User:Mac Davis/GSP

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[edit] 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.

[edit] Process

Figure 1. Steps 1-3. Secant line constructed
Figure 1. Steps 1-3. Secant line constructed

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

[edit] 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 (XA & YA).
    2. Construct parameter h.
    3. Calculate XA+h and f(XA+h)
    4. Select XA+h and f(XA+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 XA 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.

[edit] Derivative

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

[edit] 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.

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

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