Chapter 3 - Section 3.10 - Linear Approximations and Differentials. - 3.10 Exercises - Page 256: 25

$\sqrt[3] {1001} \approx 10.0033$

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 $\sqrt[3] {1001}$: $\sqrt[3] {1001} \approx \sqrt[3] {1000}+\frac{1}{300}$ $\sqrt[3] {1001} \approx 10+\frac{1}{300}$ $\sqrt[3] {1001} \approx 10.0033$