Linearization

Tangent lines can be used to approximate functions. Given a function that is not currently linear, it can be approximated with: $L(x)=f(a)+f^{\prime }(a)(x-a)$ This is pretty much the Tangent Observe that when x is close to a, $f(x)$ is close to the $L(x)$

Example

If you wanted to linear approximate at $f(8.1)$, then you can get the linear approximation at $L(8)$ to approximate $L(8.1)$

1. Find a point close to 8 - say 8.1
2. Get the tangent equation for points close to 8. This is $L(x)=f^{\prime }(8)(x-8)+f(8)$
3. $f(8.1)\approx L(8.1)$
4. Judge approximation is over or under by checking concavity at 8.1.
   1. Concave up → approximation under
   2. Concave down → approximation over