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The Target of Calculus Reform
Posted:
Jan 27, 2002 10:56 PM
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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 ) .
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