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

$f'(x)=\dfrac2{3x^{1/3}}\implies f'(125)=\dfrac2{3(125)^{1/3}}=\dfrac2{3\times5}=\dfrac2{15}$

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

$L(125.07)=f(125)+f'(125)(125.07-125)$
$L(125.07)=25+\dfrac2{15}(0.07)=25.009333\ldots$