6 1. The Brachistochrone 2.828 2.622 2.586 2.582 x 0 0.5 1 y 0 0.5 1 Figure 1.3. Values achieved (top right) for J[y] by a straight line, a quarter-circle, the best trial function and a cycloid. graph of the best trial function for values of x between about 0.02 and 0.55, and it slopes more gently elsewhere. But how could we have known that the cycloid is the curve that minimizes transit time—in other words, that the cycloid is the brachistochrone? At this stage, we couldn’t have: we need the calcu- lus of variations, which was first developed to solve this problem. We will start to develop it ourselves in Lecture 2. Exercises 1 1. Rotating a curve between (0, 1) and (1, 2) about the x-axis gen- erates a surface of revolution. Obtain an upper bound on the minimum value S∗ of its surface area by using the trial-function method (and a software package for numerical integration). 2. Obtain an upper bound on the minimum value J ∗ of J[y] = 1 0 y2y 2 dx subject to y(0) = 0 and y(1) = 1 by using the trial functions y = y (x) = x with 1 4 .

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