Linear Approximations

2
Let’s start with the function
x
y
at
x
= 4 because we
know what
is
4
Mysterious radiation has caused all calculators to stop
working.
It just so happens that at this time, you need to
find out the square root of 5.
The best you are going to be
able to do now is approximate it.
But how?
First of all, what is the tangent line at
x
= 4?
2
)
4
(
4
1
x
y
Now graph both the function and its tangent line…
When we make a graph of this process, you will
get a better sense of why we are doing this.

3
2
)
4
(
4
1
x
y
Notice that for
numbers close to
4, the tangent line
is very close to the
curve itself…
So lets try plugging
5 into the tangent
line equation and
see what we get…
Use the tangent line to the function
x
y
at
x
= 4
to approximate
5

4
2
)
4
(
4
1
x
y
So let’s plug 5 into
the tangent line to
get…
As it turns out…
(4, 2)
(5, 2.25)
)
5
,
5
(
25
.
2
2
)
4
5
(
4
1
y
2.23607
5
2.23607)
(5,
Notice how close the point on the line is to the curve…
So in this case, our approximation
will be a very good one as long as we
use a number close to 4.
Use the tangent line to the function
x
y
at
x
= 4
to approximate
5

5
For any function
f
(
x
),
the tangent is a close approximation of the function for
some small distance from the tangent point.
y
x
0
x
a
f
x
f
a
We call the equation of the tangent
the linearization
of the function.

6
The linearization is the equation of the tangent line, and you can use the old
formulas if you like.
Start with the point/slope equation:
1
1
y
y
m x
x
1
x
a
1
y
f
a
m
f
a
y
f
a
f
a
x
a
y
f
a
f
a
x
a
L x
f
a
f
a
x
a
linearization of
f
at
a
f
x
L x
is the standard linear approximation of
f
at
a.

7
Remember:
The linearization is just the equation of the tangent line.
The use of the term
L
(
x
) is to make it known that you are using the
tangent line to make a linear approximation of the function in
question.