I. NUMBER AND TITLE OF COURSE:

Mathematics 164 - Using Technology to Explore Calculus II

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.

III. MAJOR OBJECTIVES OF THE COURSE:

A. The students will use a Computer Algebra System to explore and of fundamental notions of calculus will be enhanced by their being able to visualize them. model concepts which are covered in Calculus II. Their understanding

B. The students will write programs to perform mathematical algorithms and to illustrate and apply calculus concepts. This will prepare them for more advanced programming tasks in subsequent courses.

C. The students will use a programmable graphing utility and Computer Algebra Systems to explore and model challenging and thought-provoking mathematical problems.

Notes: The primary technology used in this course will be a Computer Algebra System (CAS). Each numbered item typed in boldface below will be the focus of one class session. The sessions are keyed to the MAT 162 outline.

A. Review of functions and derivatives using Computer Algebra Systems.

1. Getting started

a. Sign on to a CAS.

b. Explore operations, numerical representations and approximations.

c. Introduce CAS functions; native CAS functions, user deemed functions, local and global variables, function composition, function arguments.

d. Solve equations. Find roots. Construct tables of function values.

e. Plot functions.

f. Introduce troubleshooting.

a. Limits. Difference quotients. Derivatives. Antiderivatives.

b. Numerical and symbolic sums.

c. Introduce Riemann sums.

d. Print reports.

3. Areas

a. Manipulate algebraic expressions for the purpose of comparing alternate forms of equivalent expressions, e.g., compare a function to the derivative of its antiderivative.

b. Construct Riemann sums to motivate and illustrate the use of calculus to find areas under a curve and areas enclosed by two curves.

a. Construct Riemann sums to motivate and illustrate the use of calculus to find volumes by slicing and cylindrical shells.

b. Construct Riemann sums to motivate and illustrate the use of calculus to find the length of a plane curve and the area of a surface of revolution.

5. Inverse functions

a. Plot a function and its inverse on the same graph.

b. write small programs to find the Inverse of a polynomial function. Represent functions with finite domain as sets of ordered pairs and graph such sets.

a. Explore logarithmic and exponential functions. Graph different logarithmic and exponential functions. Find the derivative of the logarithmic and exponential functions using CAS.

b. Explore hyperbolic functions and their inverses: operations, graphs, and equations. Explore trigonometric functions and their inverses: graphs and derivatives.

7. Techniques of integration using CAS (3 sessions)

a. Use CAS to enhance integration techniques covered in Calculus H: Program any techniques covered in Calculus II currently integration by parts, partial fraction decomposition to simplify the integrand, trigonometric substitution, and/or others.

b. Use CAS as a "table of integrals", that is, let the CAS do an the work. Explore limitations of the system.

a. Program lefthand and righthand approximation, the midpoint approximation, the trapezoidal approximation and Simpson's rule.

b . Use the numerical integration capabilities already programmed in the CAS.

a. Explore improper integrals using CAS.

b. Program L'Hopital's rule using CAS.

V. BIBLIOGRAPHY

M.L.Abell and J.P. Braselton, Mathematica ~, Example, Revised edition, Harcourt Brace and Company, New York, 1994.

G. L. Bradley and K.J. Smith. Calculus, Prentice Hall, Englewood Cliffs, NJ, 1995.

N. Blackman, C. Williams et all, CalcLabs with Mathematica, Brooks/Cole, Pacific Grove, California. 1995.

R.E.Crandall, Projects in Scientific Computation, Springer-Verlag, New York, 1994.

J.W.Gray, Mastering Mathematica, Harcourt Brace and Company, New York, 1994.

C.Knowll, M.Shaw, J.Johnson, and B. Evans, Discovering Calculus with Mathematica, Wiley, New York, 1 995.

W.T.Shaw and J. Tigg, Applied Mathematica, Getting started, getting it done, Addison-Wesley, New York, 1 994.

R.D.Skeel and J.B.Keiper, Elementary Numerical Computing with Mathematica, McGraw-Hill, New York 1993.

C.Smith and N. Blackman, The Mathematica Graphics Guidebook, Addison-Wesley, New York, 1995.

A.G.Sparks, J.W. Davenport, and J.P.Braselton, Calculus Labs using Mathematica, Harper-Collins College Publishers, New York, 1993.

T. Wicham-Jones, Mathematica Graphics, Springer-Verlag, New York, 1994.

S. Wolfram, Mathematica: A System for Doing Mathematics by Computer, Addison-Wesley, New York, 1991.

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 162 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 164 - Using Technology to Explore Calculus I. Weekly class session in which students use programmable graphing utilities and Computer Algebra Systems to explore the mathematics they are learning in MAT 162. Students in this course must be concurrently enrolled in (or have successfully completed) MAT 162.

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:

The current Mathematics Department Computer Laboratory in Bishop 340 will be sufficient for this course.