What is the linear approximation of the cube root of 8.5?

Question

How do I do linear approximation?

Answer 1

If you're doing this for a calculus class, then this will work. The linear approximation of a function is the tangent line to the function. In your case, you're trying to find the linear approximation at a point on the function f(x) = cuberoot(x). For a general function, f(x), the formula for the linear approximation is:



f(a)+f'(a)(x-a). The closer x is to a, the better the approximation is. Using f(x) = cuberoot(x), and a = 8, you will get a decent approximation of cuberoot(8.5). I chose a = 8 because the cube root of 8 is 2, which you can do in your head.



So: f(a) = f(8) = 2. f'(x)= 1/3(x)^(-2/3), just using the power rule for derivatives. Therefore f'(a) = f'(8) = 1/3(8)^(-2/3) = 1/12.



Since you're approximating the cube root of 8.5, now plug in those numbers and x = 8.5 into the above formula for the linear approximation.



approximation = f(8) + f'(8)(8.5 - 8) = 2 + (1/12)(.5) =2.04166666666666.



The actual value of cuberoot(8.5) is roughly 2.040827551, so the approximation is pretty close. The approximation is slightly above the real value, which should make sense because the tangent line to the curve at x = 8 is above the curve itself.

Answer 2

Linear approximation just mean you assume that something changes at a constant rate.



Like this:

cube root of 8 is 2.

cube root of 27 is 3.



The cube root of 8.5 is between these.

8.5 is 0.5 more than 8, and the whole distance from 8 to 27 is 19.

So then you calculate that 0.5 / 19 = 2.6% (about).



Then you assume that the cube root of 8.5 is a bit more than 2, by 2.6% of the distance from 2 to 3. The distance is 1, so the cube root approximation is 2 + 0.026 = 2.026.



The real cube root is 2.0408..., so this approximation is not very good.



Of course, you could use other numbers instead of 8 and 27. For example, 8 and 9.261. The cube roots are 2 and 2.1.

The linear approximation this time is 2.0397..., which is much better.

Answer 3

You're going to take two values for which you know the cube root which lie on either side of 8.5. The closer the numbers are to 8.5, the better. In this case, the numbers 8 and 27 are good candidates. Now you're going to calculate the ratio of the distance between 8 and 8.5 to the distance from 8 to 27. In other words, figure out what percentage of the distance from 8 to 27 is 8.5. Now add the same percentage of the distance from 2 to 3 (The cube root of your two endpoints) to 2. This is also termed, "linear interpolation"