h >0, then a D 1. Show that, if m(h) m(0) D bh C o(h) for h < 0, then
bD 1. Deduce that there does not exist a c with m(h) m(0) D ch C o(h) when h is allowed to be both positive and negative. Thus m is not differentiable at the point t D 0.
If we look at the real world, the correct question to ask is not whether ‘real world functions’ are differentiable, but ‘is a certain process usefully represented by a differentiable function?’ In the nineteenth century it was believed that the answer must always be yes, but the twentieth century provided examples like the hiss of loudspeakers which could be usefully considered as continuous but not differentiable.
A particularly interesting example is given by stock market prices. The reason why we know that the price of shares does not vary in a differentiable manner may be stated quite simply. If the price f (t) at time t were differentiable, then, at a certain scale, it would be essentially linear, that is to say, f (t C h) D
f(t) C ah C o(h). Let t be the present time and let h > 0. Knowing f (t h) and f (t), we can make a good estimate of a. If a > 0, we buy shares now (that is to say, at time t) and sell them at time t C h. If a < 0 we sell shares now (that is to say, at time t) and buy them at time t C h. If things were that simple, every fool would make money on the stock exchange. But we know that one fool’s gain is another fool’s loss, so that not every fool can make money on the stock exchange. Thus the proposed method cannot work and f cannot be differentiable. If the reader consults the graph of a stock market index she will see that it does indeed appear to be the case that the function pictured there is not differentiable.20
20 However, there are advanced mathematical theories which shed a great deal of light on the behaviour of share prices and provide well-paid employment for many mathematicians. Occasionally, people tell me that some of the firms who employ these mathematicians go bankrupt. I reply that, indeed, firms which employ such mathematicians sometimes go bust, but those which do not invariably do.
I said earlier that the notion of continuity is hard-wired in our make-up. The reader may feel that the same is true of differentiation. After all, the derivative
f0(t) is simply the ‘rate of change’ of f at t. However, she will look in vain for any concept resembling ‘rate of change’ in classical literature. What experience would an Ancient Roman have which required such an interpretation and how would it be possible to express a rate of change using Roman numerals or measure it without accurate clocks? Ingenious historians of science have found the germ of the idea of a rate of change in certain Medieval philosophers, but it was the work of Galileo on falling bodies (see Chapter4) which brought the idea to centre stage in mechanics. The idea then percolated outwards from physics to general mathematics and then into general ‘common sense’ aided by the invention of objects like the railway train which provided society with concrete illustrations of the abstract concept.21
Even if we are not genetically programmed with the notion of ‘rate of change’, the everyday life of our reader (who, if she has read this far, is certainly mathematically inclined) is so full of velocities, accelerations, rates of inflation and so on, that she will have a well-developed intuition about rates of change which I shall rely on in the work that follows. As with continuity, this means that more advanced work will require the theory to be rebuilt from scratch, but knowing the outline given here will make it easier for her to understand what the rebuilt theory will look like.
1.5 Tangents
We have already seen two ways of looking at differentiation. One was the idea of ‘approximation to first order’ and the second that of ‘short term prediction’. There is a third way of looking at things which draws on our geometric intuition. Suppose that we have a nice curve described, in the usual way, by y D f (x). If (x0, y0) is a point on the curve (that is to say, if y0D f (x0)), we can ask ‘which straight line looks most like the curve near (x0, y0)?’. We recall that the equation for a straight line is y D a C mx, where m is called the slope or
gradient of the line.
If we want our straight line to look ‘like the curve near (x0, y0)’, then, presumably, we want it to pass through (x0, y0), that is to say, we want y0D
aC mx0. If this is the case, a little algebra reveals that the equation of the line
21 A cynic might add that reading the daily paper shows that many, apparently educated, individuals still misunderstand the notion.
Figure 1.1 A tangent
can be rewritten as
y y0D m(x x0) or, after rearrangement,
yD f (x0) C m(x x0).
Thus we want f (x0) C m(x x0) to look as close as possible to f (x) when x is close to x0. In other words (as the reader is probably already muttering to herself), we want
f(x0C δx) f (x0) C mδx
as closely as possible. Replacing the words ‘as closely as possible’ by ‘to first order’, we get m D f0(x0). The straight line
yD f (x0) C f0(x0)(x x0) is called the tangent to the curve y D f (x) at x0.
In a work written more than two thousand years ago, Diocles recalls how ‘when Zenodorus the astronomer . . . was introduced to me, he asked me how to find a mirror surface such that when it is placed facing the sun the rays reflected from it meet a point and thus cause burning.’
Diocles was able to show that a paraboloid has this property and we shall use the ideas of the calculus to provide another demonstration. However, we shall need to think quite hard and do some fairly involved calculations.22 22 Ptolemy, King of Egypt, asked Euclid to teach him geometry. ‘O King’ replied Euclid ‘in
Egypt there are royal roads and roads for the common people, but there are no royal roads in geometry.’
Figure 1.2 Reflection of a ray of light from a straight line
It is very easy to produce very thin (that is to say line-like) beams of light which we refer to as rays. These rays travel in straight lines and it is not hard to produce an experimental set up in which they stay within a particular plane, so we may talk about things in two rather than three dimensions. Experiment shows that, if a ray of light strikes a straight line mirror, then it is reflected in such a way that the angle23of incidence θ
iequals the angle of reflection θr as shown in Figure1.2. (Note that the line BY is perpendicular to the mirror XZ.)