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CAS for visualization, unwieldy computation, and “hands-on” learning

CAS for visualization, unwieldy computation, and “hands-on” learning. Judy Holdener Kenyon College July 30, 2008. Small, private liberal arts college in central Ohio (~1600 students). Kenyon at a Glance. 12-15 math majors per year. All calculus courses taught in a

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CAS for visualization, unwieldy computation, and “hands-on” learning

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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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