Page:Calculus Made Easy.pdf/132

CHAPTER XII.

CURVATURE OF CURVES.

Returning to the process of successive differentiation, it may be asked: Why does anybody want to differentiate twice over? We know that when the variable quantities are space and time, by differentiating twice over we get the acceleration of a moving body, and that in the geometrical interpretation,

Fig. 31.	Fig. 32.

as applied to curves, ${\frac {dy}{dx}}$ means the slope of the curve. But what can ${\frac {d^{2}y}{dx^{2}}}$ mean in this case? Clearly it means the rate (per unit of length $x$) at which the slope is changing—in brief, it is a measure of the curvature of the slope.

Suppose a slope constant, as in Fig. 31.

Here, ${\frac {dy}{dx}}$ is of constant value.