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)