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

.

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

.