1
RATES OF CHANGE
AVERAGE AND INSTANTANEOUS RATES OF CHANGE
We have already considered the average rate of change back in another set of notes,
but we review the definition anyway.
DEFINITION:
The average rate of change of a function
f
(
x
) with respect to
x
over the interval from
x
0
to
x
0
+ h is
Now here is the definition of the instantaneous rate of change.
Note how it contrasts
with the average rate of change definition.
DEFINITION:
The instantaneous rate of change of f with respect to
x
at
x
0
is the
derivative
provided the limit exists.
Sometimes this derivative is called the rate of change. If the word “instantaneous”
does not appear before “rate of change” you must assume it is there.
The word “average”
is
always
expressed,
never implied.
The
instantaneous rate of chance in
f
(
x
),
namely
f '
(x), is the
instantaneous rate of change in
f
(
x
) per unit increase in
x
.
EXAMPLE 1:
The volume
V
= (4/3)
π
r
3
of a spherical balloon changes with the
radius. At what rate does the volume change with respect to the
radius when
r
= 2 ft.?
SOLUTION:
Here we are concerned with the instantaneous rate of change, so let us find
the derivative
dV
dr
.
dV
dr
=
4
3
3
r
2
=
4
r
2
When
r
= 2 ft., the volume is changing at a rate of 16
π
ft
3
/ft. This means that a small
change in
∆
r
ft in the radius would result in a change of (16
π
)(
∆
r
) cubic feet in the
volume of the sphere.
MOTION ALONG A LINE  DISPLACEMENT, VELOCITY, SPEED, AND
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 Spring '08
 Noohi
 Derivative, Rate Of Change, Velocity, instantaneous rate, Gertrude, Aventura Mall

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