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Enhancing Visualization and Engagement in Calculus with Computer Algebra Systems

This presentation by Judy Holdener from Kenyon College discusses the integration of Computer Algebra Systems (CAS) into the mathematics curriculum at a small liberal arts college in Ohio. It highlights how technology facilitates hands-on learning with dynamic visualizations, promoting creativity and engagement among students. The use of CAS in projects, such as parametric plots, allows students to explore mathematical concepts while designing their masterpieces. This approach proves beneficial for both visualization in Calculus III and fostering a collaborative learning environment.

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Enhancing Visualization and Engagement in Calculus with Computer Algebra Systems

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  1. CAS for visualization,unwieldy computation,and “hands-on” learning Judy Holdener Kenyon College July 30, 2008

  2. Small, private liberal arts college in • central Ohio (~1600 students) Kenyon at a Glance • 12-15 math majors per year • All calculuscourses taught in a • computer-equipped classroom • All math classes capped at 25 • Profs use Maple in varying degrees

  3. Lessons that introduce ideas • geometrically. Visualization in Calculus III • a CAS can be the medium for creative, hands-on pursuits! • a CAS can produce motivating pictures/animations. • Projects that involve an element of • design and a healthy competition.

  4. Parametric Plots Project • Students work through a MAPLE • tutorial in class; it guides them • through the parameterizations of • lines, circles, ellipses and functions. • The project culminates with a • parametric masterpiece. y(t) x(t)

  5. Dave Handy

  6. Nick Johnson

  7. Andrew Braddock

  8. Chris Fry

  9. AtulVarma

  10. Christopher White Oh, yeah? Define “well-adjusted”.

  11. If x(t), y(t), and f(x,y) are differentiable then f(x(t),y(t)) is differentiable and The Chain Rule for f(x, y) Actually,

  12. Example. Compute at t=1. Let z = f(x, y) = xe2y,x(t) = 2t+1 and y(t) = t2. Solution. Apply the Chain Rule:

  13. What does this number really mean?

  14. t=4 t=3 t=2 t=1 t=0 Here’s the parametric plot of: (x(t), y(t)) = (2t+1, t2).

  15. The curve together with the surface: At time t=1 the particle is here. z = f(x,y) = xe(2y)

  16. Another Example. Compute at t=1. Let f(x, y)= x2+y2 on R2,and let x(t)= cos(t) and y(t) = sin(t). Solution. Apply the Chain Rule:

  17. Note: it’s 0 for all t!!!

  18. f(x, y)=x2+ y2 (cos(t), sin(t), f(cos(t),sin(t))) (x(t), y(t))=(cos(t), sin(t))

  19. Unwieldy Computations Scavenger Hunt!

  20. Holdener J.A. and E.J. Holdener. "A Cryptographic Scavenger Hunt," Cryptologia,31 (2007) 316-323 References J.A. Holdener. "Art and Design in Mathematics," The Journal of Online Mathematics and its Applications, 4 (2004) Holdener J.A. and K. Howard. "Parametric Plots: A Creative Outlet," The Journal of Online Mathematics and its Applications, 4 (2004)

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