Problem

Find the linear approximation of the function

about a = 0. Use it to approximate the square root of 0.9.

Solution

We need the derivative of f(x):

The linear approximation is the formula

so when a equals 0,

Therefore we approximate square root of 0.9 as

Looking at the graph of the function and its linear approximation tells us a lot. The closer the value around which the approximation is made is to the desired value, the better the approximation will be. In this problem, choosing a = 0 made it easy to find L(x) but gives a fairly large error from the actual value of f(0.9). However, we can't choose a = 1, because the derivative of f is not defined there.