Tall, David

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University of Warwick institutional repository: http://go.warwick.ac.uk/wrapThis paper is made available online in accordance with publisher policies. Please scroll down to view the document itself. Please refer to the repository record for this item and our policy information available from the repository home page for further information.  To see the final version of this paper please visit the publisher’s website. Access to the published version may require a subscription. Author(s): David Tall Article Title: Visualizing Differentials in Integration to Picture the Fundamental Theorem of Calculus Year of publication: 1991 Link to published version:  Publisher statement: None  Published in Mathematics Teaching, 137, 29–32 (1991)Visualizing Differentials in Integrationto Picture the Fundamental Theorem of CalculusDavid TallMathematics Education Research CentreUniversity of WarwickCOVENTRY CV4 7ALIntroductionWhat is a differential? From my investigations asking sixth-formers arriving atuniversity about concepts in the calculus, the evidence shows a manifest confusion inthe meaning of notations such as dydx   and ò  f(x) dx. Many students repeat the receivedwisdom that  dydx   means the derivative of a function y=f(x) and should be thought of asa single indivisible symbol, not as a quotient. Those who do give dx a meaning as aseparate entity invariably talk of a “very tiny change in x” or an “infinitely small changein x” or even “the limit of d x as it tends to zero”. Meanwhile the dx in ò  f(x) dx means“with respect to x”, though students need to be willing to make the substitutiondu  = dudx  dx , to compute the integral by substitution. A little later they may be facedwith the problem to solve the differential equationdydx  = – xy (where they had been told that dydx  is an indivisible symbol) by “separating thevariables” to gety dy = – x dx(what does the dx mean here?) then put an integral sign in front to getò  y dy = – ò x dx(where presumably dx now means “with respect to x”), to obtain the solution(s)y22   = – x22   + c.Why is it that we seem to teach what should be a logical and clearly defined subject insuch a perverse and mystical way? Perhaps it is simply that we belong to amathematical community and have learned to repeat the litanies of our youth that gave- 2 -us a passport to become fully fledged members. I confess that, along with most of mycolleagues, I learned to cope with the routines and got the right answers until I ceasedto ask awkward questions about the meanings of the symbolism.A perusal of calculus textbooks in current use shows some very neat ways ofsidestepping the fundamental issues of meaning, but few seem to have a totallycoherent view of the meaning of differential that works throughout the calculus.In an earlier article1 I discussed the visual interpretation of the differential indifferentiation, but at the time had not yet appreciated a possible analogous use of thedifferential dx in the integral ò  f(x) dx.In this article I will show that there is indeed a simple meaning that can be given to dxwhich works in both differentiation and integration – a meaning that was given byLeibniz in his first publications on the calculus2,3 – a meaning that takes on a new andvigorous life in our modern computer age. For three hundred years Leibniz has beenmaligned in the English-speaking world, first as a man who stole Newton’s theory ofthe calculus, and then as someone who invented an incredibly useful but curiouslymystical notation. He deserves better treatment. All it requires is to look at the rightpictures.1. The differential in differentiationThis is an easy matter to explain and has been well represented in the literature (see, forexample, Quadling4). The derivative of a function f at a point x is found by calculatingf(x+h)–f(x)h  and considering what happens as h gets small. If it tends to a limiting value, this isdenoted by f'(x) and called the derivative of f at x. It is, of course, the gradient of thetangent to the graph at x. If dx is any real number, then dy is defined asdy = f'(x) dx.This simply says that the tangent vector is in the direction (dx, dy) (figure 1).- 3 -dxdy = f'(x) dxy = f(x)xfigure 1: dx and dy as components of the tangent vectorComputer graphics now help us visualize that, under high magnification, a suitablysmall portion of the graph of a differentiable function will look straight. Thus, when dxis suitably small, then dy is a good approximation to f(x+dx)–f(x) (figure 2).dxdyy = f(x)xfigure 2: magnifying a locally straight portion of the graph of a differentiable functionThis insight will prove valuable in linking this use of the differential with that inintegration.2. The differential in integrationLet us begin by attempting to give the differential in integration the same meaning as indifferentiation: dx represents a change in x. We will use the notation bSx=a  f(x) dx or S ba f(x) dx to denote the sum of the areas of strips width dx, height f(x) between x=a andx=b. (In Britain, the more usual symbol is bSx=a  f(x) d x, where d x denotes a smallchange in x, but this extra piece of symbolic baggage is not absolutely necessary. Itarose in a Cambridge textbook in 18035 to distinguish between the value of anincrement d x before taking a limit and an infinitesimal change dx “in the limit”. Sincethen it has been used to distinguish between a finite sum and the limit of the sum as thewidth of the strips diminish in size. We will avoid this notational difficulty shortly byreplacing the S  in the finite sum by the ò  in the limit. No other notational change will benecessary.