The calculus, as presently taught, suffers from a confusion of means and ends. It fails to draw the distinction between feasibility (what can be done with limits) and necessity (what must be done with limits). In so doing, for example, a relational understanding of the concepts entailed in the Fundamental Theorem of Calculus is supplanted with an instrumental understanding of the same. This might not be of great concern except for the position exemplified by the Calculus Reform effort that bemoans the lack of understanding and competence which accompanies the learning of calculus in both secondary and post-secondary settings. The assumption has been that the first target of reform is at the level of instruction. Perhaps the first target of reform needs to be our understanding of the calculus itself.
Preliminaries: 1. Suppose f( x ) is a continuous function on an interval [a,b] and has a tangent line at each point of its curve on the interval. The Quotient-Remainder Theorem tells us that the function can be written in the form f( x ) = Q_a^x( f )(x - a) + f( a ) ( read the quotient function as "Q sub a superscript x of f" ) and that the quotient function is itself continuous. The "off-diagonal" elements of the quotient function, Q_a^x( f ), determine the slope of the secant line from a to x. The "diagonal" elements of the quotient function, Q_a^a( f ), determine the slope of the tangent function at the point x=a. 2. The area function of a continuous function f( x ) can be represented by A_a^b( f ) = < f( x ) > (b - a). The area under the curve f( x ) over an interval from a to b is given by the area of the mean value rectangle, namely the average height of the function over the interval, < f( x ) >, times the base, (b - a).
An Alternative Version of the Fundamental Theorem of Calculus: Suppose we have a slope function, mT( ), for a given continuous curve f given by mT( f ) = Q_x^x( f ). The area under the slope function is computed by A_a^b(mT( f )) = < mT( f ) > (b-a). Now the mean value of mT( f ) across the interval [a,b] is simply the slope of the secant line crossing f( x ) on [a,b]. Thus < mT( f ) > = Q_a^b( f ). The area function is then given by A_a^b(mT( f )) = Q_a^b( f )(b-a). And by the Quotient-Remainder Theorem this reduces to A_a^b(mT( f )) = f( b ) - f( a ) .