Visualization in Mathematics using Geometer's Sketchpad

Jon Wilkin, Northern Virginia Community College.  www.nvcc.edu/home/jwilkin 

These models require the user have The Geometer's Sketchpad, version 4, installed on their computer. The software is available from the publisher at www.keypress.com 

Calculus models

tangents1 - this models a secant to a circle "becoming" a tangent line as one point on the circle defining the secant approaches the other point defining the secant.

tangents2 - this models a secant to a curve (here, a parabola) "becoming" a tangent line to the curve as h goes to 0.

mean value theorem - this models the proof of the Mean Value Theorem. A function is graphed. The interval used is from -4 to 4. A point travels this interval. The user can toggle on-off the tangent line to the graph as well as the vertical segment defined in the proof and the function defined for which Rolle's theorem applies.

parabola and point not on - this models the calculus problem of determining the equations of the tangent lines to a curve, given a point not on the curve.

sliding ladder - this models the top of a ladder sliding down a wall as the base moves away from the wall at a constant rate, a classic related rate problem

Searchlight beam - this models the beam of a rotating searchlight against a wall, showing the change in linear speed of the light on the wall as the searchlight rotates at a constant rate.

rectangle in semicircle - this models the classic problem of determining the dimensions of a rectangle of largest area that can be inscribed in a semicircle.

closest point - this models the classic problem of determining the point on a curve that is nearest to a given point not on the curve.

river on one side - this models the problem of maximizing the area of a rectangular field, having a river (or barn) on one side, while the remaining three sides have a fixed total length.

box - this models the classic problem of making a box of maximum volume from a rectangular sheet of paper by cutting out corners and folding.

max area fixed hypo - this models the problem of determining the dimensions of a right triangle of maximum are, given that its hypotenuse has a fixed length.

swim to shore then run - this models the problem of determining the minimum time to swim to shore, then running to a destination.

triangle with fixed point on hypo - this models the problem of determining the minimum area of a right triangle when a point on the hypotenuse is fixed.

norman - this models the problem of determining the dimensions of a norman window of fixed perimeter that maximizes its area.

trough -this models the problem of determined the maximum volume of a trough when the only variable is the angle the sides make with the base.

inscribed triangle - this models the problem of determining the dimensions of a triangle of maximum area inscribed in a circle.

split triangle into equal areas - this models the problem of splitting an isosceles triangle into two pieces, one a triangle, the other a trapezoid, of equal area.

polar graph - this plots polar points, one at a time, of a given polar function. The function can be changed.

polar region 1 - this sweeps out the polar region r ≥ 0, π/3≤ θ ≤ 2π/3

polar region 2 - this sweeps out the polar region -1 ≤ r ≤ 1, -π/4 ≤ θ ≤ π/4

parametric graph - this plots parametrically defined points. The parametric functions can be changed.

rectangular to parametric - two rectangular graphs x=f(t), y=g(t) graphed. Also the parametric curve (f(t), g(t) ) is graphed.

pip around corner - this models the classic calculus problem of determining the pipe of maximum length that will fit around a corner of two hallways. The width of either hallway can be changed.