Question

Find the approximate value of \(\sqrt[3]{63}\).

Hint:

To divide by 48 quickly: \(1/48\) is slightly more than \(1/50 = 0.02\). Since \(48<50\), the value is slightly larger than \(0.02\), approximately \(0.021\). This estimation helps in rejecting distant options.

Answer 1

Step 1: Understanding the Concept:
We use differentials to find the approximate value. For a function \(y = f(x)\), the value at \(x + \Delta x\) is \(f(x + \Delta x) \approx f(x) + f'(x) \cdot \Delta x\).

Step 2: Key Formula or Approach:
Let \(f(x) = x^{1/3}\).
We choose \(x = 64\) (nearest perfect cube) and \(\Delta x = -1\).

Step 3: Detailed Explanation:
1. Find \(f(x)\):
\(f(64) = (64)^{1/3} = 4\).
2. Find the derivative \(f'(x)\):
\(f'(x) = \frac{1}{3} x^{-2/3} = \frac{1}{3 \cdot (x^{1/3})^2}\).
3. Evaluate \(f'(64)\):
\(f'(64) = \frac{1}{3 \cdot (4)^2} = \frac{1}{3 \cdot 16} = \frac{1}{48}\).
4. Calculate the approximation:
\(f(63) \approx f(64) + f'(64) \cdot (-1)\).
\(f(63) \approx 4 - \frac{1}{48}\).
Since \(1/48 \approx 0.020833...\):
\(f(63) \approx 4 - 0.0208 = 3.9792\).

Step 4: Final Answer:
The approximate value is \(3.9792\).