Question

Find the approximate values of : `root(3)(28)`

Answer 1

Let f(x) = `root(3)(x)`

Then f'(x) = `d/dx(x^(1/3)) = (1)/(3)x^(-(2)/(3)) = (1)/(3x^(2/3)`

Take a = 27 and h = 1.

Then f(a) = f(27) = `root(3)(27)` = 3

and f'(a) = f'(27) = `(1)/(3(27)^(2/3)) = (1)/(3 xx 9) = (1)/(27)` = 0.03704

The formula for approximation is
f(a + h) ≑ f(a) + h.f'(a)
∴ `root(3)(28)`
= f(28) = f(27 + 1)
≑ f(27) – 1.f'(27) 
≑ 3 + 1 x 0.03704
= 3.03704
∴ `root(3)(28)` ≑ 3.03704.