

Preface 

0. 
Introduction and Derive®
Basics

This is a mini user's manual for DERIVE®,
emphasizing those features the course uses. 

Details
Course Use
Evaluation Copy
Main
Class Web Site
Publisher's
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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 massspring system with and without frictional forces. There is an
optional section which treats the RungeKutta 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. 
