Using the method of differentials, find the approximate value of `sqrt(0.24)`.

Answer

To find the approximate value of \(\sqrt{0.24}\) using the method of differentials, we can follow these steps:

### Step 1: Define the function
Let \( f(x) = \sqrt{x} \).

### Step 2: Differentiate the function
The derivative of \( f(x) \) is given by:
\[ f'(x) = \frac{1}{2\sqrt{x}} \]

### Step 3: Identify the point of approximation
We need to approximate \(\sqrt{0.24}\). The nearest perfect square to \(0.24\) is \(0.25\) (since \(0.25 = 0.5^2\)). Thus, we take:
\[ x = 0.25 \]

### Step 4: Determine \(\Delta x\)
We can express \(0.24\) in terms of \(0.25\):
\[ \Delta x = 0.24 - 0.25 = -0.01 \]

### Step 5: Apply the differential approximation formula
Using the formula:
\[ f(x + \Delta x) \approx f(x) + f'(x) \Delta x \]
we substitute \(x = 0.25\) and \(\Delta x = -0.01\):
\[ f(0.24) \approx f(0.25) + f'(0.25)(-0.01) \]

### Step 6: Calculate \(f(0.25)\)
We know:
\[ f(0.25) = \sqrt{0.25} = 0.5 \]

### Step 7: Calculate \(f'(0.25)\)
Now, we calculate \(f'(0.25)\):
\[ f'(0.25) = \frac{1}{2\sqrt{0.25}} = \frac{1}{2 \times 0.5} = 1 \]

### Step 8: Substitute values into the approximation formula
Now substituting back into the approximation formula:
\[ f(0.24) \approx 0.5 + (1)(-0.01) = 0.5 - 0.01 = 0.49 \]

### Conclusion
Thus, the approximate value of \(\sqrt{0.24}\) is:
\[ \sqrt{0.24} \approx 0.49 \]