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1
LINEARIZATION AND DIFFERENTIALS
LINEARIZATION
Consider the function
f
(
x
) =
x
2
+ 4. Determine its tangent line at the point (1, 5).
f '
(x) = 2
x
→
m =
f '
(1) = 2(1) = 2
y
 5 = 2(
x
 1)
→
y
 5 = 2
x
 2
→
y
= 2
x
+ 3
Now, let us look at the graph of the
function and its tangent line. Notice
that as you get close to the point (1, 5),
the function
f '
(x)
= x
2
+ 4 starts to
look like
y
= 2
x
+ 3. In fact, we can use
the tangent line to approximate a value
of a function at a point. Thus
f '
(x)
≈
2
x
+ 3 for
x
near 1.
y = x
2
+ 4
y
= 2
x
+ 3
DEFINITION:
If
f
is differentiable at
x
= a, then the approximating function
L
(
x
) =
f
(a) +
f '
(a)(x  a) is the linearization of
f
at a.
EXAMPLE 1:
Find the linearization of
f
(
x
) =
x
 1
at
x
= 2.
SOLUTION:
First evaluate the function at
x
= 2, and then find the first derivative and
evaluate it at
x
= 2.
Now substitute the information into the formula for the linearization.
For values of
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 Spring '08
 Noohi
 Approximation

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