A Course of Pure Mathematics: Third Edition by G. H. Hardy

Author:G. H. Hardy [Hardy, G. H.]
Language: eng
Format: epub
Publisher: Dover Publications

CHAPTER VI

DERIVATIVES AND INTEGRALS

110. Derivatives or Differential Coefficients. Let us return to the consideration of the properties which we naturally associate with the notion of a curve. The first and most obvious property is, as we saw in the last chapter, that which gives a curve its appearance of connectedness, and which we embodied in our definition of a continuous function.

The ordinary curves which occur in elementary geometry, such as straight lines, circles and conic sections, have of course many other properties of a general character. The simplest and most noteworthy of these is perhaps that they have a definite direction at every point, or what is the same thing, that at every point of the curve we can draw a tangent to it. The reader will probably remember that in elementary geometry the tangent to a curve at P is defined to be ‘the limiting position of the chord PQ, when Q moves up towards coincidence with P’. Let us consider what is implied in the assumption of the existence of such a limiting position.

In the figure (Fig. 36) P is a fixed point on the curve, and Q a variable point; PM, QN are parallel to OY and PR to OX. We denote the coordinates of P by x, y and those of Q by x + h, y + k: h will be positive or negative according as N lies to the right or left of M.

We have assumed that there is a tangent to the curve at P, or that there is a definite ‘limiting position’ of the chord PQ. Suppose that PT the tangent at P, makes an angle ψ with OX. Then to say that PT, is the limiting position of PQ is equivalent to saying that the limit of the angle QPR is when Q approaches P along the curve from either side. We have now to distinguish two cases, a general case and an exceptional one.

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