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 we know how to calculate y_new, which is the approximation we want.