Use a linear approximation to estimate (8.2)^(2/3)

Answer 1

in general
f(x) = f(x₀) + f ' (x₀)*(x-x₀),
where x<sub>0 is a value close to x and a value which we can calculate.

let f(x) = x^(2/3)
f ' (x) = (2/3)x^(-1/3) = 2/(3x^(1/3))

f(8) = 8^(2/3) = 4
f ' (8) = 2/(3(2)) = 1/3

f(8.2) = f(8) + f ' (8)*(8.2 - 8)
= 4 + (1/3)(.2)
= 4 + .2/3
= 4 + 2/30
= 122/30 or 61/15

using my calculator:
8.2^(2/3) = 4.066391931
my appr of 61/15 = 4.066666..
for an error of .000274.. , not bad</sub>