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

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

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
Select secant line AB and find its slope.
Plot point P by selecting X_A and the slope measurement AB in order, then Plot As (x,y).
Select point A and point P in order, and make a locus from the construct menu.

[edit] Derivative

Figure 2. End product.

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

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.

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User:Mac Davis/GSP

From Wikipedia, the free encyclopedia

[edit] Derivative Functions: Connection of Algebra and Calculus

[edit] Process

[edit] Secant Line

[edit] Derivative

[edit] What's going on

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