[Book Cover]

Calculus: Preliminary Edition, 1/e

Robert Decker, University of Hartford
Dale Varberg, Hamline University

Published March, 1996 by Prentice Hall Engineering/Science/Mathematics

Copyright 1996, 651 pp.
Paper
ISBN 0-13-287640-X


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Summary

A second generation reform text, Decker/Varberg is designed to motivate an intuitive understanding of calculus topics with the aid of visualization technology and rich problems and applications. Stronger than all other texts in its coverage of numeries.

Features


fully incorporates the Rule of Three (graphical, numeric and symbolic) with an emphasis on a numeric approach. (Example: Chapter 2.5; 4 and Lab 11, 6.3)
group projects (usually technology-based) throughout the book that focus on mathematical exploration and discovery, not just button pushing. These have been developed by the author through a series of NSF-sponsored workshop and technology training sessions. (Example: Lab #13, chapter 7.5; and Lab #18, chapter 9.2)
full coverage of important calculus topics that is sometimes omitted from other reform texts, such as sequences and series, mean value theorem, hyperbolic functions, parametric equations, L'Hopital's Rule, conic sections and centers of mass.
a early, careful introduction to exponential and logarithmic functions (Sect. 1) to motivate more interesting applications and modeling in subsequent chapters. (The topics are discussed further in Chapter 7.2).
the introduction of limits is followed quickly by an introduction to derivatives to help motivate an understanding of the connection between the two concepts. (Example: 1.1)
a complete chapter on Graphical and Numerical Techniques of Problem Solving (Chapter 2). The chapter includes a comprehensive discussion on how to interpret and use graphs and tables in problem solving. Also introduces local linearity; intermediate value theorem; linear curve fitting; and an expansion of exponential function topics.
special emphasis is placed throughout the book on showing the students both the value and the limitations of using technology to learn calculus (Example: Chapter 4.6).
book includes discussion and group project on dynamical systems and chaos to students that capitalizes on recent discussions and interesting applications in this diverse area. (Example: Chapter 9.1, Lab 17).


Table of Contents

    Preliminaries.

      Graphs and Equations. Functions. The Straight Line and Linear Functions. The Trigonometric Functions.

    1. Calculus: A First Look.

      Introduction to Limits Part I. Lab 1: Limits. Introduction to Limits Part II. The Derivative: Two Problems With One Theme. Exponential Functions. Logarithms and the Logarithmic Function. Chapter Review.

    2. Numerical and Graphical Techniques.

      Calculator and Computer Graphs. Lab 2: Calculator and Computer Graphs. Calculator and Computer Tables. Parameters. Lab 3: The Damped Harmonic Oscillator. Zooming and Local Linearity. Lab 4: Roots and Slopes: When and How Fast? Solving Equations. Linear Curve Fitting. Nonlinear Curve Fitting. Chapter Review.

    3. Derivatives.

      The Derivative as a Function. Lab 5: The Derivative as a Function. Rules for Finding Derivatives. Derivatives of Sines, Cosines, Logs and Exponentials. The Chain Rule. Higher-Order Derivatives. Implicit Differentiation and Related Rates. Differentials and Approximations. Chapter Review.

    4. Applications of the Derivative.

      Maxima and Minima. Lab 6: Increasing, Decreasing, and the Derivative. Monotonicity and Concavity. Local Maxima and Minima. Lab 7: Ideal Gases and Real Gases. More Max-Min Problems. Applications from Economics. Graphing with Parameters. The Mean Value Theorem. Chapter Review.

    5. The Integral.

      Antiderivatives (Indefinite Integrals). Introduction to Differential Equations. Lab 8: The Draining Can. Area and Reimann Sums. Lab 9: Area and Distance. The Definite Integral and Numerical Integration. Lab 10: Area Functions. The Fundamental Theorem of Calculus. More Properties of the Definite Integral. Substitution. Chapter Review.

    6. Applications of the Integral.

      The Area of a Plane Region. Volumes of Solids: Slabs, Disks, Washers. Length of a Plane Curve. Lab 11: Arc Length. Work. Moments, Center of Mass. Chapter Review.

    7. Transcendental Functions and Differential Equations.

      Inverse Functions and Their Derivatives. A Different Approach to Logarithmic and Exponential Functions. Lab 12: Falling Objects. General Exponential and Logarithmic Functions. Exponential Growth and Decay. Numerical and Graphical Approaches to Differential Equations. Lab 13: Planning Your Retirement. The Trigonometric Functions and Their Inverses. The Hyperbolic Functions and Their Inverses. Chapter Review.

    8. Techniques of Integration.

      Substitution and Tables of Integrals. Integration by Parts. Lab 14: Integration by Parts. Some Trigonometric Integrals. Integration of Rational Functions. Lab 15: Population Models. Indeterminate Forms. Improper Integrals. Lab 16: Probability and Improper Integrals. Chapter Review.

    9. Infinite Series.

      Infinite Sequences and Dynamical Systems. Lab 17: Discrete Population Models. Infinite Series. Lab 18: Bouncing Balls and Infinite Series. Altering Series, Absolute Convergence. Taylor's Approximation to Functions. Lab 19: Taylor Series and Fourier Series. The Error in Taylor's Approximation. Lab 20: Newton's Method. General Power Series. Lab 21: The Gamma Function and Taylor Series. Approximations. Operations on Power Series. Chapter Review.

    10. Conics, Polar Coordinates and Parametric Curves.

      Conic Sections. Translation of Axes. The Polar Coordinate System and Graphs. Technology and Graphs of Polar Equations. Lab 22: Graphs in Polor Coordinates. Calculus in Polar Coordinates. Lab 23: Orbits of the Planets. Plane Curves: Parametric Representation. Chapter Review.


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