Mathematics

Problem

The coordinates of a point on a circle of radius

a

are given by

x = a\cos t,\; y = a\sin t

. Determine

\dfrac{d^{2}y}{dx^{2}}

in terms of

t

and hence find its value when

t = \dfrac{\pi}{2}

Intuitive explanation

You're working with a parametric curve, so the key is to use the chain rule twice—first to find $\frac{dy}{dx}$ by dividing derivatives with respect to $t$ , then to find the second derivative by differentiating that ratio again with respect to $x$ . The second derivative tells you how the slope itself is changing, which relates to the curvature of the circle. Once you have the general formula, substituting $t = \frac{\pi}{2}$ pinpoints the curvature at the top of the circle.

Intuitive explanation

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