In the following find the approximate values, using differentials :
`sqrt(49.3)`

Answer

To find the approximate value of \(\sqrt{49.3}\) using differentials, we can follow these steps:

### Step 1: Identify the function and the point of approximation
Let \(y = f(x) = \sqrt{x}\). We want to approximate \(f(49.3)\).

### Step 2: Choose a nearby point
Choose a point \(x = 49\) which is close to \(49.3\). We will use this point for our approximation.

### Step 3: Calculate \(f(49)\)
Calculate \(f(49)\):
\[ f(49) = \sqrt{49} = 7 \]

### Step 4: Calculate \(dx\)
Determine the change in \(x\):
\[ dx = 49.3 - 49 = 0.3 \]

### Step 5: Find the derivative \(f'(x)\)
Calculate the derivative of \(f(x)\):
\[ f'(x) = \frac{1}{2\sqrt{x}} \]
Now, evaluate \(f'(49)\):
\[ f'(49) = \frac{1}{2\sqrt{49}} = \frac{1}{2 \cdot 7} = \frac{1}{14} \]

### Step 6: Calculate \(dy\)
Using the formula \(dy = f'(x) \cdot dx\):
\[ dy = f'(49) \cdot dx = \frac{1}{14} \cdot 0.3 = \frac{0.3}{14} = \frac{0.3}{14} \approx 0.02142857 \]

### Step 7: Calculate the approximate value of \(f(49.3)\)
Now, we can find \(f(49.3)\) using:
\[ f(49.3) \approx f(49) + dy = 7 + 0.02142857 \approx 7.02142857 \]

### Final Result
Thus, the approximate value of \(\sqrt{49.3}\) is:
\[ \sqrt{49.3} \approx 7.021 \]

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