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

Figure 1. Steps 1-3. Secant line constructed

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

Secant Line

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

Derivative

Figure 2. End product.

1. Select $f(x)=-0.11x^{3}+.72x^{2}+.23x+1.5\,$ 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: $-0.11x^{3}+.72x^{2}+.23x+1.5\,$

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.

Figure 3. The Tangent Line (brown) is the limit of the Secant Line (purple/pink) as h tends to 0.

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 (X_A,f(x)) and (X_A+h,f(X_A+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 X_A. 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.