Questions: The linear approximation of √55 is □

Solution Steps

To find the linear approximation of \(\sqrt{55}\), we can use the linear approximation formula for a function \(f(x)\) at a point \(a\), which is \(f(a) + f'(a)(x - a)\). Here, \(f(x) = \sqrt{x}\), and we choose \(a\) to be a number close to 55 for which we can easily compute the square root, such as 49 or 64. We then compute \(f(a)\), \(f'(a)\), and substitute \(x = 55\) into the linear approximation formula.

We define the function \( f(x) = \sqrt{x} \) and choose the point of approximation \( a = 49 \), which is close to \( 55 \) and has a known square root.

We compute \( f(49) \): \[ f(49) = \sqrt{49} = 7 \]

The derivative of \( f(x) \) is: \[ f'(x) = \frac{1}{2\sqrt{x}} \] Evaluating this at \( a = 49 \): \[ f'(49) = \frac{1}{2\sqrt{49}} = \frac{1}{14} \]

Using the linear approximation formula: \[ L(x) = f(a) + f'(a)(x - a) \] we substitute \( x = 55 \): \[ L(55) = 7 + \frac{1}{14}(55 - 49) \] Calculating \( 55 - 49 = 6 \): \[ L(55) = 7 + \frac{1}{14} \cdot 6 = 7 + \frac{6}{14} = 7 + \frac{3}{7} \] Converting \( 7 \) to a fraction: \[ L(55) = \frac{49}{7} + \frac{3}{7} = \frac{52}{7} \]

Final Answer