Math 16B – S06 – Supplementary Notes 1
The Derivative and Local Linear Approximation
For a function
g
(
x
) of one variable, the value
g
(
a
) of the derivative of
g
at the point
x
=
a
is
the slope of the tangent line to the graph of
g
at the point (
a, g
(
a
)). Near that point, the tangent
line is a “good” approximation to the graph, in the following sense. The equation of the tangent
line is
y
=
g
(
a
) +
g
(
a
)(
x

a
)
.
The difference
(1)
R
(
x
) =
g
(
x
)

g
(
a
)

g
(
a
)(
x

a
)
is the error you get at
x
when you approximate
g
by the function whose graph is the tangent line
(see Figure 1
.
1). The approximation is good near
a
in the sense that
R
(
x
) is small for
x
near
a
compared to
x

a
, vanishingly small in the limit:
(2)
lim
x
→∞
R
(
x
)
x

a
= 0
.
To see this, simply divide (1), the equality that defines
R
(
x
), by
x

a
, to get
R
(
x
)
x

a
=
g
(
x
)

g
(
x
)
x

a

g
(
a
)
.
As
x
approaches
a
, the difference quotient on the right side of the equality approaches
g
(
a
), so the
ratio on the left side tends to 0.
Among all straight lines through the point (
a, g
(
a
)), the tangent line gives the best linear approx
imation near
a
to the graph of
g
. For the line with slope
c
=
g
(
a
), the error in the approximation
at
x
, divided by
x

a
, has the limit
g
(
a
)

c
, not 0, as
x
approaches
a
.
The relation (2), while it says you can expect the error in a certain approximation to be small,
does not tell you how small.
It does not tell you, for example, how close
x
should be to
a
to
guarantee that the approximation is accurate to, say, three decimal places.
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 Fall '10
 pederson
 Derivative, pH, lim

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