Linearization's are a method to approximate a function using a tangent line at an easily calculable point.

From the graph above you can see that is a very good approximation when values of are close to . As becomes farther away from then the approximation begins to have increased errors.

By the definition of concavity, if is concave up then will underestimate, alternatively if is concave down will overestimate. This is because at its core linearization is just a tangent line at a point.

In the previous example is concave up and the linearization underestimates the true value of the function.

Calculating

The function can be approximated by calculating a tangent line at the point where the linearization is being built.

Example

Given the function create a linearization () at and then use this linearization to estimate at