Answer :
Final answer:
Using linear approximation, the estimated values for the function f(x) are f(16) \\u2248 71, f(14) \\u2248 77, and f(18) \\u2248 68. option A
Explanation:
To estimate the value of the function f(x) for values near 15 using the given information that f(15) = 74 and f'(15) = -3, we use the concept of linear approximation. This method is based on the idea that if we know the value of a function and its derivative at a point, we can make an approximate prediction for the function's value at points close to the known point.
The basic formula for linear approximation is:
f(x) ≈ f(a) + f'(a) * (x - a)
Using this formula:
- To estimate f(16), we calculate 74 + (-3) * (16 - 15), which equals 71.
- To estimate f(14), we calculate 74 + (-3) * (14 - 15), which equals 77.
- To estimate f(18), we calculate 74 + (-3) * (18 - 15), which equals 68.
The correct estimates based on this method are therefore: f(16) ≈ 71, f(14) ≈ 77, and f(18) ≈ 68, which corresponds to option (a).