# Tangent Lines as Approximation

### From www.norsemathology.org

## Contents |

## Tangent Lines 101

There are certainly two things that you should know about tangent lines:

- They're graphs of linear functions, and
- They do the best job of approximating a function at the point of tangency of any other linear function.

Tangent lines are generalized by Taylor Series as Approximations.

## The Math

If we want the tangent line at the point (*a*,*f*(*a*)), it's the graph of

This makes perfect sense, showing that *L*(*x*) matches *f*(*x*) in terms of both the function values and derivatives at *x* = *a*:

## Application: Newton's Method

One of the most important applications of the linearization is in root-finding: that is, finding zeros of a non-linear function.

Given function , and a guess for a root . We would first check to see if . If it is, we're done; otherwise, we might try to **improve** the guess. How so?

Use the linearization , and find a zero of it:

This is Newton's method, an iterative scheme for improving the approximation of a root: we need a good approximation for the root, , and then

Then we compute

and so on, computing a sequence which we hope will converge on the true root .

### For example:

- Let ; then
- Guess: (pretty bad guess!)
- Improved guess:
- Now do it again! After another go,
- Once more:
- Once more:
- Once more: .

We check, and find that -- egads, we've found one! We've found a root:

It's like a miracle... but it's just mathematics!