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 0.0
and 0.55, and it slopes more gently elsewhere.
But how could we have known that the cycloid is the curv
that minimizes transit time—in other words, that the cycloid is th
brachistochrone? At this stage, we couldn’t have: we need the calcu
lus of variations, which was first developed to solve this problem. W
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 th
minimum value
S∗
of its surface area by using the trial-functio
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 function
y = y (x) = x with
1
4
.
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