Find the value of the `sqrt199` correct to 4 decimal places

Answer

To find the value of \( \sqrt{199} \) correct to four decimal places, we can use the approximation method based on the binomial theorem. Here’s a step-by-step solution:

### Step 1: Identify the base value and the increment
We know that \( 199 \) is close to \( 196 \), which is a perfect square. We can express \( 199 \) as:
\[ 199 = 196 + 3 \]
Here, \( x = 196 \) and \( \Delta x = 3 \).

### Step 2: Define the function
We define the function \( f(x) = \sqrt{x} \). We will use this function to find \( f(199) \).

### Step 3: Calculate \( f(x) \) at the base value
Calculate \( f(196) \):
\[ f(196) = \sqrt{196} = 14 \]

### Step 4: Calculate the derivative \( f'(x) \)
The derivative of \( f(x) = x^{1/2} \) is:
\[ f'(x) = \frac{1}{2} x^{-1/2} = \frac{1}{2\sqrt{x}} \]
Now, evaluate \( f'(196) \):
\[ f'(196) = \frac{1}{2\sqrt{196}} = \frac{1}{2 \cdot 14} = \frac{1}{28} \]

### Step 5: Apply the approximation formula
Using the approximation formula:
\[ f(x + \Delta x) \approx f(x) + f'(x) \Delta x \]
Substituting the values:
\[ f(199) \approx f(196) + f'(196) \cdot 3 \]
\[ f(199) \approx 14 + \frac{1}{28} \cdot 3 \]

### Step 6: Calculate \( \frac{3}{28} \)
Now, calculate \( \frac{3}{28} \):
\[ \frac{3}{28} \approx 0.1071 \]

### Step 7: Add the values
Now, add this to \( 14 \):
\[ f(199) \approx 14 + 0.1071 = 14.1071 \]

### Final Answer
Thus, the value of \( \sqrt{199} \) correct to four decimal places is:
\[ \sqrt{199} \approx 14.1071 \]

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