In the following find the approximate values, using differentials :
`sqrt(360).`

Answer

To find the approximate value of \(\sqrt{360}\) 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 360
We choose \(x = 361\) because it is a perfect square, and we know \(\sqrt{361} = 19\).

### Step 3: Determine \(dx\)
Since we are approximating \(\sqrt{360}\), we have:
\[ dx = 360 - 361 = -1 \]

### Step 4: Calculate \(y\) at \(x = 361\)
Now, we find:
\[ y = f(361) = \sqrt{361} = 19 \]

### Step 5: Find the derivative \(dy/dx\)
To find \(dy\), we need to calculate the derivative of \(f(x)\):
\[ f'(x) = \frac{1}{2\sqrt{x}} \]
Thus,
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{361}} = \frac{1}{2 \cdot 19} = \frac{1}{38} \]

### Step 6: Calculate \(dy\)
Now we can find \(dy\) using the formula:
\[ dy = f'(x) \cdot dx \]
Substituting the values we have:
\[ dy = \frac{1}{38} \cdot (-1) = -\frac{1}{38} \]

### Step 7: Approximate \(\sqrt{360}\)
Now, we can approximate \(\sqrt{360}\) using:
\[ \sqrt{360} \approx y + dy = 19 - \frac{1}{38} \]
Calculating \(19 - \frac{1}{38}\):
\[ 19 - \frac{1}{38} \approx 19 - 0.0263 \approx 18.9737 \]

### Final Answer
Thus, the approximate value of \(\sqrt{360}\) is:
\[ \sqrt{360} \approx 18.9737 \]

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