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 = 28 = 27 + 1 = a + h

Here, 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 × 0.03704

= 3.03704

∴ `root(3)(28)` ≑ 3.03704