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The calculus is still about limits?
Posted:
Jan 6, 2002 12:11 AM


At this moment the outstanding issue seems to be that "Calculus is still about limits."
Some 50 years ago in his book An Introduction to Mathematics Alfred North Whitehead warns
"The study of mathematics is apt to commence in disappointment ... The reason for the failure of the science to live up to its expectations is that its fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances."
The calculus is not about limits. The calculus is about the study of continuous change  changes in quantity, rates of change, and ultimately their reciprocal determination. Limits are simply "the technical procedure which has been invented to facilitate their exact presentation in particular instances." The utility of a calculus on polynomial functions is that it allows us to critically examine the calculus on a function domain where a particular technical procedure, the use of limits, is not necessary. We must be careful not to conflate the means with the ends that they are intended to facilitate. In order to evaluate the validity of the claim that limits do not sneak in the back door let us take another look at the previous response to the algebraic quadrature of the parabola "without limits" which raised the issue.
"However, one hidden assumption is that the area under a curve whose heights are k times the heights of a second curve is k times the area under that second curve. We think this is obvious geometrically because we think of areas as limits of areas of rectangles. Archimedes would prove such a statement by using similarity and exhaustion, and we would use limits of Riemann sums, but it does require a proof of this nature. Thus, the limits are hidden but there." But this is not the only way to think about area. Here is another.
The principle of trichotomy tells us that, for all rectangles having the same base as a given parabolic right triangle, the area of any one of these rectangles is either greater than, equal to, or less than the area of the parabolic triangle. The rectangle whose area is equal to the area of the parabolic right triangle we call the mean value rectangle. Just as we associate a slope function with the existence of tangent lines to a curve, in an analogous fashion, we can associate an area function, say A( f ), with the existence of a mean value rectangle for a curve over a given interval.
There is a "hidden" assumption behind the algebraic quadrature of the parabola and it revolves around this geometric notion of a mean value rectangle and its algebraic representation as an area function. The key property used in this quadrature calculation is the linearity of the area function. If A( f ) is the area under the curve over a specified interval then the property we used is A( af+bg ) = aA( f )+bA( g ). Can the linearity of the area function only be justified by the use of limits? No. The properties of the area function can also be justified from the geometric properties of the mean value rectangle. In order to do this we need to introduce a little terminology. We will call the original rectangle of base b and height b^2 the principal rectangle. In a sense this principal rectangle will act as a standard or benchmark. We will measure all other areas in comparison to it. The area of the mean value rectangle or what is the same thing, the area of the parabolic right triangle, will then be some fraction of the area of the principal rectangle.
A uniform vertical scaling of the principal rectangle by a factor r creates the following changes. ( h > h' = rh ) The heights of the mean value rectangle and the curve are scaled by the same factor. While the absolute vertical dimensions of the figures have changed, the relative heights are invariant. ( y/h = y'/h' ) A uniform horizontal scaling of the principal rectangle produces an analogous change. ( b > b' = sb and x/b = x'/b' ) Now this uniform scaling of the principal rectangle can be thought of in the following way. In effect the scaling exchanges a tiling of the principle rectangle using unit squares with a tiling using r by s rectangles. This exchange of tiles is onetoone and in each substitution the relative sizes of all portions of figures falling within the tile is left invariant. Consequently the absolute areas are scaled by a factor of rs. ( A > A' = rsA )
The difference between these two approaches to area is this. The construction of area as approximating rectangles is focused on determining bounds on the actual area under the curve. And ultimately, limits are required to force these bounds to deliver an equality. The area as a mean value rectangle is not being used to determine bounds or measures of the area directly. There are other algebraic means for accomplishing that task. In this approach it is necessary to determine the scaling properties of figures with polynomial boundaries by relating them to the scaling properties of figures with rectilinear boundaries. It is just these properties of scaling that are captured in the linearity of the area function and used in the quadrature of the parabola. The trick here is to recognize the pragmatic difference between the existence problem (of a mean value rectangle) and the construction problem (of approximating the area of a figure by rectangles).
One quick, little piece of related geometry: All parabolic triangles of the same hypotenusetype are similar. Suppose we start with a ptriangle whose hypotenusetype is given by y=ax^2. Let this ptriangle be situated in a principal rectangle of base b and height b^2. Now scale the rectangle and consequently the ptriangle by a factor s so that its base goes from b to b' = sb and its height goes from b^2 to (b')^2 = (sb)^2. We see that the relative positions of all horizontal and vertical displacements of the hypotenuse from the origin are invariant. That is, x/b = x'/b' and y/b^2 = y'/b'^2. The shape and relative size of the ptriangle within the principal rectangle are scale invariant, and consequently, the area of the ptriangle scales like s^3. Hence the factor of 8 in the previous proof.
Unfortunately in the conventional presentation of the calculus the mean value rectangle never acquires the prominence in the integral calculus that the tangent line does in the differential. It is instructive to reconsider the relationship between fundamental ideas such as mean value rectangles and tangent lines and their associated technical procedures, namely limits of Riemann sums and secant lines and to note the differences and similarities in their conceptual development as course material. There are striking differences in the conceptual architecture of the differential calculus and the integral calculus as it is usually taught. There are striking costs associated with these differences.
Bill

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