Question:
Approximate value of \( \sqrt[3]{345} \), when it is calculated with the application of derivatives, is
Approximate value of \( \sqrt[3]{345} \), when it is calculated with the application of derivatives, is
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Always pick the closest known integer power point (\( 7^3 = 343 \)) as your reference base. Keeping your increment \( \Delta x \) small ensures the accuracy of your first-order differential approximation.
Updated On: Jun 8, 2026
- \( 7.013 \)
- \( 7.025 \)
- \( 7.001 \)
- \( 7.003 \)
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The Correct Option is A
Solution and Explanation
Concept:
To find the approximate value using differentials, we use the first-order Taylor approximation formula:
\[
f(x + \Delta x) \approx f(x) + f'(x) \cdot \Delta x
\]
We choose a perfect cube number close to 345 for our base point \( x \). The closest perfect cube is \( 343 = 7^3 \).
Step 1: Defining function values and parameters.
Let \( f(x) = x^{1/3} \).
• Base point \( x = 343 \implies f(343) = 343^{1/3} = 7 \)
• Shifting increment \( \Delta x = 345 - 343 = 2 \)
Step 2: Evaluating the derivative at the base point.
Find the general derivative of the function: \[ f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3(x^{1/3})^2} \] Substitute \( x = 343 \): \[ f'(343) = \frac{1}{3(7)^2} = \frac{1}{3 \times 49} = \frac{1}{147} \]
Step 3: Computing the approximate value.
\[ f(345) \approx f(343) + f'(343) \cdot \Delta x = 7 + \frac{1}{147} \times 2 = 7 + \frac{2}{147} \] Performing the long division for the fractional term: \[ \frac{2}{147} \approx 0.0136 \] Adding this to our base integer: \[ 7 + 0.0136 = 7.0136 \approx 7.013 \] This matches option (A) perfectly.
Step 1: Defining function values and parameters.
Let \( f(x) = x^{1/3} \).
• Base point \( x = 343 \implies f(343) = 343^{1/3} = 7 \)
• Shifting increment \( \Delta x = 345 - 343 = 2 \)
Step 2: Evaluating the derivative at the base point.
Find the general derivative of the function: \[ f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3(x^{1/3})^2} \] Substitute \( x = 343 \): \[ f'(343) = \frac{1}{3(7)^2} = \frac{1}{3 \times 49} = \frac{1}{147} \]
Step 3: Computing the approximate value.
\[ f(345) \approx f(343) + f'(343) \cdot \Delta x = 7 + \frac{1}{147} \times 2 = 7 + \frac{2}{147} \] Performing the long division for the fractional term: \[ \frac{2}{147} \approx 0.0136 \] Adding this to our base integer: \[ 7 + 0.0136 = 7.0136 \approx 7.013 \] This matches option (A) perfectly.
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