Problem: Use second degree Taylor polynomial to estimate cos(0.5).
Estimate the error of this approximation.

Solution: Before we find the suitable Taylor polynomial for the function f (x) = cos(x), we have to decide where its center should be. We need a number a that is close to the given point 0.5 and is "nice" with respect to cosine. The obvious candidate is a = 0. Thus we will find the Taylor polynomial of degree 2 with this center. The general formula is

We first find the necessary derivatives and then substitute the center a into them.

We substitute into the formula for T₂.

Now we use this to estimate cos(0.5).

It remains to estimate the error of this approximation. For this we use the Lagrange form of the remainder.

Thus we can say that

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