# Differential Equations

# Tangent Line Approximations (Again)

Tangent line approximation can also be called **local linearization**, **linear approximation**, and probably a bunch of other names. The important thing is that you're *using a line to approximate a curve*.

Here's a reminder of how tangent line approximation works:

Suppose we know the value of *f* at a particular value of *x*:

and we want to know the value of *f* at a nearby value of *x*:

We draw the tangent line to *f* at the point we know:

Then we find the value on the tangent line at the nearby value of *x*:

This tangent line value is close to the value we actually wanted.

Using the language from the last section, let's talk this through again.

We know y_{old}:

We want to know the value of *f* if we change *x* by Δ *x*:

We draw the tangent line to *f* at the point we know:

and we use the tangent line to find y_{new}, which is close to the value we actually wanted.

### Sample Problem

Another kind of problem you may run into is the kind where you're given a formula for the function *f* and asked to use a tangent line to approximate *f* at some particular value *x*^{*}.

To do this, pick some *x* = a close to *x*^{*}. Choose a so that *f* (*a*) and *f* '(*a*) are easy to calculate. Then we have y_{old} = *f* (*a*), Δ *x* = *x*^{*} - *a*, and slope = *f* '(*a*).

From there you know how to calculate y_{new}, which is the approximation you want.