use linear approximation to estimate (8.06)^(2/3)

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justaguide | College Teacher | (Level 2) Distinguished Educator

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Consider the function `f(x) = x^(2/3)` . When x is close to a value a, `f(x) ~~ f(a) + f'(a)*(x - a)`

For `f(x) = x^(2/3)` and a = 8

`f(x) ~~ f(8) + (2/3)*x^(-1/3)*(x - 8)`

If x = 8.06

`8.06^(2/3) ~~ 8^(2/3) + (2/3)*8^(-1/3)*(8.06 - 8)`

`~~ 4 + (2/3)*(1/2)*(0.06)`

`~~ 4 + 0.02`

`~~ 4.02`

The approximate value of `8.06^(2/3)` = 4.02

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