Question

Use a linear approximation to estimate the number (125.07)2/3

Answer 1

Let
`f(x)=x^(2/3)`. You have
`f(125)=(125)^(2/3)=(5^3)^(2/3)=5^2=25`.

Taking the derivative, you get


`f'(x)=\frac2{3x^(1/3)}\implies f'(125)=\frac2{3(125)^(1/3)}=\frac2{3*5}=\frac2{15}`

The linear approximation to
`f(x)` near
`x=125` is given by


`L(x)=f(125)+f'(125)(x-125)\approx f(x)`

So the value of
`(125.07)^(2/3)` can be estimated as


`L(125.07)=f(125)+f'(125)(125.07-125)`

`L(125.07)=25+\frac2{15}(0.07)=25.009333\ldots`