Math 1a
Linear Approximations
Fall, 2009
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$
%
Linear Approximation
A linear approximation is
dy
dx
≈
Δ
y
Δ
x
. If
y
=
f
(
x
), then this can be rewritten as
f
0
(
a
)
≈
Δ
y
Δ
x
=
y

f
(
a
)
x

a
or
y
≈
f
(
a
) +
f
0
(
a
)(
x

a
)
near the point (
x,y
) = (
a,f
(
a
)). This is the tangent line:
y
=
f
(
a
) +
f
0
(
a
)(
x

a
).
Thus the linear approximation is: “near
x
=
a
,
y
is approximately the tangent line
to the curve at
x
=
a
.”
1
(a) Find the equation of the tangent line to
y
= ln(
x
) at
x
= 1.
(b) Use the equation you found in part (a) to estimate ln(1
.
1).
(c) Write down a linear estimate ln(1 +
x
)
≈
L
(
x
) for small values of
x
.
2
(a) Estimate
e

0
.
2
using the linear approximation near
x
= 0.
(b) Give the full linear approximation for
e
x
near zero.
3
(a) Write down the linear approximation for
√
x
near
x
= 9.
(b) Estimate
√
8
.
94 and
√
9
.
12.
(c) Can you tell whether the estimates you obtained in parts (a) and (b) are too high or too
low?
Hint:
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 Fall '09
 BenedictGross
 Approximation, Linear Approximation

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