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 \( \boxed{9.0041} \).