Calculus: Mathematics and Modeling, Updated Preliminary Edition

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:
Modeling Change. Drugs in the Body. Patterns of Accumulation. A Model for Natural Growth. Modeling Motion. Linear Functions as Models. Solving Equations. Measuring Change. @AHEADS 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.@CHAPTER = 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 Appoximation 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.