Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY CALCULUSf(x)GxFigure 2.3 The graph of a function f(x) that has a general point of inflectionat the point G.Now, we examine each stationary point in turn. For x =3,d 2 f/dx 2 = 30. Since this ispositive, we conclude that x = 3 is a minimum. Similarly, for x = −2, d 2 f/dx 2 = −30 andso x = −2 is a maximum. ◭So far we have concentrated on stationary points, which are defined to havedf/dx = 0. We have found that at a stationary point of inflection d 2 f/dx 2 isalso zero and changes sign. This naturally leads us to consider points at whichd 2 f/dx 2 is zero and changes sign but at which df/dx is not, in general, zero. Suchpoints are called general points of inflection or simply points of inflection. Clearly,a stationary point of inflection is a special case for which df/dx is also zero.At a general point of inflection the graph of the function changes from beingconcave upwards to concave downwards (or vice versa), but the tangent to thecurve at this point need not be horizontal. A typical example of a general pointof inflection is shown in figure 2.3.The determination of the stationary points of a function, together with theidentification of its zeros, infinities and possible asymptotes, is usually sufficientto enable a graph of the function showing most of its significant features to besketched. Some examples for the reader to try are included in the exercises at theend of this chapter.2.1.9 Curvature of a functionIn the previous section we saw that at a point of inflection of the functionf(x), the second derivative d 2 f/dx 2 changes sign and passes through zero. Thecorresponding graph of f shows an inversion of its curvature at the point ofinflection. We now develop a more quantitative measure of the curvature of afunction (or its graph), which is applicable at general points and not just in theneighbourhood of a point of inflection.As in figure 2.1, let θ be the angle made with the x-axis by the tangent at a52