Final answer:
The instantaneous rate of change of the function f(x) at x=2 is found by calculating its derivative and evaluating it at that point. The calculation reveals that the rate of change is 2, making the correct answer option C).
Step-by-step explanation:
To find the instantaneous rate of change of the function f(x) = (x^2-2)/(x-1) at x=2, we need to calculate its derivative and evaluate it at x=2. The derivative f'(x) represents the instantaneous rate of change of the function at any point x.
The derivative of f(x) is found using the quotient rule. For f(x) = (x^2-2)/(x-1), let's denote u = x^2 - 2 and v = x - 1, then f'(x) = (v*u' - u*v') / v^2. Calculating u' = 2x and v' = 1 and substituting them into the derivative formula gives us:
f'(x) = ((x - 1)(2x) - (x^2 - 2)(1)) / (x - 1)^2
Simplifying this expression, we find:
f'(x) = (2x^2 - 2x - x^2 + 2) / (x - 1)^2 = (x^2 - 2x + 2) / (x - 1)^2
Evaluating the derivative at x=2 yields:
f'(2) = (2^2 - 2*2 + 2) / (2 - 1)^2 = (4 - 4 + 2) / 1^2 = 2
Therefore, the instantaneous rate of change of the function at x=2 is 2, which corresponds to option C).