Final answer:
To estimate √0.95 using the linear approximation of f(x) = √x + 1 at x = 0, we find the derivative of f(x), which is 1/(2√x). Then we use the linear approximation formula to calculate the estimate.
Step-by-step explanation:
To estimate the value of √0.95 using the linear approximation of f(x) = √x + 1 at x = 0, we first need to find the derivative of f(x). The derivative of f(x) = √x + 1 with respect to x is 1/(2√x). Evaluating the derivative at x = 0 gives us 1/(2√0) = 1/2. Now we can use the linear approximation formula: f(x) ≈ f(a) + f'(a)(x - a), where a = 0. Plugging in the values, we get f(x) ≈ f(0) + f'(0)(x - 0) = √0 + (1/2)(x - 0) = 1/2x.
Next, we substitute x = 0.95 into the linear approximation: f(0.95) ≈ 1/2(0.95) = 0.475.
Therefore, the estimate for √0.95 using the linear approximation is approximately 0.475.