Calculus: Mathematics and Modeling

by Ellis, Robert; Bauldry, William C.; Fiedler, Joe; Giordano, Frank R.; Judson, Phoebe T.; Lodi, Ed; Vitray, Richard

• ISBN13: 9780201338607
• ISBN10: 0201338602
• Edition: 2nd
• Format: Paperback
• Copyright: 1999-01-01
• Publisher: Pearson College Div

Summary

The first generation of calculus reformers exploited emerging technologies and the theme of multiple representations of functions. These pioneers also demonstrated effective, innovative teaching techniques, including collaborative learning, writing, discovery, and extended problem solving. Calculus: Mathematics and Modeling introduces a second generation of calculus reform, combining the lessons of the first generation with advances in differential equations through the use of discrete dynamical systems. This teaching philosophy requires a computational environment in which students can move smoothly between symbolic, numeric, graphic, and textual contexts. The text requires use of a computer algebra-capable graphing calculator.

Table of Contents

Calculus: Mathematics and Modeling	1	(2)
PART I CALCULUS --- THE MATHEMATICS OF CHANGE	3	(388)
Modeling Change	5	(62)
Drugs in the Body	5	(10)
Patterns of Accumulation	15	(10)
A Model for Natural Growth	25	(11)
Investigating Limits	36	(10)
Continuous and Piecewise Continuous Models	46	(21)
Summary	59	(8)
Measuring Change	67	(60)
Analyzing a Discrete Function	67	(11)
Contructing Models from Patterns	78	(11)
Divided Differences and Average Rate of Change	89	(9)
Rate of Change	98	(13)
Rate of Change as a Function	111	(16)
Summary	118	(9)
The Derivative: A Tool for Measuring Change	127	(100)
The Derivative	127	(14)
Rules, Rules, and More Rules	141	(11)
The Chain Rule	152	(7)
Finding Features of a Continuous Function	159	(17)
Properties of Continuous Function	176	(5)
Optimization: Finding Global Extrema	181	(12)
Implicit Differentiation and Its Applications	193	(10)
Modeling Motion with Parametric Equations	203	(10)
Partial Derivatives	213	(14)
Summary	219	(8)
The Definite Integral: Accumulating Change	227	(84)
Rate and Distance	227	(10)
Sums of Products	237	(9)
Error Bounds for the Left and Right Endpoint Methods	246	(8)
The Definite Integral	254	(8)
Other Methods and Their Error Bounds	262	(8)
Applications	270	(17)
The Fundamental Theorem of Calculus, Part I	287	(24)
Summary	296	(7)
The Truth About Limits	303	(8)
The Integral: Theory, Applications, and Techniques	311	(80)
Rate, Accumulation, and the Fundamental Theorem of Calculus, Part II	311	(8)
The Antiderivative Concept	319	(9)
Finding Antiderivatives Using Properties and Formulas	328	(6)
Applications	334	(14)
Using the Chain Rule in Finding Antiderivatives	348	(8)
Techniques of Integration	356	(35)
Sumamry	362	(9)
Further Techniques of Integration	371	(20)
PART II MODELING WITH CALCULUS	391	(240)
Modeling with the Derivative	393	(60)
One Day in the Life of a Modeler	393	(7)
Warming and Cooling	400	(6)
Population Modeling	406	(7)
Euler's Method	413	(7)
Slope Fields	420	(7)
Errors in the Model Construction	427	(26)
Terminology	435	(1)
Summary	436	(7)
Existence and Uniqueness Theorems	443	(1)
Uniqueness	443	(3)
Existence	446	(4)
The General Existence and Uniqueness Theorem	450	(3)
Solving Differential Equations	453	(42)
Integration and Separation of Variables	454	(7)
Linear Differential Equations	461	(8)
Errors in Euler's Method	469	(5)
Improving Euler's Method	474	(7)
Advanced Numeric Techniques	481	(14)
Summary	487	(8)
Modeling with Systems	495	(60)
Spirals of Change: You Are What You Eat	496	(10)
Modeling	506	(9)
Numerical Solutions: Iteration and Euler's Method	515	(13)
Symbolic Solutions of Systems of Differential Equations	528	(8)
Bungee Jumping	536	(19)
Summary	542	(13)
Power Series: Approximating Functions with Functions	555	(38)
Polynomial Approximation of Functions	555	(5)
Using Polynomial Approximations	560	(5)
How Good Is a Good Polynomial Approximation?	565	(8)
Convergence of Series	573	(9)
Power Series Solutions of Differential Equations	582	(11)
Summary	587	(6)
Optimization of Functions of Two Variables	593	(38)
Optimization with Two Variables	593	(5)
Vectors, Lines and Planes	598	(9)
Tangent Vectors and Tangent Lines in Three Dimensions	607	(7)
Tangent Planes	614	(8)
The Gradient Search	622	(9)
Summary	628	(3)
APPENDIX A TI-89 Computer Algebra System Tutorial	631	(22)
A.1 The TI Computer Algebra System Tutorial	631	(18)
A.2 Troubleshooting: Things that Go Bump in the Night	649	(4)
APPENDIX B TI-92 Computer Algebra System Tutorial	653	(22)
B.1 The TI Computer Algebra System Tutorial	653	(19)
B.2 Troubleshooting: Things that Go Bump in the Night	672	(3)
APPENDIX C Solving Equations with the TI Calculator	675	(10)
APPENDIX D The SlopeFld Program	685	(4)
Index	689