Calculus Concepts
Using Derive
for Windows

Ralph S. Freese and David A. Stegenga
Professors of Mathematics, University of Hawaii

2000 by Prentice-Hall
Upper Saddle River, NJ 07458

ISBN 0-13-085152-3

Table of Contents

0. Introduction and Derive Basics
This is a mini user's manual for DERIVE, emphasizing those features the course uses.
  Course Use
  Evaluation Copy
  Class Web Site
  Publisher's Web Site
1. Curve Sketching
Chapters 1, 2, and 3 teach the students how to use DERIVE and introduces graphing, solving equations both exactly and approximately, finding the maximum and the minimum both with graphing and calculus. In this chapter, various transformations of a function and the effect on it's graph are examined. The growth of the exponential function and resulting scaling problems in viewing it's graph are examined and several techniques are discussed for dealing with this problem.
2. The Derivative
The tangent line is viewed geometrically using a sequence of secant lines. The method of approximating a function near a point using the tangent line is demonstrated.
3. Basic Algebra and Graphics
This lab covers some of the basic concepts from first semester calculus, such as where functions are increasing, concavity, limits, etc. The geometric significance of the derivative is discussed along with the determination of critical points and inflection points both graphically and using calculus.
4. Curve Fitting
This chapter covers fitting linear, quadratic and more general polynomial curves, to data and solving simultaneous linear equations. One problem compares a linear population model with an exponential model. A table with US population sizes for the last 200 years lets students compare the model with actual growth. A brief introduction to quadratic spline functions is also given along with an application to approximating the sine function.
5. Finding Roots Using Computers
This lab explores Newton's method for finding roots. It is a nice application of differential calculus and the geometry associated with it. It serves as a simple introduction to dynamical systems and includes such topics as fixed points, attractors, super attractors, cycles, chaos and fractals. The exercises give the students several chances to explore interesting problems at various skill levels.
6. Numerical Integration Techniques
Various methods of numerical integration are covered including comparisons of error estimates between the rectangular, trapezoidal and Simpson methods. This lab clearly demonstrates the advantages of efficient computational algorithms. The importance of error estimates is stressed. An advanced section studies an integral where the standard error method gives no information and how it can be handled both analytically and geometrically.
7. Taylor Polynomials
This chapter deals with Taylor polynomials with remainder and Taylor series. Graphing the sine function and several of its Taylor polynomials gives a dramatic demonstration of how well and over what range these polynomials approximate the function. Intervals of convergence are also dramatically illustrated.
8. Series
This chapter covers infinite series emphasizing evaluation techniques. Several techniques are examined for estimating the error in approximating a series by its partial sums.
9. Approximating Integrals
This chapter applies the previous two chapters to the problem of approximately evaluating definite integrals.
10. Polar and Parametric Curves
This highly graphical topic is very appropriate for a computer lab course. The basic concepts can be well illustrated. As an example, the cycloid generated by a rolling wheel is one of the problems.
11. Differential Equations
This chapter covers differential equations in more detail manner than is usually done in the first year of calculus and, if the more advanced parts are covered, would be suitable for second year students. Nevertheless, it still concentrates primarily on traditional population growth and related areas such as Newton's Law of Cooling. Direction fields are also covered and the more advanced sections include Euler's approximation method. The Verhulst population model makes a nice demonstration of direction fields and gives the students a glimpse into modeling techniques.
12. Harmonic Motion
This chapter is an elementary treatment of second order differential equations with constant coefficients. The main example is the mass-spring system with and without frictional forces. There is an optional section which treats the Runge-Kutta method for finding numerical solutions.
Appendix A: Utility Files
Over a dozen utility functions are defined throughout the book. These functions are all in a file the students can conveniently download. This appendix explains how these functions work and gives the source code. The utility file can be downloaded for free.


Appendix B: Instructor's Manual
Suggestions are given on how to use this course either as a supplement to a calculus course or as a separate lab course. For each chapter, hints are given to the instructor on things that the students may have trouble with, what topics should be emphasized and possible advanced demonstrations.

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