Use differential to approximate `(66)^(1//3)`.

Answer

To approximate \( 66^{1/3} \) using differentials, we can follow these steps:

### Step 1: Define the function
Let \( f(x) = x^{1/3} \). We want to find \( f(66) \).

### Step 2: Identify a nearby point
We need to find a value of \( x \) close to 66 for which we know the cube root. The nearest perfect cube is 64, since \( 4^3 = 64 \). Thus, we can take \( x = 64 \).

### Step 3: Calculate \( h \)
We define \( h \) as the difference between our target value and the nearby value:
\[ h = 66 - 64 = 2 \]

### Step 4: Compute the derivative
Next, we find the derivative \( f'(x) \):
\[ f'(x) = \frac{1}{3} x^{-2/3} = \frac{1}{3} \cdot \frac{1}{x^{2/3}} \]

### Step 5: Evaluate the derivative at the nearby point
Now we evaluate the derivative at \( x = 64 \):
\[ f'(64) = \frac{1}{3} \cdot \frac{1}{64^{2/3}} \]
Since \( 64^{1/3} = 4 \), we have:
\[ 64^{2/3} = (64^{1/3})^2 = 4^2 = 16 \]
Thus,
\[ f'(64) = \frac{1}{3} \cdot \frac{1}{16} = \frac{1}{48} \]

### Step 6: Use the differential approximation
Using the differential approximation formula:
\[ f(66) \approx f(64) + f'(64) \cdot h \]
We know \( f(64) = 4 \) and \( h = 2 \):
\[ f(66) \approx 4 + \left(\frac{1}{48}\right) \cdot 2 \]
Calculating this gives:
\[ f(66) \approx 4 + \frac{2}{48} = 4 + \frac{1}{24} \]

### Step 7: Convert \( \frac{1}{24} \) to decimal
Calculating \( \frac{1}{24} \) gives approximately \( 0.04167 \):
\[ f(66) \approx 4 + 0.04167 = 4.04167 \]

### Final Result
Thus, the approximate value of \( 66^{1/3} \) is:
\[ \boxed{4.04167} \]