Difficult AP Calc BC Bonus Questions (trick?)

TheMathProf
Tue, 25 Sep 2007 18:59:12 +0000

Well, f(3) is a constant (since it is equal to 5), and accordingly, d/dx[f(3)] = 0.

I can’t imagine that f '(x) is (-infinity, infinity). That seems like the domain of f '(x), and possibly the range of f '(x), but not the function itself.

One key idea that your teacher might be hinting at is the idea that f '(3) and d/dx[f(3)] are not the same thing. The idea is that the first is saying find the derivative of f(x) and evaluate it at x = 3, while the second is saying find the derivative of the constant f(3). But the question as phrased seems to be looking for something more sophisticated than that.

I’m 99% certain that the answer to the second question is therefore 0 = 0.

I’m also 99% certain that you have insufficient information to answer the first question (at least as it is relayed here). Telling you the location on one point on f(x) doesn’t give you enough information to say anything about the rate of change of the function. Does f '(x) exist? Sure, and you know that because f(x) is differentiable for all x (which incidentally, makes the remark about f(x) being continuous a redundant remark). Can you determine what f '(x) is? I would venture not. There’s no function to take the derivative of. There’s no hint as to the type of function that f(x) is. And one point does not a derivative function make.

So unless there is more to the question than what is given, I would say that there is insufficient information to answer the first.