In the following find the approximate values, using differentials :
`sqrt(0.17)`

Answer

To find the approximate value of \(\sqrt{0.17}\) using differentials, we can follow these steps:

### Step 1: Define the function
Let \(y = \sqrt{x}\). We want to find the approximate value of \(y\) when \(x = 0.17\).

### Step 2: Choose a point near 0.17
Choose a value of \(x\) that is easy to compute and close to \(0.17\). We can take \(x = 0.16\) since \(\sqrt{0.16} = 0.4\).

### Step 3: Calculate \(dx\)
Now, we find \(dx\):
\[ dx = 0.17 - 0.16 = 0.01 \]

### Step 4: Differentiate the function
Next, we need to find the derivative of \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{x}} \]
At \(x = 0.16\):
\[ \frac{dy}{dx} = \frac{1}{2\sqrt{0.16}} = \frac{1}{2 \cdot 0.4} = \frac{1}{0.8} = 1.25 \]

### Step 5: Calculate \(dy\)
Now, we can find \(dy\):
\[ dy = \frac{dy}{dx} \cdot dx = 1.25 \cdot 0.01 = 0.0125 \]

### Step 6: Approximate \(\sqrt{0.17}\)
Now we can approximate \(\sqrt{0.17}\):
\[ \sqrt{0.17} \approx y + dy = 0.4 + 0.0125 = 0.4125 \]

### Final Answer
Thus, the approximate value of \(\sqrt{0.17}\) is:
\[ \sqrt{0.17} \approx 0.4125 \]

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