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Re: [apcalculus] Introducing the Fundamental Theorem of Calculus
Posted:
Mar 5, 2009 10:02 AM


The simplest case is paradigmatic. AreaSlope Theorem I: The slope, m, of the graph of the area, A, under a horizontal line, H, is the original horizontal line. m( A( H ) ) = H. AreaSlope Theorem II: The graph of the area, A, under the slope graph of a straight line,SL, is a straight line parallel to the original straight line  which means that the change in the area graph is equal to the change in the original straight line across the same displacement. A  SL delta A = delta SL In the general case, consider the area "under" the graph of a continuous function f(x). The tangent line to the area graph is a linear area function. The equation of the area graph's tangent line can be written in pointslope form: AT(x)=m(xa)+A(a). The areaslope term of that tangent line, by that I mean the term m(xa), corresponds to a rectangle of base (xa) and height m. Since the linear component of the area function is produced by the constant component of the graph f(x), the slope is equal to the height of the function, f(a). So at each point x=a, the slope of the tangent line to the area graph is the height of the original function at the specified point, f(a). I leave it to others to dot all the "i"s and cross all the "t"s. Bill Crombie In a message dated 3/4/2009 9:05:52 A.M. Eastern Standard Time, rexboggs5@optusnet.com.au writes:
Does anyone have a neat way of introducing/teaching this topic? The approach in my text is quite long and I think there must be a better, more intuitive way.
Cheers
Rex
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