In the following find the approximate values, using differentials :
`sqrt(0.50)`.

Answer

To find the approximate value of \( \sqrt{0.50} \) using differentials, we can follow these steps:

### Step 1: Identify the function and the point of approximation
Let \( y = \sqrt{x} \). We will approximate \( \sqrt{0.50} \) using a nearby value of \( x \) that is easy to compute. We choose \( x = 0.49 \) because it is close to \( 0.50 \) and its square root is easy to calculate.

### Step 2: Calculate the value of \( y \) at \( x = 0.49 \)
Now we find:
\[ y = \sqrt{0.49} = 0.7 \]

### Step 3: Calculate \( dy \)
To find \( dy \), we first need to compute the derivative \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x}} \]
Now, substituting \( x = 0.49 \):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{0.49}} = \frac{1}{2 \times 0.7} = \frac{1}{1.4} = 0.7142857142857143 \]

### Step 4: Determine \( dx \)
Next, we find \( dx \):
\[ dx = 0.50 - 0.49 = 0.01 \]

### Step 5: Calculate \( dy \)
Now we can calculate \( dy \):
\[ dy = \frac{dy}{dx} \cdot dx = 0.7142857142857143 \times 0.01 = 0.007142857142857143 \]

### Step 6: Approximate \( \sqrt{0.50} \)
Finally, we can approximate \( \sqrt{0.50} \) using \( y + dy \):
\[ \sqrt{0.50} \approx y + dy = 0.7 + 0.007142857142857143 \approx 0.7071428571428571 \]

### Conclusion
Thus, the approximate value of \( \sqrt{0.50} \) is:
\[ \sqrt{0.50} \approx 0.7071 \]

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