I still have no voice, so Wendy (another calculus teacher) will be lecturing today. Yes, she always dresses up this extravagantly for Halloween.

Transcription

1 I still have no voice, so Wendy (another calculus teacher) will be lecturing today. Yes, she always dresses up this extravagantly for Halloween. I will be grading your tests over the weekend and I will post the grades as soon as I m finished. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October / 11

2 Today s Agenda Upcoming Homework : The Mean Value Theorem Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October / 11

3 Upcoming Homework Written HW I: Section 3.7, #2,4,6,10,12,28,38. Section 4.1, #28,32,36,46,48,50. Due 11/2/2015. WeBWorK HW #18:, due 11/4/2015 Written HW J:, #10,12,16,24. Due 11/6/2015. WeBWorK HW #19: Sections 4.3 and 4.4, due 11/9/2015 Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October / 11

4 Rolle s Theorem Let f be a function that satisfies the following three hypotheses: 1 f is continuous on the closed interval [a, b] 2 f is differentiable on the open interval (a, b) 3 f (a) = f (b) Then there is a number c in (a, b) such that f (c) = 0. Discussion Question: How is Rolle s Theorem different from the Extreme Value Theorem that we discussed on Monday? Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October / 11

5 Rolle s Theorem Let f be a function that satisfies the following three hypotheses: 1 f is continuous on the closed interval [a, b] 2 f is differentiable on the open interval (a, b) 3 f (a) = f (b) Then there is a number c in (a, b) such that f (c) = 0. Discussion Question: How is Rolle s Theorem different from the Extreme Value Theorem that we discussed on Monday? Answer: First of all, the Extreme Value Theorem does not require hypotheses (2) and (3). In addition, the Extreme Value Theorem only guarantees that a maximum and minimum are attained on [a, b] the max and min could occur at an endpoint, and the derivative might not even be 0 there! (Example: f (x) = x on [0, 1].) Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October / 11

6 Here are some visual examples of Rolle s Theorem from your textbook: Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October / 11

7 The Mean Value Theorem is a more general case of Rolle s Theorem, so it applies in many more situations. The Mean Value Theorem Let f be a function that satisfies the following hypotheses: 1 f is continuous on the closed interval [a, b] 2 f is differentiable on the open interval (a, b) Then there is a number c in (a, b) such that f (c) = f (b) f (a), b a or equivalently, f (b) f (a) = (f (c))(b a). Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October / 11

8 Notice that if a function f satisfies the hypotheses of the Mean Value Theorem, and in addition f (a) = f (b), then there exists c (a, b) such that f (c) = f (b) f (a) b a = 0 b a = 0, so Rolle s Theorem is simply a special case of the Mean Value Theorem. The proofs of both Rolle s Theorem and the Mean Value Theorem can be found in of your textbook; they are worth reading if you are interested in proofs. However, it is more important for our class s purpose that you learn applications of these theorems. Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October / 11

9 Here are some visual examples of the Mean Value Theorem from your textbook: Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October / 11

10 Let s try some simple examples of both Rolle s Theorem and the Mean Value Theorem. In each of the following problems, identify whether Rolle s Theorem or the Mean Value Theorem should be used, and find the value c (a, b) that satisfies the conclusion of the theorem. 1 f (x) = 2x 2 3x + 1 on [0, 2] 2 f (x) = x 1 3x on [0, 9] 3 f (x) = ln x on [1, 4] Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October / 11

11 The following is a classic example of an application of the Mean Value Theorem. Michael s car is equipped with an EZ-pass (a sensor in the car that sends data to your registered account whenever you cross a toll booth, and is tied to your credit card to automatically charge you tolls without needing to stop and pay). Michael is driving along a turnpike (an expressway on which a toll is charged) that has EZ-pass sensors spaced 50 miles apart. Michael passes the first sensor at 3:52 pm and he passes the second sensor at 4:37 pm. The speed limit on the turnpike is 60 mph. When Michael returns home, he finds that a speeding ticket has been charged to his account. He wants to fight the speeding ticket in court, but his wife, a mathematician, tells him that fighting the ticket would be futile. Why was she justified in saying this? Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October / 11

12 Here are some further example problems for Rolle s Theorem and the Mean Value Theorem. For each of the following problems,. 1 Find all numbers c (a, b) that satisfy the conclusion of the relevant theorem. 1 f (x) = cos(2x) on [π/8, 7π/8] 2 f (x) = x on [0, 4] 3 f (x) = x 3 x 2 6x + 2 on [0, 3] 4 f (x) = e x on [0, 2] 2 Let f (x) = (x 3) 2. Show that there is no value of c in (1, 4) such that f (4) f (1) = (f (c))(4 1). Why does this not contradict the Mean Value Theorem? Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October / 11