2
Instantaneous Rate of Change:
The Derivative
Suppose that
y
is a function of
x
, say
y
=
f
(
x
). It is often necessary to know how sensitive
the value of
y
is to small changes in
x
.
EXAMPLE 2.1
Take, for example,
y
=
f
(
x
) =
√
625
−
x
2
(the upper semicircle of
radius 25 centered at the origin). When
x
= 7, we find that
y
=
√
625
−
49 = 24. Suppose
we want to know how much
y
changes when
x
increases a little, say to 7.1 or 7.01.
In the case of a straight line
y
=
mx
+
b
, the slope
m
= Δ
y/
Δ
x
measures the change in
y
per unit change in
x
. This can be interpreted as a measure of “sensitivity”; for example,
if
y
= 100
x
+ 5, a small change in
x
corresponds to a change one hundred times as large
in
y
, so
y
is quite sensitive to changes in
x
.
Let us look at the same ratio Δ
y/
Δ
x
for our function
y
=
f
(
x
) =
√
625
−
x
2
when
x
changes from 7 to 7
.
1. Here Δ
x
= 7
.
1
−
7 = 0
.
1 is the change in
x
, and
Δ
y
=
f
(
x
+ Δ
x
)
−
f
(
x
) =
f
(7
.
1)
−
f
(7)
=
radicalbig
625
−
7
.
1
2
−
radicalbig
625
−
7
2
≈
23
.
9706
−
24 =
−
0
.
0294
.
Thus, Δ
y/
Δ
x
≈ −
0
.
0294
/
0
.
1 =
−
0
.
294.
This means that
y
changes by less than one
third the change in
x
, so apparently
y
is not very sensitive to changes in
x
at
x
= 7.
We say “apparently” here because we don’t really know what happens between 7 and 7
.
1.
Perhaps
y
changes dramatically as
x
runs through the values from 7 to 7
.
1, but at 7
.
1
y
19
20
Chapter 2 Instantaneous Rate of Change: The Derivative
just happens to be close to its value at 7. This is not in fact the case for this particular
function, but we don’t yet know why.
One way to interpret the above calculation is by reference to a line. We have computed
the slope of the line through (7
,
24) and (7
.
1
,
23
.
9706), called a
chord
of the circle.
In
general, if we draw the chord from the point (7
,
24) to a nearby point on the semicircle
(7 + Δ
x, f
(7 + Δ
x
)), the slope of this chord is the socalled
difference quotient
slope of chord =
f
(7 + Δ
x
)
−
f
(7)
Δ
x
=
radicalbig
625
−
(7 + Δ
x
)
2
−
24
Δ
x
.
For example, if
x
changes only from 7 to 7.01, then the difference quotient (slope of the
chord) is approximately equal to (23
.
997081
−
24)
/
0
.
01 =
−
0
.
2919. This is slightly less
steep than the chord from (7
,
24) to (7
.
1
,
23
.
9706).
As the second value 7 + Δ
x
moves in towards 7, the chord joining (7
, f
(7)) to (7 +
Δ
x, f
(7 + Δ
x
)) shifts slightly. As indicated in figure 2.1, as Δ
x
gets smaller and smaller,
the chord joining (7
,
24) to (7 + Δ
x, f
(7 + Δ
x
)) gets closer and closer to the
tangent line
to the circle at the point (7
,
24). (Recall that the tangent line is the line that just grazes
the circle at that point, i.e., it doesn’t meet the circle at any second point.) Thus, as Δ
x
gets smaller and smaller, the slope Δ
y/
Δ
x
of the chord gets closer and closer to the slope
of the tangent line. This is actually quite difficult to see when Δ
x
is small, because of the
scale of the graph. The values of Δ
x
used for the figure are 1, 5, 10 and 15, not really very
small values. The tangent line is the one that is uppermost at the right hand endpoint.
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 Fall '08
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 Calculus, Derivative, Rate Of Change, lim, ... ...

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