In the following find the approximate values, using differentials :
`sqrt(50)`

Answer

To find the approximate value of \( \sqrt{50} \) using differentials, we can follow these steps:

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

### Step 2: Choose a value of \( x \) close to 50
We choose \( x = 49 \) because \( \sqrt{49} = 7 \) is easy to compute and close to \( \sqrt{50} \).

### Step 3: Calculate \( dy \)
To find \( dy \), we need to differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x}} \]

### Step 4: Evaluate the derivative at \( x = 49 \)
Now, we substitute \( x = 49 \):
\[ \frac{dy}{dx} \bigg|_{x=49} = \frac{1}{2\sqrt{49}} = \frac{1}{2 \times 7} = \frac{1}{14} \]

### Step 5: Determine \( dx \)
We have \( dx = 50 - 49 = 1 \).

### Step 6: Calculate \( dy \)
Now we can find \( dy \):
\[ dy = \frac{dy}{dx} \cdot dx = \frac{1}{14} \cdot 1 = \frac{1}{14} \]

### Step 7: Approximate \( \sqrt{50} \)
Now we can approximate \( \sqrt{50} \):
\[ \sqrt{50} \approx y + dy = 7 + \frac{1}{14} \]
Calculating \( \frac{1}{14} \) gives approximately \( 0.0714 \):
\[ \sqrt{50} \approx 7 + 0.0714 = 7.0714 \]

### Final Answer
Thus, the approximate value of \( \sqrt{50} \) is \( 7.0714 \).

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