In the following find the approximate values, using differentials :
`sqrt(0.0037).`

Answer

To find the approximate value of \(\sqrt{0.0037}\) using differentials, we can follow these steps:

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

### Step 2: Choose a point close to \(0.0037\)
We will choose \( x = 0.0036 \) because it is close to \( 0.0037 \) and its square root is easier to calculate.

### Step 3: Calculate \( \Delta x \)
We find \( \Delta x \):
\[ \Delta x = 0.0037 - 0.0036 = 0.0001 \]

### Step 4: Calculate the derivative
Next, we need to find the derivative of \( f(x) \):
\[ \frac{dy}{dx} = f'(x) = \frac{1}{2\sqrt{x}} \]

### Step 5: Evaluate the derivative at \( x = 0.0036 \)
Now, we will evaluate the derivative at \( x = 0.0036 \):
\[ f'(0.0036) = \frac{1}{2\sqrt{0.0036}} = \frac{1}{2 \times 0.06} = \frac{1}{0.12} = \frac{25}{3} \approx 8.3333 \]

### Step 6: Calculate \( \Delta y \)
Now we can find \( \Delta y \) using the formula:
\[ \Delta y = f'(x) \cdot \Delta x \]
Substituting the values we have:
\[ \Delta y = \frac{25}{3} \cdot 0.0001 = \frac{25 \times 0.0001}{3} = \frac{0.0025}{3} \approx 0.0008333 \]

### Step 7: Find the approximate value of \( \sqrt{0.0037} \)
Now we can approximate \( \sqrt{0.0037} \):
\[ \sqrt{0.0037} \approx \sqrt{0.0036} + \Delta y \]
We know \( \sqrt{0.0036} = 0.06 \):
\[ \sqrt{0.0037} \approx 0.06 + 0.0008333 \approx 0.0608333 \]

### Final Answer
Thus, the approximate value of \( \sqrt{0.0037} \) is:
\[ \sqrt{0.0037} \approx 0.0608333 \]

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