I have a big test tomorrow and as I was reviewing, I encountered the following confusions. I hope that someone can help me out. I really appreciate for your help! 1) The answer is NO. But when I differentiate both sides and using the fundamental theorem of calculus, I get f(x)=e^x, which is WRONG. Why does the theorem give me the wrong answer? 2) The example says that at x=-r, u=-pi/2 (but why not 3pi/2, or 7pi/2?) Clearly, sin (-pi/2) = sin (3pi/2) = -1. Say, if I use 3pi/2 (instead of -pi/2), my final answer would be -pi(r^2)/2, which is not the correct answer...why is this happening? What is the problem? 3) For what values of c does the equation ln x = c(x^2) have exactly one solution? Justify fully. Clearly c<0 is part of the answer. For the case of c>0, there is one case where the curves y=ln x and y=c(x^2) intersect at only one point. But I need one more equation to solve for 2 variables. If the 2 curvesthey have a COMMON tangent at this point, then I got my second equation and I can solve for c. But how can I justify that at that point, they have a COMMON tangent (i.e. same slope)? I have no idea... Can someone please explain?