Use linear approximations to estimate the following quantities. Choose a value of a to produce a small error.
√146

Verified step by step guidance

1. Identify a function that can be approximated, in this case, f(x) = √x.
2. Choose a value of 'a' that is close to 146 and for which the square root is easy to calculate; a good choice is a = 144, since √144 = 12.
3. Calculate the derivative of the function f(x) = √x, which is f'(x) = (1/2) * x^(-1/2).
4. Evaluate the derivative at the chosen point a = 144 to find f'(144).
5. Use the linear approximation formula: f(x) ≈ f(a) + f'(a)(x - a) to estimate √146 by substituting x = 146 and the values of f(a) and f'(a).