The slope of an arbitrary function f(x) at x is interpreted as the slope of a straight line tengential to f(x) at x. This interpretation of the slope comes ultimately from the mathematical definition of the derivative as a limit Dx -> 0 of the Forward Difference formula.
Maple provides a powerful environment to illustrate this interpretation of the slope. In the following, the equation for the tangential straight line
y(x) = m x + b
is defined and plotted at various location along f(x). This allow to check "by eyes" the validity of the slope interpretation.
The steps are:
- calculate exactly the slope of f(x) - the Skewed Mexican Hat function.
- set the slope m = slope of f(x) at x = X (arbitrary position); therefore m = m(X)
- solve for b such that y(X) = f(X); therefore b = b(X)
- plot some of the tangential lines for various values of X
- plot these lines together with f(x) to illustrate that the lines are really tangential to f(x) for any X
Here is the actual Worksheet.
Section 5.4 
Chapter 5 
Section 5.6
TOC
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