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)