Mathematica-Based Computer Laboratories for the Enhancement Of a Calculus Sequence Mike O’Leary and Raouf Boules Mathematics Department Towson University ICTCM, 2000

Background (1) zTowson is a comprehensive university located in the Baltimore metropolitan area zSAT mid range score of entering freshman class in Fall 2000 is (Verbal), and (Math) zMain textbook in use: Stewart’s Calculus: Early transcendental functions, Brooks/Cole 1999

Background (2) zAudience: Mainly math, math education, computer science, physics and chemistry (lesser degree) zSemester system zClass meets for 4 hours per week for lectures and one hour in a computer laboratory

Philosophy (1) zMaterial supplements the mathematics of an otherwise traditional course in calculus, and is NOT a substitute for it (more on this later) zOnly small set of built-in functions used in building small programs (Modules) for graphical demonstration or to bypass tedious repetitive calculations

Philosophy (2) zNo prior knowledge of Mathematica assumed zMaterial is based on interactive text (A computer document from which symbolic, numerical, and graphic tools can be invoked. The results of these computations can be pasted into the document so that each learner has an individual record of his or her explorations)

Philosophy (3) zStudents are encouraged to experiment and make discoveries zStudents may work in small groups but submit a lab report individually zStudents are expected to write answers, many comments, observations,…etc

Mathematica Role: Black Box or Teaching Tool (1) zHistory of Mathematica and its Use: yWolfram: Born in 1959 and received his PH.D. in Physics from Caltec in 1979 yBegan in 1979 constructing a CAS to solve his own problems in physics yMathematica released in 1988 by Wolfram Research, Inc

Mathematica Role: Black Box or Teaching Tool (2) zHistory of Mathematica and its Use (cont’d): yBecame popular first in the research fields of science, engineering, and economics where it was typically used as a black box to solve complicated problems yLater it entered the scene of education as people realized its potentials for demonstrations and as its front end became more user friendly

Mathematica Role: Black Box or Teaching Tool (3) zExample 1: Using Newton’s method in solving a nonlinear equation in a single variable  Black box approach: use the command FindRoot (This also uses Newton’s method) yApproach adopted: see next demonstration

Mathematica Role: Black Box or Teaching Tool (4) zExample 2: Numerical integration  Black box approach: Use the command NIntegrate (This uses a sophisticated numerical adaptive quadrature algorithm) yApproach adopted: build a module to show the graphical meaning of the tarpezoidal, and Simpson’s rules; another assesses the error in each as function of number of partitions and asks for conjectures