2 LIMITS

CHAPTER 2LIMITSGROUP WORK 3: The Twin ProblemWhen students see this problem, there is a good chance that they will disagree among themselves about theanswer. Let them argue for a while. Ideally, they will come up with the idea of using the Intermediate ValueTheorem to prove that Dr. Stewart was correct. If they don’t, this may need to be given to them as a hint.Another hint they may need is that the Intermediate Value Theorem deals with a single continuous function,whereas the problem is talking about two functions, Stewart’s temperature and Shasta’s temperature. Theywill have to figure out a way to find a single function that they can use. Encourage them to write up a solutionto the exact degree of rigor that will be expected of them on homework and exams; this is a good opportunityto convey the course’s expectations to the students.Answer: Let S (t) and O (t) be Dr. Stewart’s and Shasta’s temperatures at time t. NowletT (t) = S (t)−O (t).T (t) is continuous (being a difference of continuous functions), T (0) >0 (Dr. Stewart is warmer at first),and T ( f ) < 0(where f represents the end of the vacation; Shasta is warmer at the end). Therefore, by theIVT, there exists a time a at which T (a) = 0 and hence S (a) = O (a).GROUP WORK 4: Swimming to the ShoreEmphasize to the students that they are not trying to find x, but simply trying to prove its existence. As in theTwin Problem, a first hint might be to use the IVT, and a second could be to find a single continuous functionof x.It is probably best to do this activity after the students have seen the solution to the ice fisherman problemabove, or the Twin Problem.Answer: Let D (x) = d (P x , A) − d (P x , B). D (x) is continuous, D (A) > 0, and D (B) < 0. Therefore,by the IVT, there is a place where D (x) = 0.HOMEWORK PROBLEMSCore Exercises: 3, 7, 10, 16, 21, 32, 38, 47, 57, 61Sample Assignment: 3, 6, 7, 9, 10, 16, 21, 29, 32, 35, 38, 40, 43, 44, 47, 53, 57, 60, 61, 65Exercise D A N G3 × ×6 ×7 × ×9 ×10 ×16 × ×21 × ×29 × ×32 ×35 ×Exercise D A N G38 × ×40 × ×43 ×44 × ×47 ×53 × × ×57 × ×60 ×61 ×65 × ×72