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13.3 Rates of Change
Suppose in the given figure, I want to find the shortest path from
a to a point p on the segment
and back to
d. Any such path will be uniquely defined by giving any one of the six
numbers:
Here are as shown in the figure below:
For a given point p, any of the six numbers
is a
function of any of the others. For example, we have is a function of
and is a function of , since for and we have
Also is a function of , since by similar triangles
and hence
We have
and
I refer to
as the rate of change of with respect to and to
as the rate of
change of
with respect to . Note that the `` "'s in ``
'' and ``
''
represent
different functions. In the first case
and in the
second
case
. Here
is positive,
indicating that increases when increases, and
is negative, indicating that decreases when increases.
I want to find the path for which is shortest; i.e., I want to find the
minimum value of . I can think of and as being functions of ,
and
then the minimum value will occur when
; i.e.,

(13.16) 
Now , so
; i.e.,
and
, so
, i.e.,
Equation (13.16) thus says that for the minimum path
; i.e.,
, and hence
.
Thus the minimizing path satisfies the reflection condition, angle of
incidence
equals angle of reflection. Hence the minimizing triangle will make
and
similar, and will satisfy
so
or
This example was done pretty much as
Leibniz would have done it. You should compare
the solution given here to your solution of exercise 13.15A.
The problem in the last example was solved by Heron (date uncertain, sometime
between 250 BC and 150 AD)
as follows[26, page 353].
Imagine the line
to be a mirror. Let and
denote the images of and in the mirror,
i.e. and are perpendicular to
and dist
= dist
, dist
= dist
. Consider any path going from
a to a point p on the mirror, and then to d.
Then triangle and triangle
are congruent, and hence
and hence the paths apd and have equal lengths.
Now the shortest path is a straight line, which makes
the angles and are vertical angles, which are equal.
Hence the shortest path makes the angle of incidence equal to
the angle of reflection, as we found above by calculus.
Remark: We can think of velocity as being rate of change of position with
respect
to time.
13.17
Exercise.
Consider a conical tank in the shape of a right circular cone with
altitude
and
diameter
as shown in the figure.
Water flows into the tank at a constant rate of 10 cubic ft./minute. Let
denote
the height of the water in the tank at a given time
. Find the rate of
change of
with respect to
. What is this rate when the height of the water is
?
What can you say about
when
is nearly
zero?
13.18
Exercise.
A particle
p moves on the rim of a
wheel
of radius 1 that rotates about the origin at constant angular speed
, so that
at time
it is at the point
. A light at
the
origin causes
p to cast a shadow at the
point
on a wall two feet from the center of the wheel.
Find the rate of
change of
with respect to time. You should ignore the speed of light, i.e.
ignore the time it takes light to travel from the origin to the wall.
Next: 14. The Inverse Function
Up: 13. Applications
Previous: 13.2 Optimization Problems.
Index
Ray Mayer
20070907