Use $\Delta y \approx f^{\prime}(x) \Delta x$ to find a decimal approximation of the radical expression.
\[
\sqrt{107}
\]
What is the value found using $\Delta y \approx f^{\prime}(x) \Delta x$ ?
$\sqrt{107} \approx \square$ (Round to three decimal places as needed.)

Step 1 ：We are given the function \(f(x) = \sqrt{x}\) and we want to find an approximation for \(f(107)\).

Step 2 ：We choose a value of \(x\) close to 107 for which we know the exact value of \(f(x)\). A good choice is \(x = 100\), because \(\sqrt{100} = 10\).

Step 3 ：The derivative of \(f(x) = \sqrt{x}\) is \(f^{\prime}(x) = \frac{1}{2\sqrt{x}}\).

Step 4 ：We use the linear approximation formula with \(x = 100\), \(\Delta x = 107 - 100 = 7\), and \(f^{\prime}(100) = \frac{1}{2\sqrt{100}} = \frac{1}{20}\).

Step 5 ：Applying the linear approximation formula \(\Delta y \approx f^{\prime}(x) \Delta x\) gives us \(\Delta y = 0.35\).

Step 6 ：Adding this to our initial \(y\) value (which is \(f(100) = 10\)) gives us an approximation for \(f(107)\), which is \(10 + 0.35 = 10.35\).

Step 7 ：Final Answer: The decimal approximation of \(\sqrt{107}\) using the linear approximation formula is \(\boxed{10.35}\).