3. The Punchline We have now computed three different area functions. Fill in the blanks: f (1) = 4 A (x) = A' (x) = f (t) =2t +2 B (x) = B' (x) = f (t) =2t + 2 C(x) = C' (x) = You should notice a very interesting fact about the derivatives of the area functions - a fundamentally beautiful property. What is it? 4. We are going to define one final function, D (x) = So 0.2t sin (cos (sint) ) dt. f (1 ) + Don't worry about trying to find a simple formula for D (x). But, using our amazing fact, fill in the last blank: D' (x) = Not only is the fact that we've discovered a surprising one; as we shall see in Section 5.4, it is also extremely useful