AP Calculus. Derivatives and Their Applications. Presenter Notes

Stuent Stuy Session Presenter Notes Thank you for agreeing to present at one of NMSI s Saturay Stuy Sessions. We are grateful you are sharing your time an expertise with our stuents. Saturay mornings can be a tough sell for stuents, so we encourage you to incorporate strategies an techniques to encourage stuent movement an engagement. Suggestions for ifferent presentation options are inclue in this ocument. If you have any questions about the content or about presentation strategies, please contact Mathematics Director Charla Holzbog at cholzbog@nms.org or AP Calculus Content Specialist Karen Miksch at kmiksch@nms.org. The material provie contains many release AP multiple choice an free response questions as well as some AP like questions that we have create. The goal for the session is to let the stuents experience a variety of both types of questions to gain insight on how the topic will be presente on the AP exam. It is also beneficial for the stuents to hear a voice other than their teacher in orer to help clarify their unerstaning of the concepts. Suggestions for presenting: The vast majority of the stuy sessions are on Saturay an stuents an teachers are coming to be WOWe! We want activities to engage the stuents as well as prepare them for the AP Exam. The following presenter notes inclue pacing suggestions (you only have 50 minutes!), solutions, an recommene engagement strategies. Suggestions on how to prepare: The notes/summaries on the last page(s) are for reference. We want the stuents time uring the session to be focuse on the questions as much as possible an not taking or reaing the notes. As the questions are presente uring the session, you may wish to refer the stuents back to those pages as neee. It is not our intent for the sessions to begin with a lecture over these pages. As you prepare, work through the questions in the packet noting the level of ifficulty an topic or skill require for the questions. Design a plan for what questions you woul like to cover with the group epening on their level of expertise. Some groups will be reay for the tougher questions while other groups will nee more guiance an practice on the easier ones. Create an easy, meium, an har listing of the questions prior to the session. This will allow you to ajust on the fly as you get to know the groups. In most instances, there will not be enough time to cover all the questions in the packet. Use your jugement on the amount of questions to cover base on the stuents interactions. Remember to inclue both multiple choice an free response type questions. Discussions on test taking strategies an scoring of the free response questions are always great to inclue uring the ay. The concepts shoul have been previously taught; however, be prepare to teach the topic if you fin out the stuents have not covere the concept prior in class. In sessions where multiple schools come together, you might have a mixture of stuents with an without prior knowlege on the topic. You will have to use your best jugement in this situation. Consier working through some free response questions before the multiple choice questions, or flipping back an forth between the two types of questions. Sometimes, if free response questions are save for the last part of the session, it is possible stuents only get practice with one or two of them an most stuents nee aitional practice with free response questions. Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 1

Derivatives an Their Applications Stuent Session-Presenter Notes This session inclues a reference sheet at the back of the packet. We suggest that the presenter spens few minutes only on the ifferentiation rules an oes not spen time going over the whole reference sheet, but may point it out to stuents that it is available to refer to if neee. We suggest that stuents will work in pairs (epening on the size of the class)- arrange the esks prior to the start of the session. We have intentionally inclue more material than can be covere in most Stuent Stuy Sessions (this one is especially long) to account for groups that are able to answer the questions at a faster rate. Use our presenter notes an your own jugment, base on the group of stuents, to etermine the orer an selection of questions to work in the session. Be sure to inclue a variety of types of questions (multiple choice, free response, calculator, an non-calculator) in the time allotte. Notice in the solutions guie the questions are categorize as 3, 4, or 5 inicating a typical question of the ifficulty level (DL) for a stuent earning these qualifying scores on the AP exam. I. 5 Minute Introuctory Activity When stuents enter the room, ask them to complete the reference sheet about the erivatives as they wait a few minutes for the start of the session. When the session starts, isplay the answers an let the stuents first mark an then correct any mistakes. 1 Ask stuents to take the erivatives of u? an using power an u? chain rules. Remin them these will appear in the AP exam more often than other raical an rational functions, an it is beneficial to memorize these two rules as well. Let the stuents to work in their pairs on problems #1, 3, 5 an 6. Please walk aroun to actively monitor them an also etermine the level of the group. Check the answers. II. 15 minutes Differentiation Rules Practice Moel numeric ifferentiation using questions FR 30 an FR 31. Compare an contrast these two questions. Remin stuents that when estimating the erivative using the given numeric ata, they nee to use the closest possible ata points (question 31 has three acceptable answers), must show the ifference quotient, use ~ symbol, in noncalculator questions they on t nee to reuce/simplify their answers an in most cases must inicate the units an explain the meaning of their answers. Let the stuents to work on problems #15, 16, 20, 25, 34 an 37 in pairs in a speeating activity: Spee Dating: (or inner/outer circle) Assign half of the stuents to a particular station (A,B,C, etc.). The other stuents are assigne to begin the speeating activity at a specific station an then will rotate through the other stations after an allotte amount of time for a multiple choice (presenter calls out the question to be Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 3

