Let’s find the Taylor polynomial for

First, take the derivatives of

Now, form the Taylor series about c = 4

This is equal to 2.02484567314. The calculator value is 2.02484567313, giving a difference of 1x10-11.

The predicted error is given by

.

This gives the estimate of  if we let z = 4.

We choose z = 4, because this is the value of z in the interval [4,4.1] which gives the largest error estimate.  Thus, the Taylor remainder exceeds the actual error, as it should.