##
Linear Approximations

This approximation is crucial to many known numerical techniques
such as Euler's Method to approximate solutions to ordinary
differential equations. The idea to use linear approximations
rests in the closeness of the tangent line to the graph of the
function around a point.

Let *x*_{0} be in the domain of the function *f*(*x*). The equation
of the tangent line to the graph of *f*(*x*) at the point
(*x*_{0},*y*_{0}), where
*y*_{0} = *f*(*x*_{0}), is

If *x*_{1} is close to *x*_{0}, we will write
,
and we will approximate
by the point
(*x*_{1},*y*_{1}) on the tangent line given by

If we write
,
we have

In fact, one way to remember this formula is to write *f*'(*x*) as
and then replace *d* by .
Recall that, when *x* is close to *x*_{0}, we have

**Example.** Estimate
.

Let
.
We have
.
Using the
above approximation, we get

We have

So
.
Hence

or
.
Check with your calculator and
you'll see that this is a pretty good approximation for
.
**Remark.** For a function *f*(*x*), we define the **differential** *df* of *f*(*x*) by

**Example.** Consider the function
*y* = *f*(*x*) = 5*x*^{2}. Let
be an increment of *x*. Then, if
is the
resulting increment of *y*, we have

On the other hand, we obtain for the differential *dy*:

In this example we are lucky in that we are able to compute
exactly, but in general this might be impossible. The
error in the approximation, the difference between *dy*(replacing *dx* by )
and ,
is
,
which is small compared to .
**Exercise 1.** Use linear approximation to approximate

**Answer.**

**Exercise 2.** Use linear approximation to approximate

**Answer.**

**
**

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