Video: Finding the Linear Approximation of a Root Function

Find the linear approximation of the function 𝑓(π‘₯) = √(π‘₯) at π‘₯ = 4.

02:03

Video Transcript

Find the linear approximation of the function 𝑓 of π‘₯ equals the square root of π‘₯ at π‘₯ equals four.

Linear approximation, sometimes called the tangent line approximation, is a way of approximating values of 𝑓 of π‘₯ as long as we stay fairly near a given number π‘₯ equals π‘Ž. The formula we use is 𝑙 of π‘₯ equals 𝑓 of π‘Ž plus 𝑓 prime of π‘Ž times π‘₯ minus π‘Ž. In our question, 𝑓 of π‘₯ is equal to the square root of π‘₯. So we’ll begin by working out what 𝑓 prime of π‘₯ is, so we can use that to evaluate 𝑓 prime of π‘Ž.

Now remember, 𝑓 prime of π‘₯ is the first derivative of the function with respect to π‘₯. It’s sensible to write the square root of π‘₯ as π‘₯ to the power of a half before finding its derivative. Then we use the general result that the derivative of π‘₯ to the power of 𝑛 for real nonzero constants 𝑛 is 𝑛 times π‘₯ to the power of 𝑛 minus one. And we see that 𝑓 prime of π‘₯ is a half times π‘₯ to the power of a half minus one. That’s a half times π‘₯ to the power of negative one-half.

We’re trying to work out the linear approximation of our function at the point where π‘₯ is equal to four. This is the value of π‘Ž in our equation, so we let π‘Ž be equal to four. Then we see that 𝑓 of π‘Ž is 𝑓 of four and 𝑓 prime of π‘Ž 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 π‘Ž is two and 𝑓 prime of π‘Ž is a quarter. Then π‘₯ minus π‘Ž is π‘₯ minus four. We distribute our parentheses. And we get two plus a quarter π‘₯ minus a quarter times four which is two plus a quarter π‘₯ minus one. Simplifying and we obtain the tangent line approximation to be equal to a quarter π‘₯ plus one.

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