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Question
Find the approximate values of : `root(5)(31.98)`
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Solution
Let f(x) = `root(5)(x)`
Then f'(x) = `d/dx(x^(1/5))`
= `(1)/(5)x^(-4/5)`
= `(1)/(5x^(4/5)`
Take a = 2 and h = – 0.02.
Then f(a) = f(32) = `root(5)(32)` = 2
f'(a) = f'(32) = `(1)/(5(32)^(4/5)`
= `(1)/(5 xx 16)`
= `(1)/(80)`
= 0.0125
The formula for approximation is
f(a + h) ≑ f(a) + h.f'(a)
∴ `root(5)(31.98)`
= f(31.98)
= f(32 – 0.02)
≑ f(32 – 0.02.f'(32)
≑ 2 – 0.02 x 0.0125
≑ 2 – 0.000250
= 1.99975
∴ `root(5)(31.98)` ≑ 1.99975.
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