I. NUMBER AND TITLE OF COURSE

Mathematics 163 - Using Technology to Explore Calculus I

II. REASONS FOR ADDITION TO PRESENT CURRICULA

A. There is a need for students to explore and model mathematics using currently available technology such as programmable graphing calculators and computer algebra systems. Research shows that this is best done with hands-on activities and projects.

B. When we reviewed our mathematics program in 1993, our outside evaluators strongly encouraged us to incorporate more technology into our calculus sequence as is the norm in many current calculus programs.

A. The students will use a programmable graphing utility to explore and model concepts which are covered in Calculus I. Students' understanding of fundamental notions of calculus will be enhanced by their being better able to visualize them.

B. The students will write programs to experiment with mathematical algorithms. This will prepare them for more advanced programming tasks in subsequent courses.

C. The students will use a programmable graphing utility to explore and model multiple representations of challenging and thought-provoking mathematical problems.

Notes: Each numbered item typed in boldface below will be the focus of one class session. The sessions are keyed to the MAT 161 outline.

A. Functions - notation, domain, range, finding zeros, finding limits

1. Basic graphing technology skills - Part I

a. Solve problems using the numerical capabilities of a graphing calculator.

b. Do multistep computations, noting the importance of parentheses. Include computations which involve manipulation of complex numbers and fractions and conversions between units.

c. Introduce function graphing, define graphing windows, trace function curves, and evaluate functions.

d. Note domain and range of functions across multiple representations.

a. Use the various ZOOM options available on graphing utilities.

b. Display complete graphs of functions being alert to hidden behavior.

c. Techniques of graph formatting.

d. Graphical solutions of equations and inequalities.

a. Explore limits numerically and graphically.

b. Use a program to set up a table of values for a function. Use this to help compare numeric, algebraic, and graphic representations of functions.

a. Introduce programming syntax for use with the graphing utility.

b. Write a bisection algorithm program and use it to find zeros of a function.

a. Explore more limit problems numerically, graphically, and algebraically.

b . Introduce the definition of the derivative and write a program which explores f(a+h) - f(a) / h for successively smaller values of h.

B. Derivatives and applications of derivatives

6. Derivatives - Part I

a. Explore numerical derivatives

b. Explore relationships between the graphs of y = f(x), y = f'(x) and y = f'(x). and y = f''(x)

a. Compare graphical and algebraic approaches to finding and using the derivative.

b . Use a programmable graphing utility calculator to verify results found algebraically.

c. Use graphing utility to calculate derivatives when algebra fails or is too cumbersome.

a. Explore the Mean Value Theorem with the aid of a program and a graphical interpretation.

b. Explore the idea of linearization of a function near a particular point in its domain.

a. Explore applied optimization problems with aid of a programmable graphing utility.

a. Explore more applied optimization problems.

b. Explore the behavior of nontrivial functions with the aid of the first and second derivative tests.

a. Write a program to perform Newton's method for the solution of equations.

b. Use Newton's method to solve a variety of problems.

c. Illustrate when Newton's method works and when it fails.

a. Write programs to approximate areas using Riemann sums.

b. Write a program to approximate areas using the trapezoidal rule.

c. Write a program to approximate areas using Simpson's method.

a. Generate sequences and series.

b. Graph conic sections.

c. Graph polar and parametric curves.

A. BOOKS

Beckmann, Charlene, E. and Theodore, A. Sundstrom (1992) Exploring Calculus with a Graphing Calculator. Reading, MA: Addison-Wesley.

Bradley, Gerald, L. and Smith, Karl J. (1995) Calculus. Englewood Cliffs, NJ: Prentice Hall.

Cohen, Jack and Hagin, Frank (1995) Calculus Explorations with TI Calculators. Englewood Cliffs, NJ: Prentice Hall.

Cohen, Marcus, Edward D. Gaughan, Arthur Knoebel, Douglas Kurtz, and David Pengelley (1991). Student Research Projects in Calculus. Washington, DC: Mathematical Association of America,

Fraga, Robert, ed. (1993). Calculus Problems for a New Century. MAA Notes, Vol.28. Washington, DC: The Mathematical Association of America.

Hughes-Hallett, Deborah, Andrew. M. Gleason, et al. (1994). Calculus. New York: John Wiley.

Jackson, M.B. and Ramsey, J.R., eds. (1993). Problems for Student Investigation. MAA Notes, Vol. 29. Washington, DC: Mathematical Association of America

Leinbach, L.C., ed. (1991). The Laboratory Approach to Teaching Calculus. MAA Notes. Vol 20. Washington, DC: Mathematical Association of America

McCarter, Joan H. (1992). Discovering Calculus with Graphing, Calculators. New York: John Wiley and Sons, Inc.

Ostebee, Arnold and Paul Zorn.(1995). Calculus from Graphical Numerical. and Symbolic Points of

View. Volumes I and II (Preliminary Edition). New York: Saunders College Publishing.

Smith, Robert, T. and Roland B. Minton.(1993). Discovering Calculus with the TI-81 and the TI-85. New York: McGraw-Hill, Inc. .

Solow, Anita, ed.(1994). Learning by Discovery: A Lab Manual for Calculus. MAA Notes. Vol 27. Washington, DC: Mathematical Association of America

Thomas, George B., Jr. and Ross L. Finney (1996). Calculus and Analytic Geometry 19th Edition). Reading, MA: Addison-Wesley.

The College Mathematics Journal (MAA)

Mathematics Magazine (MAA)

PRIMUS (Problems, Resources, and Issues in Mathematics Undergraduate Studies)

VI. PRESENTATION AND EVALUATION

Students will complete activities during each session and will turn in weekly reports. The course grade will be based on the quality of the weekly reports.

VII. PREREQUISITES

Students must be concurrently enrolled in, or have successfully completed, MAT 161 or its equivalent.

VIII. CREDIT 1 credit: (1: 0)

IX. STATEMENT OF APPROVAL

This course proposal was examined in accordance with recommended procedures and was approved by the Mathematics Department on December 19, 1995

X. CATALOG DESCRIPTION

MATH 163 Using Technology to Explore Calculus I - Weekly class session in which students use a programmable graphing utility to explore the mathematics they are reaming in MAT 161. Students in this course must be concurrently enrolled in (or have successfully completed) MAT 161.

XI. STATEMENT OF QUALIFICATIONS OF FACULTY WHO WILL TEACH THE COURSE

This course may be taught by any member of the Mathematics department who has experience using graphing utilities and Computer Algebra Systems to do mathematics. Among the current faculty who have acquired such expertise are:

Joaquin Carbonara, Ph.D Mathematics

Daniel Cunningham, Ph.D Mathematics

James Guyker, Ph.D Mathematics

And others.

XII. SUPPORT SERVICES REQUIRED:

In order to implement this course, the Mathematics Department will need access to classrooms which are properly wired for the use of computer and calculator overhead units. These must be rooms in which students can see to work while overhead units are displaying information on a screen in the front of the room.