Final answer:
To estimate f(0.9), we use the linear approximation formula f(a) + f'(a)(x - a). For a) 0.9^2, the estimate is 0.8, and for b) √0.9, the estimate is 0.95. Options c) and d) are 1 and not applicable, as we can estimate using the given functions.
Step-by-step explanation:
To use linear approximation to estimate the value of f(0.9), we need to know the function f(x) and its derivative at a point close to x = 0.9. Assuming that f(x) can be approximated by a linear function near x = 1, we then apply the formula:
f(a) + f'(a)(x - a)
Let's apply this formula to each of the given options:
- a) 0.9² means f(x) = x² so f'(x) = 2x. At x = 1, f(1) = 1 and f'(1) = 2. The linear approximation for f(0.9) would be 1 + 2(0.9 - 1) = 0.8.
- b) √0.9 suggests f(x) = √x. Thus, f'(x) = 1/(2√x). At x = 1, f(1) = 1 and f'(1) = 0.5. The linear approximation for f(0.9) is 1 + 0.5(0.9 - 1) = 0.95.
- c) f(1) is already provided and is simply 1 for both functions above.
- d) Not possible to estimate without knowing the function f(x), this option doesn't hold since we've successfully estimated for both given functions.
Therefore, the best answers would involve utilizing the derived linear approximations from a) and b) depending on function that f(x) refers to in the question. The choice between a) and b) depends on the defined f(x).