Geometric Series Descartes, and the Fundamental Theorem of Calculus Jeff D. Farmer The Mathematical Gazette, Vol. 84, No. 500. (Jul., 2000), pp. 227-231. Stable URL: http://links.jstor.org/sici?sici=0025-5572%28200007%292%3A84%3A500%3C227%3AGSDATF%3E2.0.CO%3B2-9 The Mathematical Gazette is currently published by The Mathematical Association.

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Geometric series

Descartes, and the fundamental theorem of calculus JEFF D. FARMER This note describes a possible approach to the direct calculation of both derivatives and integrals of power functions using rather simple algebra and limits that could be easily accessible to students beginning a calculus course. The calculations provide a powerful example of the fundamental theorem, and could be used to allow students to discover the fundamental theorem from the example. The main calculation, which is not found in many calculus textbooks and is due to Descartes, is described by Maor [ 2 ] . While this approach may seem like the popular American 'reform' calculus, one may find a somewhat similar approach (presenting integral calculus from first principles) in Tom Apostol's classic and lovely text [I]. Algebraic preliminaries

One can claim that calculus students need to know all sorts of algebra. Algebra, being fundamentally more difficult than calculus, then ends up serving as a filter for the course. Nevertheless, there is some truth to the claims, and for the approach here a little algebra is required. To wit, one needs the following formula: where r is real and n is any positive integer. While one might take the opportunity to introduce proof by mathematical induction in establishing this fact, it is probably best simply to allow the students to convince themselves by working through examples for the first few values of n, or to demonstrate the formula by shifting and subtracting:

from which one may solve for S. Once the formula is established, one should also point out the equivalent formula:

In preparation, it might also be advisable to have the students practise applying the formula to situations where some power of a variable (such as 2')appears in place of r. This will prepare the students for all the calculations involved in the sequel. A few limits

The calculations in this example do involve a few limits, although these may probably be approached heuristically. The main requirement is to

228

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establish the formula for the sum of a geometric series. It may be natural, in this age of calculators, to combine this with a discussion of the equivalence between fractions and repeating decimals, in order to establish a sense among students of the utility of the formula 1 - J"' 1 - iflrl < I. 1 + r + r 2 + . . . = lim n+.. 1 - r I - r

In addition, students should be aware that for a fixed power p,

Differentiation of xP The first use that can be made of the formulas above involves differentiation. Consider the function f (t) = P ( p a positive integer). To find the derivative at a value x, one needs to find the limit of the difference quotient with points approaching x. For this purpose, we may consider r to be a number 'rather' close to 1 (students may think of numbers such as 0.9, 0.99 or 1.1, 1.01, etc.). If we then think of x as being any positive number, we have the following picture:

The derivative calculation is now simple: f' (x)

= lim

- f (r.1

X-YX

I

=

f(4

limF'(1 r

+r+

lim x" - (rxy = limg-I.- 1 - r " r-ll x - rx r+ 1 1 - r r2 + ... + F1) = pXP-'.

=

i 1

This is similar to the approach used to calculate derivatives of power functions in reform textbooks such as [3], but has added advantages which we will see when considering the fundamental theorem. Descartes and integration We now come to the main use of our algebraic facts in this context, namely a direct calculation of the area under the curve f ( t ) = tP between 0 and x, for any positive number x. This calculation, according to Maor [ 2 ] ,is due to Descartes, and it nicely gives the formula for the integral without

GEOMETRIC SERIES

229

excessive trouble. The key is to use unequal intervals* (actually geometrically equivalent ones), specifically as pictured: 9' PxP rZpxp

etc

etc

r2x

rx

x

The point is to get an upper estimate of the area using an infinite number of intervals of decreasing size, starting from the right, with end points x , rx, r2x, ... , where r is a positive number less than one. If the area under these rectangles can be calculated, it is clearly going to be greater than the area under the curve. Indeed, the calculation is not difficult: c ~ d
-

+ ( r x - r 2 x ) ? 2 + (r2x - r3x)r2'2 +..

