User:Mac Davis/GSP

=Derivative Functions: Connection of Algebra and Calculus= Derivative functions can be difficult to learn and understand. If you learn it yourself instead of being lectured, its a lot easier. Socratic method.

Download Sketchpad file here.

Process
The first step is to graph a third degree polynomial function. I chose $$ -0.11 x^3 + .72 x^2 +.23 x + 1.5\,$$.

Secant Line

 * 1) Construct a secant line by the following
 * 2) Constructing first one point, a, anywhere on the function plot f, and measuring it's abscissa and ordinate (XA & YA).
 * 3) Construct parameter h.
 * 4) Calculate XA+h and f(XA+h)
 * 5) Select XA+h and f(XA+h) in order, to plot point B by using Plot as (x,y) from the "graph menu".
 * 6) Construct the line joining A and B
 * 7) Select secant line AB and find its slope.
 * 8) Plot point P by selecting XA and the slope measurement AB in order, then Plot As (x,y).
 * 9) Select point A and point P in order, and make a locus from the construct menu.

Derivative

 * 1) Select $$f(x) = -0.11 x^3 + .72 x^2 +.23 x + 1.5\,$$ and produce its derivative.
 * 2) Plot the derivative.

What's going on
In this beautiful work of art, the only thing we made that does not rely on something else, is the original function: $$ -0.11 x^3 + .72 x^2 +.23 x + 1.5\,$$

A is a point on the function, and B is a point on the function that is a given distance away from point A at all times, which changes depending on how large or small parameter h is. The secant line is an approximation of the tangent line, and moves depending on where A and B are on the original function.

Between steps one and two under Derivative, we see the what I call the minus-droppy rule applied. The derivative's function's, $$f'(x)\,$$'s, form is $$f'(x) = -3ax^2 + 2bx +c\,$$ because the original was in form $$f(x) = ax^3 + bx^2 +cx + d\,$$. Each exponent of $$x\,$$ is dropped as a coefficient, and 1 is subtracted from it's place in the superscript as well as $$d\,$$ being dropped, hence "minus-droppy!" The best part about self-teaching is that you get to make up your own names for things! Okay, I don't know what its name is, but I am sure it is some kind of important rule.

The larger parameter h is, the farther away A and B are from each other on the function, and also, the more error there is in the locus, an approximation of the derivative. When h is 0, the secant line becomes a tangent line because A and B are overlapping (therefore the tangent line is the limit of the secant line, see right). When h is 0, all points on the locus match with all points on the derivative.

When h=0, the secant line is undefined, but as h —> 0, the secant line approaches to the tangent. Because h cannot equal 0, the definition must be $$ \lim_{h \to 0}Secant=Tangent\,$$. AB [coordinates (XA,f(x)) and (XA+h,f(XA+h)) respectively] approach the derivative

Point A and Point P at all y values, have equal x values, while A is attached to $$f(x)\,$$ and P to $$f'(x)\,$$, because they share XA. P's y coordinates come from the slope of line AB. In the ideal condition that h is zero and AB is a tangent line, slope of AB is actually the slope of each point in the original function, and the locus is the derivative.