2 LIMITS

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CHAPTER 2<strong>LIMITS</strong>GROUP 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
GROUP WORK 1, SECTION 2.5Exploring Continuity⎧⎨ cx 2 if x < 11. Are there values of c and m that make h (x) = 4 if x = 1⎩−x 3 + mx if x > 1and m, or explain why they do not exist.continuous at x = 1? Find c{ 0 if x is rational2. Recall the function f (x) =x 2 if x is irrational(a) Do you believe that f (x) is continuous at x = 0? Why or why not?(b) What is f (0)?(c) What is limx→0f (x)?(d) Use parts (b) and (c) either to revise your answer to part (a), or to prove that your answer is correct.73
Page 1 and 2: 2 LIMITS2.1 THE TANGENT AND VELOCIT
Page 3: SECTION 2.1THE TANGENT AND VELOCITY
Page 8 and 9: 2.2 THE LIMIT OF A FUNCTIONSUGGESTE
Page 10 and 11: CHAPTER 2LIMITSGROUP WORK 2: The Sh
Page 12 and 13: CHAPTER 2LIMITS3. Answers will vary
Page 14 and 15: GROUP WORK 2, SECTION 2.2The Shape
Page 16 and 17: GROUP WORK 2, SECTION 2.2The Shape
Page 18 and 19: 2.3 CALCULATING LIMITS USING THE LI
Page 20 and 21: CHAPTER 2LIMITSGROUP WORK 3: The Sq
Page 22 and 23: GROUP WORK 2, SECTION 2.3Fixing a H
Page 24 and 25: 2.4 THE PRECISE DEFINITION OF A LIM
Page 26 and 27: CHAPTER 2LIMITSGROUP WORK 2: The Di
Page 28 and 29: GROUP WORK 1, SECTION 2.4A ‘‘Ji
Page 30 and 31: GROUP WORK 2, SECTION 2.4The Dire W
Page 32 and 33: Consider the following functions:GR
Page 34 and 35: GROUP WORK 4, SECTION 2.4The Signif
Page 36 and 37: CHAPTER 2LIMITS• Start by stating
Page 40 and 41: Exploring Continuity3. Consider the
Page 42 and 43: The following are graphs of y = f (
Page 44 and 45: GROUP WORK 4, SECTION 2.5Swimming t
Page 46 and 47: CHAPTER 2LIMITS3. Find the vertical
Page 48 and 49: CHAPTER 2LIMITS7. Let⎧ √⎨ 3
Page 50: CHAPTER 2LIMITS6. (a) L = 0√0.01(

2 LIMITS

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