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by Wade Ellis, Bill Bauldry and Joe Fiedler

Edition: 99Copyright: 1999

Publisher: Addison-Wesley Longman, Inc.

Published: 1999

International: No

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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.

- This text represents the second generation of calculus reform, combining first-generation calculus reform philosophy with advances in differential equations, through the use of discrete dynamical systems.
- A rigorous approach to the content requires the use of a computer algebra capable graphing calculator.
- Concepts are applied to science, engineering, economics, and math.
- A full 50 percent of the exercises require written interpretation of results.
- Applied problems and exercises are used to introduce the development of topics.
- Group problems (for in-class use) and exercises (for out of class use) are used as examples.
- Discovery Activities, using trial and error; conjecturing solutions from tables; graphs; and symbols, are used within the exercise sets.
- One or two Extended Projects are included in each chapter.
- "Interludes," covering non-essential theoretical, technical, and historical topics, are included.
- Chapter Summary remarks contain a statement of what was learned, and why it was included.

Modeling Change

Drugs in the Body

Patterns of Accumulation

A Model for Natural Growth

Modeling Motion

Linear Functions as Models

Solving Equations

Measuring Change

Measuring Change: Regular Data

Analyzing a Discrete Function

From Discrete to Continuous Functions

Measuring Change: Irregular Data

Rate of Change

From Average Rate of Change to Instantaneous Rate of Change

Rate of Change as a Function

The Derivative: A Tool for Measuring Change

The Derivative at a Point and the Idea of a Limit

The Derivative as a Function

Rules, Rules, and More Rules

Finding Features of a Continuous Function

Optimization: Finding Global Extrema

Implicit and Parametric Differentiation

Partial Derivatives

The Definite Integral: Accumulating Change

Rate and Distance

Sums of Products

Error Bounds for the Left and Right Endpoint Methods

The Definite Integral

Other Methods and Their Error Bounds

Applications

The Truth About Limits

The Fundamental Theorem of Calculus and its Uses

Rate and Accumulation

The Antiderivative Concept and The Fundamental Theorem of Calculus, Part I

The Fundamental Theorem of Calculus, Part II

Applications

Using the Chain Rule in Finding Antiderivatives

Techniques of Integration

Further Techniques of Integration

Models and Derivatives

One Day in the Life of a Modeler

Population Modeling

Euler's Method

Slope Fields

Symbolic and Numeric Solutions

Integration and Separation of Variables

Linear Differential Equations

Errors in the Model Construction

Errors in Model Analysis: Euler's Method

Advanced Numeric Techniques

Existence and Uniqueness Theorems

Uniqueness

Existence

The General Existence and Uniqueness Theorem

Modeling With Systems

Spirals of Change: You Are What You Eat

Modeling

Numerical Solutions: Iteration and Euler's Method

Symbolic Solutions of Systems of Differential Equations

Bungee Jumping

Power Series: Approximating Functions with Functions

Polynomial Approximation of Functions

Using Polynomial Approximations

How Good Is a Good Polynomial Approximation?

Convergence of Series

Power Series Solutions of Differential Equations

Optimization of Functions of Two Variables

Optimization with Two Variables

Vectors, Lines, and Planes

Tangent Vectors and Tangent Lines in Three Dimensions

Tangent Planes

The Gradient Search

Appendix: Handheld Computer Algebra System Tutorial

The Computer Algebra System Tutorial

Troubleshooting: Things that Go Bump in the Night

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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.

- This text represents the second generation of calculus reform, combining first-generation calculus reform philosophy with advances in differential equations, through the use of discrete dynamical systems.
- A rigorous approach to the content requires the use of a computer algebra capable graphing calculator.
- Concepts are applied to science, engineering, economics, and math.
- A full 50 percent of the exercises require written interpretation of results.
- Applied problems and exercises are used to introduce the development of topics.
- Group problems (for in-class use) and exercises (for out of class use) are used as examples.
- Discovery Activities, using trial and error; conjecturing solutions from tables; graphs; and symbols, are used within the exercise sets.
- One or two Extended Projects are included in each chapter.
- "Interludes," covering non-essential theoretical, technical, and historical topics, are included.
- Chapter Summary remarks contain a statement of what was learned, and why it was included.

