In the following find the approximate values, using differentials :
`sqrt(1.037)`.

Answer

To find the approximate value of \(\sqrt{1.037}\) using differentials, we can follow these steps:

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

### Step 2: Identify the point of approximation
We choose \( x = 1 \) because it is close to \( 1.037 \) and we know the square root of 1:
\[ \sqrt{1} = 1. \]

### Step 3: Calculate \(\Delta x\)
Calculate \(\Delta x\):
\[ \Delta x = 1.037 - 1 = 0.037. \]

### Step 4: Find the derivative
Now, we need to find the derivative of \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x}}. \]
At \( x = 1 \):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{1}} = \frac{1}{2}. \]

### Step 5: Use the differential to approximate \(\Delta y\)
Using the formula for differentials:
\[ \Delta y \approx \frac{dy}{dx} \cdot \Delta x. \]
Substituting the values we have:
\[ \Delta y \approx \frac{1}{2} \cdot 0.037 = 0.0185. \]

### Step 6: Calculate the approximate value of \( y \)
Now, we can find the approximate value of \( y \):
\[ y \approx \sqrt{1} + \Delta y = 1 + 0.0185 = 1.0185. \]

### Conclusion
Thus, the approximate value of \(\sqrt{1.037}\) is:
\[ \sqrt{1.037} \approx 1.0185. \]

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