- 4 -3. Visualizing the Fundamental Theoremy=f(x)xdxf(x)a bfigure 3 :  The area sum  S ba  f(x) dxTo understand the relationship between the use of dx here and that in differentiation, allwe have to do is to look at the right picture. Figure 3, which for so long has been thestandard view is not the right one for the fundamental theorem. We must look not at thegraph of y=f(x), but at the graph of y=I(x) where I'(x)=f(x). We will assume for themoment that such a function I can be found, returning to the conditions under which itexists in the next section. Given such an I where I'(x)=f(x) we simply draw thecorresponding picture for y=I(x) with the same subdivision of the interval [a,b] intosub-intervals (figure 3).y=I(x)dxdyI(b)I(a)a x bdy = I’(x)dxfigure 4: The sum  S ba  f(x) dx as a sum of lengths  S  dyAt each point x of the subdivision we draw the tangent to the curve I(x), then thecorresponding increment to the tangent isdy = I'(x) dx = f(x) dx.- 5 -Thus the sum S ba  f(x) dx is seen as the sum of the lengths S  dy where each dy is thevertical component of the tangent vector to the graph of y=I(x). The sum S  dy is thesum of the vertical line segments, and, provided that the dx are taken to be small so thatthe graph is relatively straight from x to x+dx, then this is approximately equal to theincrement to the graph, I(x+dx)–I(x). Adding together the increments to the graph fromx=a to x=b simply gives I(b)–I(a). Thus adding together the vertical steps dy may insome sense approximate to I(b)–I(a). Figure 4 gives a fair indication of this idea. It failsin part because we had to make the strips fairly wide to see what is going on and this,in turn, means that the graph may be so curved in a strip that dy is clearly different fromI(x+dx)–I(x). But imagine it instead as having a large number of strips, and imagine apart of the graph being magnified to see its local straightness with a few strips next toeach other (figure 5).by=I(x)aI(a)I(b)dydxfigure 5: looking closely at the summation processNow it should be possible to imagine that as the lengths dx get smaller, the sum of thelengths S  dy approximates to I(b)–I(a).The picture only gives a sense of what is going on. But the zooming-in process shouldhint at something more. As one zooms in, the curved graph gets less curved. Studentswho use a graph-plotting program readily appreciate this phenomenon. It was the firstthing that the first students to play with Graphic Calculus observed without anyprompting.If we accompany the zooming-in process by taking a smaller value of dx, thecorresponding value of dy more closely approximates the step up to the curve. Weshould be able to see that the relative error in the difference between the value of dy- 6 -and the actual step to the curve gets less. In this way we get a hint that the errors arenow small in proportion to the vertical step, so adding them together, the total error isalso now small in proportion to the total vertical step. In other words we may begin tosee that, as the lengths dx get smaller, the sum of the vertical steps, S  dy, gets closer toI(b)–I(a).The symbol ò ba  f(x) dx is used to represent the limit of S ba  f(x) dx as (the maximumsize of) dx gets small. The argument just given provides a powerful intuition as to whythis limit is likely to be: òba  f(x) dx = I(b)–I(a).4. Proving the Fundamental TheoremAlthough we may now have a sense of why the fundamental theorem of calculus mightbe true, this does not yet constitute a proof. For instance, what are the conditions on thefunction f which would guarantee that there is a function I such that I' = f ? Until wecan establish this we cannot even draw figures 4 and 5, because we cannot be sure ofthe existence of the function I. I have earlier shown how one might visualize the likelyproperties required of f by looking at the graph of f in a different way6,7. I suggest thatinteresting pictures might occur by maintaining a constant y-range whilst taking a muchsmaller x-range. For instance, figure 6 shows the graph of y=sinx with the samey-range in each (–3 to 3) but x-range being changed from –3 to 3 down to 1 to 1.01.What happens is that the graph in the second case is pulled flat by the stretching of athin x-range to fill the computer window.             