The first theorem is a mildly weaker form of the Fundamental Theorem of Calculus. Caratheodory used the Quotient-Remainder Theorem as the basis for defining the derivative and its properties. This same theorem lies at the heart of the Fundamental Theorem of Calculus.
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 subscript 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). 3. For a given continuous function, f( x ), we assume the existence of the slope of the area function, A_a^t( f ) . The area function is here considered to be a function of t, the upper value of the interval.
We will apply the Quotient-Remainder Theorem to an area function based at an origin, o, which is then a function of its upper value, t. This yields A_o^t( f ) = Q_a^t( A )(t-a) + A_o^a( f ) or A_o^t( f ) - A_o^a( f ) = Q_a^t( A )(t-a). Now it is also the case that A_o^t( f ) - A_o^a( f ) = A_a^t( f ) = < f( x ) > (t - a) . Comparing expressions for the quotient function yields Q_a^t( A ) = < f( x ) > . The "diagonal elements" of the quotient function determine the slope function for the tangent line. Q_t^t( A ) = < f( x ) > = f( t ) . The mean value of f( x ) on the degenerate interval [t,t], a point, is just the function evaluated at t. Thus mT(A_a^t( f )) = f( t ) .
And again the other way ...
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 ) .
So why such an emphasis on doing a version of the calculus without limits? To teach students the calculus without teaching them also the true power and limitations of Algebra, to teach students the calculus without also teaching them the true conditions under which the power of Analysis is absolutely indispensable is to do a disservice to Algebra and to Analysis as disciplines of inquiry, and ultimately a disservice to the education of the young.
The first target of reform needs to be our understanding of the calculus itself.