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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 - PowerPoint PPT Presentation
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CAS for visualization,unwieldy computation,
and “hands-on” learningJudy Holdener
Kenyon CollegeJuly 30, 2008
Kenyon at a Glance
• Small, private liberal arts college in central Ohio (~1600 students)• 12-15 math majors per year
• All calculus courses taught in a computer-equipped classroom
• Profs use Maple in varying degrees
• All math classes capped at 25
Visualization in Calculus III
• Projects that involve an element of design and a healthy competition.
• Lessons that introduce ideas geometrically.
a CAS can produce motivating pictures/animations.
a CAS can be the medium for creative, hands-on pursuits!
x(t) y(t)
• Students work through a MAPLE tutorial in class; it guides them through the parameterizations of lines, circles, ellipses and functions.
Parametric Plots Project
• The project culminates with a parametric masterpiece.
Dave Handy
Nick Johnson
Andrew Braddock
Chris Fry
Atul Varma
Christopher White
Oh, yeah? Define “well-adjusted”.
The Chain Rule for f(x, y)
If x(t), y(t), and f(x,y) are differentiable then f(x(t),y(t)) is differentiable and
dtdy
yf
dtdx
xf
dtdf
Actually,
dtdy
ytytxf
dtdx
xtytxf
dttytxdf
))(),(())(),(())(),((
Example.
Let z = f(x, y) = xe2y, x(t) = 2t+1 and y(t) = t2.
Compute at t=1. dt
tytxdf ))(),((
dtdy
yf
dtdx
xf
dtdf
Solution.Apply the Chain Rule:
yxeyf 22
ye
xf 2
2
dtdx t
dtdy 2
)2(2)2( 22 txeedtdy
yf
dtdx
xf
dtdz yy
yxeyf 22
ye
xf 2
2
dtdx t
dtdy 2
)2()12(2)2(22 22 tete
dtdz tt
22 222 )48(2 tt ette 222 )248( tett
45.10314 2
1
edtdz
t
What does this numberreally mean?
Here’s the parametric plot of: (x(t), y(t)) = (2t+1, t2).
t=1
t=2
t=3
t=4
t=0
z = f(x,y) = xe(2y)
The curve together with the surface:
At time t=1 the particle is here.
Another Example.Let f(x, y)= x2+y2 on R2, and let x(t)= cos(t) and y(t) = sin(t).
Compute at t=1. ))(),(( tytxfdtd
dtdy
yf
dtdx
xf
dtdf
Solution.Apply the Chain Rule:
xxf 2
y
yf 2
tdtdx sin t
dtdy cos
dtdy
yf
dtdx
xf
dtdf
tytx cos2sin2
tttt cossin2sincos2
0 Note: it’s 0 for all t!!!
f(x, y)=x2+ y2
(x(t), y(t))=(cos(t), sin(t))
(cos(t), sin(t), f(cos(t),sin(t)))
Unwieldy Computations
Scavenger Hunt!
References
Holdener J.A. and E.J. Holdener. "A Cryptographic Scavenger Hunt," Cryptologia, 31 (2007) 316-323
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)