I. NUMBER AND TITLE OF COURSE

Mathematics 264 - Using Technology to Explore Calculus III

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 Computer Algebra System to explore and model concepts which are covered in Calculus III. Their understanding of fundamental notions of calculus will be enhanced by their being able to better visualize them.

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: Each numbered item typed in boldface below will be the focus of one class session. The sessions are keyed in to the MAT 263 outline.

A. Sequences/Series

1. Numerical and graphical representations of sequences

a. Explore convergence by plotting the pairs (n, an).

b. For y = f(x) where f(n) = an, apply l'Hopital's rule and the first derivative test (for the Monotone Convergence Theorem) to investigate convergence as in MAT 163 and MAT 164.

c. Program recursive sequences such as the Fibonacci sequence, and determine limits and closed forms.

2. Convergence of infinite sums

a. Depict infinite series graphically (e.g., geometric diagrams and harmonic dominos).

b. Estimate the sum of a convergent series by programming error estimates for special types of series. As a consequence, approximate well-known irrational numbers to any degree of accuracy.

c. Use numerical integration techniques from MAT 164 to check convergence by the integral test.

3. Numerical and graphical depictions of power series

a. Confirm accuracy of Maclaurin series representations by animating plots of successive partial sums against the plot of the given function.

b. By varying the center x=a, determine graphically the best Taylor series for a given function over a finite interval.

c. Program remainder estimates to investigate polynomial approximations of a given function numerically.

B . Other graphing forms and techniques

4. Polar coordinates and parametric equations

a. Graph parametric representations of curves.

b. Explore CAS options for customizing graphs.

c. Visualize and determine the area between polar curves.

5. Conic sections and quadric surfaces

a. Write programs to express conic sections and quadric surfaces in standard form.

b. Graph conic sections and quadric surfaces. Investigate quadric surfaces further by rotating their graphs and by plotting their level curves.

C. Vectors in the plane and in space

6. Vector operations

a. Verify the laws of dot and cross product symbolically.

b. Visualize and compute the area of a triangle and the volume of a parallelepiped.

c. Animate the parametric plot of a line in vector form (as x(t) moves along the line).

7. Lines and planes

a. Graph lines and planes, and determine the angles between them.

b. Program the vector methods for measuring distance from a point to a line, and from a point to a plane.

8. Vector calculus

a. Investigate limits, derivatives and integrals of vector-valued functions symbolically.

b. Analyze the motion of a particle along a curve (i.e., graph, and calculate velocity, acceleration, curvature, as well as unit tangent and unit normal vectors).

D. Partial differentiation

9. Visual and numerical representations of higher dimensional functions

a. Graph z = f(x, y) with various plotting options. Compare with several level curves.

b. Investigate limits and continuity of functions of several variables.

10. Differentiation in higher dimensions

a. Estimate partial derivatives graphically, and compute Hem symbolically.

b. Verify the chain rules symbolically.

c. Plot a function along with its tangent plane and normal line at a given point.

11. Optimization

a. Determine and classify the critical points of a given function by the second derivative test as well as by graphical inspection.

b. Solve constrained optimization problems with Lagrange multipliers.

E. Multiple integration

12. Volumes and surface areas

a. Compute multiple integrals numerically as well as symbolically.

b. Visualize and estimate volumes and surface areas of solids determined by functions over rectangular and nonrectangular regions.

c. Convert multiple integrals to polar, cylindrical, or spherical coordinates in order to achieve 12(a) and 12(b) above.

13. Integrals over curves and surfaces

a. Symbolically integrate scalar functions over curves and the tangential and normal components of a vector field along a curve.

b. Integrate scalar functions over surfaces and vector fields over oriented surfaces.

c. Program computing the area of any polygon using Green's theorem.

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

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

G.L. Bradley and K.J. Smith, Calculus, Prentice Hall, Englewood Cliffs, New Jersey, 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. New York, 1995.

Evans, Discovering Calculus with Mathematica, Wiley,

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

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

C.Smith and N. Blachman, 1995.

Me Mathematics Graphics Guidebook Addison-Wesley, New York,

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

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

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

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

MAT 164; students must be concurrently enrolled in, or have successfully completed, MAT 263.

1 credit: (1:0)

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

MATH 264 - Weekly class session in which students use a programmable graphing utility and Computer Algebra Systems to explore the mathematics they are learning in MAT 263. Students in this course must be concurrently enrolled in (or have successfully completed) MAT 263.

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.

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