Question

The approximate value of $\sqrt[3]{730}$ obtained by the application of derivatives is:

• A. $9.0041$
• B. $9.01$
• C. $9.006$
• D. $9.05$

Hint:

Use linear approximation for small changes in $x$. - The derivative helps approximate functions near known values.

Answer 1

The Correct Option is A

Step 1: Use the approximation formula
Using the approximation formula: $$f(a + h) \approx f(a) + h f'(a),$$ where $f(x) = \sqrt[3]{x} = x^{\frac{1}{3}}$. Step 2: Compute the derivative
$$f'(x) = \frac{1}{3} x^{-\frac{2}{3}}.$$ Choosing $a = 729$, since $\sqrt[3]{729} = 9$, and $h = 1$: $$f'(729) = \frac{1}{3} (729)^{-\frac{2}{3}} = \frac{1}{3} \times \frac{1}{81} = \frac{1}{243}.$$ Step 3: Compute approximation
$$f(730) \approx f(729) + 1 \times f'(729).$$ $$= 9 + \frac{1}{243} \approx 9.0041.$$ Thus, the correct answer is $\boxed{9.0041}$.