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 2015 1 / 11
Today’s Agenda • Upcoming Homework • Section 4.2: The Mean Value Theorem Lindsey K. Gamard, ASU SoMSS MAT 265: Calculus for Engineers I Friday, 30 October 2015 2 / 11
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: Section 4.2, due 11/4/2015 • Written HW J: Section 4.2, #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 2015 3 / 11
Section 4.2 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 2015 4 / 11
Section 4.2 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 2015 4 / 11
Section 4.2 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 2015 5 / 11
Section 4.2 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 2015 6 / 11
Section 4.2 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 ) 0 = b − a = 0 , b − a 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 Section 4.2 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 2015 7 / 11
Section 4.2 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 2015 8 / 11
Section 4.2 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 ) = 2 x 2 − 3 x + 1 on [0 , 2] 2 f ( x ) = √ x − 1 3 x 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 2015 9 / 11
Section 4.2 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 2015 10 / 11
Section 4.2 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. f ( x ) = cos(2 x ) on [ π/ 8 , 7 π/ 8] 1 f ( x ) = √ x on [0 , 4] 2 f ( x ) = x 3 − x 2 − 6 x + 2 on [0 , 3] 3 f ( x ) = e − x on [0 , 2] 4 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 2015 11 / 11
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