)
Chapter 8 Project	680	(2)
Appendix A Essentials of Algebra	682	(6)
Appendix B Calculator Programs	688	(5)
Appendix C Selected Proofs	693	(9)
Appendix D Photo and Illustration Credits	702
Answers to Odd-Numbered Exercises	1	(1)
Index	1

Excerpts

Audience In preparing to write this text, we talked with many colleagues who teach a brief or applied math course to find out if they experienced the same difficulties in teaching this course_ that we have encountered. What we learned is that while there is some similarity in the topics covered and in how much time is spent on each area, there is remarkable uniformity in the needs of students who enroll in these diverse courses. Professors at Community Colleges, Universities, and Liberal Arts Colleges all told us that their students are generally unmotivated, unsure of their algebra skills, uncomfortable with translating English into mathematics, and unschooled in how to set up problems for solution. Armed with this knowledge, we prepared the second edition ofBrief Calculusto address these fundamental needs. As with other applied calculus textbooks, our text may be used in either a one or two term course for students majoring in economics, business, or social or behavioral sciences. We have organized the topics for maximum flexibility so that the text may be adapted to any college or university's curriculum. However, that is where the similarity ends. We have crafted this book around five key principles designed to address students' needs: Present the mathematics in language that students can read and understand Teach good problem solving techniques and provide ample practice Use real data applications to keep it interesting Provide timely reinforcement of algebra and other essential skills Let instructors decide whether to incorporate technology Present the Mathematics in Language That Students Can Read and Understand By writing this text in a conversational, easy-to-read style, we strive to evoke the one-on-one communication of a tutorial session. When students find that they can understand the clear presentation and follow the interesting, real world examples, we believe they will get into the habit of reading the text. Although we have written a text that is accessible, we have been careful not to sacrifice the proper depth of coverage and necessary rigor required of applied calculus. We are confident both objectives have been met. Teach Good Problem Solving Techniques and Provide Ample Practice Problem Solving Sections A new feature of the Second Edition is the addition of dedicated problem solving sections. Before explaining how to solve an entire class of problems, we provide a special section that demonstrates how to apply the appropriate mathematical tools to analyze a given type of problem. For example Section 1.2,Introduction to Problem Solving,introduces our general approach to problem solving and then explores mathematical models and their properties, and how numerical solutions to mathematical models are interpreted. Section 2.3--Problem Solving: Rates of Changeconnects the average rate of change and secant line slope to instantaneous rate of change and tangent line slope using numerical and graphical techniques. Section 6.4--Problem Solving: Integral Calculus and Total Accumulationexplains how the definite integral can be used, given a rate of change function, to determine a continuous sum. Many of the exercises in these problem solving sections prepare students for subsequent sections because they introduce exercises that are solved later in the textbook. Problem Solving Method New to the second edition,our clearly developed problem solving method is the single most distinctive, and user-friendly feature of the book. Frequently, applied mathematics instructors hear students comment that "I don't even know how to begin this problem." or "If the problem was just set up for me, I could solve it." Because skills such as setting up problems and writing the solution in its proper context can be a major challenge for applied cal