Answer
$\sqrt[3] {1001} \approx 10.0033$
Work Step by Step
Let $y = \sqrt[3] x$
$\frac{dy}{dx} = \frac{1}{3}x^{-2/3}$
$\frac{dy}{dx} = \frac{1}{3~x^{2/3}}$
$dy = \frac{1}{3~x^{2/3}}~dx$
Let $x = 1000$ and let $dx = 1$
$dy = \frac{1}{3~(1000^{2/3})}~(1)$
$dy = \frac{1}{300}$
We can find an approximation for $\sqrt[3] {1001}$:
$\sqrt[3] {1001} \approx \sqrt[3] {1000}+\frac{1}{300}$
$\sqrt[3] {1001} \approx 10+\frac{1}{300}$
$\sqrt[3] {1001} \approx 10.0033$