Table of Contents
#### Part I: The Calculus of Change

#### Part II: Modeling with Calculus

Modeling Change

Drugs in the Body

Patterns of Accumulation

A Model for Natural Growth

Modeling Motion

Linear Functions as Models

Solving Equations

Measuring Change

Measuring Change: Regular Data

Analyzing a Discrete Function

From Discrete to Continuous Functions

Measuring Change: Irregular Data

Rate of Change

From Average Rate of Change to Instantaneous Rate of Change

Rate of Change as a Function

The Derivative: A Tool for Measuring Change

The Derivative at a Point and the Idea of a Limit

The Derivative as a Function

Rules, Rules, and More Rules

Finding Features of a Continuous Function

Optimization: Finding Global Extrema

Implicit and Parametric Differentiation

Partial Derivatives

The Definite Integral: Accumulating Change

Rate and Distance

Sums of Products

Error Bounds for the Left and Right Endpoint Methods

The Definite Integral

Other Methods and Their Error Bounds

Applications

The Truth About Limits

The Fundamental Theorem of Calculus and its Uses

Rate and Accumulation

The Antiderivative Concept and The Fundamental Theorem of Calculus, Part I

The Fundamental Theorem of Calculus, Part II

Applications

Using the Chain Rule in Finding Antiderivatives

Techniques of Integration

Further Techniques of Integration

Models and Derivatives

One Day in the Life of a Modeler

Population Modeling

Euler's Method

Slope Fields

Symbolic and Numeric Solutions

Integration and Separation of Variables

Linear Differential Equations

Errors in the Model Construction

Errors in Model Analysis: Euler's Method

Advanced Numeric Techniques

Existence and Uniqueness Theorems

Uniqueness

Existence

The General Existence and Uniqueness Theorem

Modeling With Systems

Spirals of Change: You Are What You Eat

Modeling

Numerical Solutions: Iteration and Euler's Method

Symbolic Solutions of Systems of Differential Equations

Bungee Jumping

Power Series: Approximating Functions with Functions

Polynomial Approximation of Functions

Using Polynomial Approximations

How Good Is a Good Polynomial Approximation?

Convergence of Series

Power Series Solutions of Differential Equations

Optimization of Functions of Two Variables

Optimization with Two Variables

Vectors, Lines, and Planes

Tangent Vectors and Tangent Lines in Three Dimensions

Tangent Planes

The Gradient Search

Appendix: Handheld Computer Algebra System Tutorial

The Computer Algebra System Tutorial

Troubleshooting: Things that Go Bump in the Night

Publisher Info

Publisher: Addison-Wesley Longman, Inc.

Published: 1999

International: No

Published: 1999

International: No

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.

- This text represents the second generation of calculus reform, combining first-generation calculus reform philosophy with advances in differential equations, through the use of discrete dynamical systems.
- A rigorous approach to the content requires the use of a computer algebra capable graphing calculator.
- Concepts are applied to science, engineering, economics, and math.
- A full 50 percent of the exercises require written interpretation of results.
- Applied problems and exercises are used to introduce the development of topics.
- Group problems (for in-class use) and exercises (for out of class use) are used as examples.
- Discovery Activities, using trial and error; conjecturing solutions from tables; graphs; and symbols, are used within the exercise sets.
- One or two Extended Projects are included in each chapter.
- "Interludes," covering non-essential theoretical, technical, and historical topics, are included.
- Chapter Summary remarks contain a statement of what was learned, and why it was included.

Modeling Change

Drugs in the Body

Patterns of Accumulation

A Model for Natural Growth

Modeling Motion

Linear Functions as Models

Solving Equations

Measuring Change

Measuring Change: Regular Data

Analyzing a Discrete Function

From Discrete to Continuous Functions

Measuring Change: Irregular Data

Rate of Change

From Average Rate of Change to Instantaneous Rate of Change

Rate of Change as a Function

The Derivative: A Tool for Measuring Change

The Derivative at a Point and the Idea of a Limit

The Derivative as a Function

Rules, Rules, and More Rules

Finding Features of a Continuous Function

Optimization: Finding Global Extrema

Implicit and Parametric Differentiation

Partial Derivatives

The Definite Integral: Accumulating Change

Rate and Distance

Sums of Products

Error Bounds for the Left and Right Endpoint Methods

The Definite Integral

Other Methods and Their Error Bounds

Applications

The Truth About Limits

The Fundamental Theorem of Calculus and its Uses

Rate and Accumulation

The Antiderivative Concept and The Fundamental Theorem of Calculus, Part I

The Fundamental Theorem of Calculus, Part II

Applications

Using the Chain Rule in Finding Antiderivatives

Techniques of Integration

Further Techniques of Integration

Models and Derivatives

One Day in the Life of a Modeler

Population Modeling

Euler's Method

Slope Fields

Symbolic and Numeric Solutions

Integration and Separation of Variables

Linear Differential Equations

Errors in the Model Construction

Errors in Model Analysis: Euler's Method

Advanced Numeric Techniques

Existence and Uniqueness Theorems

Uniqueness

Existence

The General Existence and Uniqueness Theorem

Modeling With Systems

Spirals of Change: You Are What You Eat

Modeling

Numerical Solutions: Iteration and Euler's Method

Symbolic Solutions of Systems of Differential Equations

Bungee Jumping

Power Series: Approximating Functions with Functions

Polynomial Approximation of Functions

Using Polynomial Approximations

How Good Is a Good Polynomial Approximation?

Convergence of Series

Power Series Solutions of Differential Equations

Optimization of Functions of Two Variables

Optimization with Two Variables

Vectors, Lines, and Planes

Tangent Vectors and Tangent Lines in Three Dimensions

Tangent Planes

The Gradient Search

Appendix: Handheld Computer Algebra System Tutorial

The Computer Algebra System Tutorial

Troubleshooting: Things that Go Bump in the Night