Find the linear approximation of the function 𝑓 of 𝑥 equals the square root of 𝑥 at 𝑥 equals four.
Linear approximation, sometimes called the tangent line approximation, is a way of approximating values of 𝑓 of 𝑥 as long as we stay fairly near a given number 𝑥 equals 𝑎. The formula we use is 𝑙 of 𝑥 equals 𝑓 of 𝑎 plus 𝑓 prime of 𝑎 times 𝑥 minus 𝑎. In our question, 𝑓 of 𝑥 is equal to the square root of 𝑥. So we’ll begin by working out what 𝑓 prime of 𝑥 is, so we can use that to evaluate 𝑓 prime of 𝑎.
Now remember, 𝑓 prime of 𝑥 is the first derivative of the function with respect to 𝑥. It’s sensible to write the square root of 𝑥 as 𝑥 to the power of a half before finding its derivative. Then we use the general result that the derivative of 𝑥 to the power of 𝑛 for real nonzero constants 𝑛 is 𝑛 times 𝑥 to the power of 𝑛 minus one. And we see that 𝑓 prime of 𝑥 is a half times 𝑥 to the power of a half minus one. That’s a half times 𝑥 to the power of negative one-half.
We’re trying to work out the linear approximation of our function at the point where 𝑥 is equal to four. This is the value of 𝑎 in our equation, so we let 𝑎 be equal to four. Then we see that 𝑓 of 𝑎 is 𝑓 of four and 𝑓 prime of 𝑎 is 𝑓 prime of four. Of course, 𝑓 of four is just the square root of four, which is two. And 𝑓 prime of four is a half times four to the power of negative one-half, which is equivalent to saying a half times one over the square root of four. Now, the square of four is two, so a half times one over the square root of four is a quarter.
Let’s substitute everything we have into our formula for the tangent line approximation. 𝑓 of 𝑎 is two and 𝑓 prime of 𝑎 is a quarter. Then 𝑥 minus 𝑎 is 𝑥 minus four. We distribute our parentheses. And we get two plus a quarter 𝑥 minus a quarter times four which is two plus a quarter 𝑥 minus one. Simplifying and we obtain the tangent line approximation to be equal to a quarter 𝑥 plus one.