Midterm 2 be sure to verify the hypotheses are satis ed: complete the following problems. You may use any result from class you like, but if you cite a theorem: this is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted: in order to receive full credit, please show all of your work and justify your answers. You do not need: if you need extra room, use the back sides of each page or the scratch paper provided at the end of the exam. If you must use extra paper, make sure to write your name on it and attach it to this exam. Do not unstaple or detach pages from this exam. to simplify your answers unless speci cally instructed to do so: please sign the following: On my honor, i have neither given nor received any aid on this examina- tion.