Figure 6 : The graph of y=sinx, pulled out flatIf one calculates the area under a flat graph like this, the area from x to x+h isapproximately f(x)h. This represents a change in area from A(x) to A(x+h), so- 7 -A(x+h)–A(x) »  f(x)h.This suggests that we may haveA(x+h)–A(x)h   »  f(x),and perhaps, as h fi  0, we might getA'(x)=f(x).What kind of function, when stretched out horizontally near x=x0, looks flat ? If wesuppose this means that the graph lies in a pixel representing a height f(x)± e , then weneed to know that, given such an e  > 0, then we can find a small enough x interval, sayx± d , so that when t lies between x– d  and x+ d , then f(t) lies between f(x)– e  and f(x)+ e .In other words, a natural condition for the function to satisfy the fundamental theoremis that it be continuous in the formal sense:Given any e >0, a d >0 can be found such that whenever x– d  < t <x+ d  weknow that f(x)– e  < f(t) < f(x)+ e .For such a function if the width of a strip h is taken positive and less than d  then thevalue of f(t) will lie between f(x)–e  and f(x)+e  throughout the strip and so(f(x)– e )h  < A(x+h)–A(x) < (f(x)+ e )h.HenceA(x+h)–A(x)h  is sandwiched between f(x)– e  and f(x)+ e . A similar argument holds for h negative. Ase  is arbitrary, this is the formal definition thatlime fi 0  A(x+h)–A(x)h    = f(x),i.e. A'(x) = f(x).Note that this only requires the continuity of f(x), not that it be differentiable. Thereader who has completed a university mathematics degree will doubtless know thistheorem. But can you see why it is true with your inner eye?It is easy to simulate the idea on the computer. Figure 7 shows the blancmangefunction, which is everywhere continuous and nowhere differentiable. Differentiability- 8 -is exhibited by seeing that when magnifying the graph it never looks straight (figure 7)and continuity by stretching the graph horizontally to see it pull out flat (figure 8) 8,9.Figure 7 : The blancmange function being magnified near x=1/3Figure 8 : The blancmange function being stretched horizontally from x=0.499 to x=0.501Figure 9 shows two graphs10: one – the smooth one – is the numerical area a(x) underthe blancmange function, calculated using the mid-ordinate rule with strips width 0.1.This is smooth and (within the accuracy of the approximation) locally straight (as canbe seen by magnifying it). The other looks like the blancmange function, but it is not. Itis the (numerically calculated) gradient function of the numerical area. It is only anapproximation, but it clearly intimates that the area function a(x) is differentiable once toget the blancmange function. Thus the area function a(x) is differentiable everywhereonce but nowhere twice...- 9 -Figure 8 : The area function for the blancmange and its (approximate) derivativeUsing Real Functions and Graphs 10 on the Archimedes computer, it is possible todraw (a numerical approximation to) the area function for a(x), which is a graph b(x)whose derivative is a(x) and whose second derivative is the blancmange function. Thisbuilds an approximation to a graph which is everywhere differentiable twice, butnowhere three times, even though its second derivative is continuous. It is relativelyboring to draw, yet illustrates the concept which is otherwise hard to imagine: thepossibility of a graph differentiable twice but not three times, or, more generally, ntimes but not n+1 times.Leibniz would have never visualized such monsters. In fact, when they appeared in thenineteenth century, they were generally regarded as unintuitive. But, given appropriatecomputer environments for exploration, their properties become more natural and theybecome integral parts of the theory. We can now see continuity arising as a naturalingredient of the Fundamental Theorem of the Calculus rather than being a theoreticalpre-requisite imposed for some more esoteric reason. And, at last, we see the intuitionof Leibniz, which was exhibited in the very first published articles on the calculus, yetvilified for three hundred years in the English press, appearing as a natural conceptionin a modern computer approach.- 10 -References                                                1.  Tall D. O. 1985: “Tangents and the Leibniz notation”, Mathematics Teaching, 11248-52.2. Leibniz G. W. 1684: “Nova methodus pro maximis et minimis, itemquetangentibus, qua nec fractas, nec irrationales quantitates moratur, & sinulare proillis calculi genus”, Acta Eruditorum 3,  467-473.3. Leibniz G. W. 1686: “De geometria recondita et analysi indivisibilium atqueinfinitorum”, Acta Eruditorum 5.4 . Quadling D.A. 1955: Mathematical Analysis, O.U.P., Oxford.5. Woodhouse R. 1803: The Principles of Analytical Calculation, Cambridge.6. Tall D. O. 1982: “The blancmange function, continuous everywhere butdifferentiable nowhere”, Mathematical Gazette, 66, 11-22.7 .Tall D.O. 1986: Graphic Calculus II (for BBC, Nimbus, Archimedes computers),Glentop Press, London.8. Tall D. O. 1986: Graphic Calculus I (for BBC, Nimbus, Archimedes computers),Glentop Press, London.9. Tall D. O., Blokland P. & Kok D. 1990: A Graphic Approach to the Calculus, (forIBM compatibles) Sunburst, Pleasantville, NY.10. Tall D. O. 1991: Real Functions and Graphs, (picture drawn using the Archimedescomputer), C.U.P., Cambridge.

1684: “Nova methodus pro maximis et minimis, itemque tangentibus, qua nec fractas, nec irrationales quantitates moratur, & sinulare pro illis calculi genus”,

1686: “De geometria recondita et analysi indivisibilium atque infinitorum”,

1803: The Principles of Analytical Calculation,

A Graphic Approach to the Calculus, (for IBM compatibles) Sunburst,

Graphic Calculus I (for BBC, Nimbus, Archimedes computers),

Real Functions and Graphs, (picture drawn using the Archimedes computer), C.U.P.,

Tangents and the Leibniz notation”,

The blancmange function, continuous everywhere but differentiable nowhere”,

Visualizing differentials in integration to picture the fundamental theorem of calculus

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Visualizing differentials in integration to picture the fundamental theorem of calculus

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