cas for visualization, unwieldy computation, and “hands-on” learning judy holdener kenyon...
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CAS for visualization,unwieldy computation,
and “hands-on” learning
Judy Holdener
Kenyon College
July 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
dt
dy
y
f
dt
dx
x
f
dt
df
Actually,
dt
dy
y
tytxf
dt
dx
x
tytxf
dt
tytxdf
))(),(())(),(())(),((
Example.
Let z = f(x, y) = xe2y, x(t) = 2t+1 and y(t) = t2.
Compute at t=1. dt
tytxdf ))(),((
dt
dy
y
f
dt
dx
x
f
dt
df
Solution.Apply the Chain Rule:
yxey
f 22ye
x
f 2
2dt
dxt
dt
dy2
)2(2)2( 22 txeedt
dy
y
f
dt
dx
x
f
dt
dz yy
yxey
f 22ye
x
f 2
2dt
dxt
dt
dy2
)2()12(2)2(22 22 tete
dt
dz tt 22 222 )48(2 tt ette
222 )248( tett
45.10314 2
1
edt
dz
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. ))(),(( tytxfdt
d
dt
dy
y
f
dt
dx
x
f
dt
df
Solution.Apply the Chain Rule:
xx
f2
yy
f2
tdt
dxsin t
dt
dycos
dt
dy
y
f
dt
dx
x
f
dt
df
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)