Xp+ l

1+r+r2+

... + # "

If we let r + 1 from below, we obtain the usual formula. If students are slightly uncomfortable with the limit (and after all, Descartes did not share our modem formulation), one can make a more delicate argument by bracketing the area with a lower estimate as well:??

rPxP r2pxp

etc

etc

r2x

rx

x

* This approach was also used by Ferrnat (c 1640) and Gregory of St Vincent (1647)

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230

L ~ d >r (x-rx)S.?+

-

( r x - h ) r z P g +( r 2 x - I ~ X ) ~ ~ ~ Y + . . .

y+' f

1 + r + r 2 + ... +rj" Considering that the inequality is true for all values of r which are strictly less than one, we see that there is only one possible value for the area represented by the integral, namely that which is given by the usual limiting process. The fundamental theorem There are three ways in which this example nicely illustrates the relationships which are rigorously described by the fundamental theorem. One comes from simply observing the outcomes; the formula obtained for the derivative of a power function may be seen to represent the exact inverse (as a process) of the one obtained from the area calculation. This indeed could lead students to conjecture the essence of the fundamental theorem for themselves. In addition, in examining the algebra behind this calculation, the reasons that the derivative should have an exponent that is one lower than the original function, and that the integral should have a power one higher, are particularly clear. One can see in this example exactly where this happens in the calculation. Since one power of x appears in the denominator of the difference quotient, and one power of x also appears in the width of each interval, one can see exactly why the 'power rule' is what it is. The second observation above now leads to a nice heuristic view of what is 'really' happening in the fundamental theorem: calculating a difference quotient involves subtracting and then dividing, whereas calculating (an estimate for) the area under a curve (using rectangles) involves multiplying and then adding - the exact opposite process (students may need to be reminded of their shoes and socks in this connection!). The limiting procedures involved in both integration and differentiation then serve to turn exactly opposite discrete processes into exactly opposite continuous ones. There is a moral to this story. When we organize mathematics into a cuniculum, we too often lose important ideas that led some of the original discoverers into the subject. It should not surprise us that such ideas can often be useful to us and our students as well.

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References 1. Tom Apostol, Calculus, Volume I. Blaisedell Publishing (1961). 2. Eli Maor, e: The story of a number, Princeton University Press (1998). 3. David A. Smith and Lawrence C. Moore. Calculus: modeling and application, D. C. Heath (1996). JEFF D. FARMER Department of Mathematical Sciences, University of Northern Colorado, Greeley, CO 80639 USA Life-spans again According to the Council for Aid to Education, former students now constitute the single largest source of donations - ahead of foundations and corporations. Over the next ten years, demographic trends suggest they will become ever more important: in the 1960s there were only 1.9 alumni per student, but the ratio has risen to 3.5, and by 2010 it is expected to be around 5 per student. From the Cambridge Alumni Magazine No 27 Easter Term by Mark King who comments that even 15 years seems a short expected life span for graduates. Can't slimmers take away? We're delighted with this move because food that's misleadingly trumpeted as '85 per cent fat free', like the Go Ahead Caramel crisp biscuits pictured, is, in fact, 15 per cent fat - which is quite high. It's higher than the Tesco Strawbeny cheesecake which doesn't make any such claims. From Which Magazine August 1999, sent by A. Robert Pargeter, who asks who is being misled? 100 - 85 = 15. Rule of rating The Sail Training Association's rule of rating allows the largest square-riggers to compete with the smaller vessels. It has been worked out by a mathematical formula which has been developed over the years and its success is put down to the fact that it is kept secret. The rule of rating for the tall ships is different from others used in sailing races in that a vessel which does not cross the finishing line can still win. The crew of a smaller, traditional vessel may have sailed more skilfully and worked harder than a crew of a light displacement yacht, for example, so a formula is used to calculate an estimated time which can be used in the race results. From the Shetland Times, 6 August 1999, sent in by Andrew Evans. An imaginary polar bear tale or tail Now, for swimming in figure eights, well you can't beat the polar bear I was watching the other day. Eight eight eight eight, just like a repeating decimal, you'd have thought he was the square root of minus something-or-other. From The burglar who thought he was Bogart, by Lawrence Block, sent in by Frank Tapson.