Linear approximation

Question

Hello, I have correctly solved part a of the following question:

a) find linear approx of (1+5x)^1/7 at x=2 and b)then use it to approx (11.005)^1/7

I found that the linear approx is 0.0914*x+1.22but I do not know how to approx (11.005)^1/7

Please assist me if you can!

Answer 1

Consider asking yourself the question "for what value of x does 1+5x = 11.005"?

Realize that f(2) = [1 + 5(2)]^1/7 or (11)^1/7.

And f(2.001) = [1+ 5(2.001)]^1/7 = 11.005^1/7

So, if your linearization is L(x) = .0914x + 1.22 (you might not have rounded off quite correctly there), then L(2.001) should give a fair approximation for f(11.005).

Answer 2

All you have to do now is plug in (11.005)^(1/7) into the equation you found in part A.