HSU Mathematics Department
Colloquium
April 2, 1998
Solving Calculus Problems without
Calculus:
Some Historical and Contemporary Approaches
by
Martin Flashman
I. Tangent and Extreme
Problems

The Tangent Problem in Euclid Tangent to a
circle.

Euclid's Proof

Application of the proof to construction of a tangent.
(Use GSP.)

What this means with coordinates.

The Tangent Problem in Archimedes Tangent to a
parabola

Archimedes statements  Missing proofs.
(Use GSP.)

Application of the statement to construction of a tangent.
(Use GSP.)

What this means with coordinates.

A coordinate based proof and a maximization/minimization
problem.

A Minimization Problem. Old MacDonald's Farm
(Use GSP.)

Exploration with Geometers' Sketchpad.

Locus of areas yields dynamic view of solution.

Changing the constraint leads to conjecture on the general
solution.

Variables lead to a parabola and the problem solution (sans
calculus).

Generalizations of this problem with GSP.
Triangle (1 point), rectangle (2 point) , ...
? (Use GSP.)

Closest point on a curve to a given point.

Euclid: Line.

Euclid: Circle.

Archimedes: Parabola.

Exploration with GSP.
(Use GSP.)

The 2 Locations 1 Utility on Line Connection
Problem.

Exploration with GSP. (Use
GSP.)

Locus of lengths yields view of solution.
(Use GSP.)

The reflection solution. (Use
GSP.)

Some generalizations:

3 locations, 4 locations,.... (Use
GSP.)

Connection on a curve. (Use
GSP.)

The Runner/Swimmer Problem

Exploration with GSP. (Use
GSP.)

Locus of time yields view of solution.

Comparing Solutions

A generalization: Triatholon Problem?

The Least Squares Problem

Exploration
with JavaSP.

Locus of sums yields view of solution.

Another Quadratic.

Some generalizations.  Other curves.
II. Area and Velocity Problems

Area of a Parallelogram and Triangle in Euclid. Web site. (Math
371)

Area of a Circle in Euclid. Web site. (Math 401)

Area of a parabola in Archimedes. (Math 401)

Position of an object with constant acceleration
 Oresme and Galileo (Math 401)

Transformation Figures (Current  Work in Progress)

Visualization of uniform motion and position.
(Use GSP.)

Constant velocity. (Use
GSP.)

Visualization of uniform acceleration. (Falling objects?)
Euclid (Thanks to
D. Joyce  Clark U.)
Book III (Circles)
Proposition
16.
The straight line drawn at right angles to the diameter of
a circle from its end will fall outside the circle, and into the space between
the straight line and the circumference another straight line cannot be
interposed, further the angle of the semicircle is greater, and the remaining
angle less, than any acute rectilinear angle.
Corollary
to 16. From this it is manifest that the straight line drawn at right
angles to the diameter of a circle from its end touches the circle.
Proposition
17.
From a given point to draw a straight line touching a given
circle.
Proposition
18.
If a straight line touches a circle, and a straight line is
joined from the center to the point of contact, the straight line so joined
will be perpendicular to the tangent.
Proposition
19.
If a straight line touches a circle, and from the point of
contact a straight line is drawn at right angles to the tangent, the center
of the circle will be on the straight line so drawn.