worke) or part of a free response question of your choice. Discuss an grae each question as the game procees, or wait until all the ates are complete to iscuss an correct. Extra practice questions on ifferentiation rules: #2, 4, 7, 9, 11, 12, 14, 24, 27, 29, 35. - Use more of these as time allows uring the spee-ating activity or have them work in pairs on some extra problems in class or avise the stuents to work on these problems on their own time when they use this packet to stuy for their AP exam. III. 10 minutes Tangent Line Approximation Practice Remin stuents that tangent line approximation may appear in the AP exam uner ifferent names as linearization or local linear approximation. Moel problems MC #13, 18, 21 an FR 32. Extra practice questions on tangent line approximation: #17, 36 IV. 10 minutes Implicit Differentiation Practice Moel problem #26 Let the stuents work in their group on the free response problem #28. After few minutes, isplay the rubric an let the stuents score their answers. Extra practice questions on implicit ifferentiation: #8, 10. V. 10 minutes Relate Rates Practice Moel problem #22 Let the stuents work in their group on the free response problem #33. After few minutes, isplay the rubric an let the stuents score their answers. Extra practice question on relate rates: #19. Upate Option: Please notice the QR Coe activity to accompany this stuy session that was create by Bryan Passwater. Note that it inclues some of the questions in Activity B that go along with the latter part of the session an contains more general questions from the packet as well as a few aitional problems. Feel free to use these as an engaging, quick culminating activity for your stuents-instructions are inclue in the file. We know this is a long packet an want to encourage our presenters to pick an choose the best questions for the group of stuents in your session. Encourage the stuents an teachers present to be sure to revisit the questions that you in t have time to cover in the classroom. Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 4

Derivatives an Their Applications Solutions Multiple Choice: 1. A (1993 AB24) DL: 4 f (x) x 2 2x 1 2 3 f (x) 2 3 x2 2x 1 1 3 2x 2 22x 2 3x 2 2x 1 1 3 f 0 4 3 4 3 2. E (1985 AB2) DL: 4 f f x 42x1 f f 4 3 2 x 24 2x 1 2 2 x 96 2x 1 x 384 1 2 3. A (AP-like) DL: 4 h x f x g 4. A (AP-like) DL: 3 5. C (AP-like) DL: 3 6. E (1988 AB18) DL: 3 82x 1 x g x 3 48 2x 1 2 1922x 1 f x g x f x f x 2 g x gx h5 3524 24 35 7 23 2 2 h x f g2x g 2x 2 h2 fg4g42 h x 2g x h3 2g 3 gx g4 2 f 3 g 3 26 2 24 5 4 5 4 9 2 90 y ; 2sin x 1 2 2 sin x 2 2 y cos x 1 2 2 2 1 2 cos x 2 Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 5

7. D (AB Sample Question #7 from AP Calculus Course an Exam Description) DL: 4 8. B (2003 AB26) DL: 3 3y 2 2x 2 2xy 6 (moving 2xy from right to the left sie of the equation will create a positive coefficient in front of the term that requires prouct rule an will avoi some potential errors) 6y y y 4x 2x y2 0 y 6y 2x 4x 2y y 4x 2y 6y 2x 4 3 22 62 23 9. B (AB Sample Question #18 from AP Calculus Course an Exam Description) DL: 3 Instantaneous rate of change is another name for the erivative. Stuents shoul use their calculators to evaluate the erivative of the given function at t 90 ays. 10. D (2008 AB16) DL: 4 cosxy 11. A (1995 BC5, appropriate for AB) DL: 4 12. A (AP-like) DL: 3 x y y 1 1 8 18 4 9 x y y 1 cosxy x y 1 cosxy h h h h f x f gxg x x requires the use of the prouct rule, x x x f f 1 1 x 2 1 f 3 1 3 gxgx g x f 1 ycos xy y cos xy gxg x gxgx gx 2 fg x 2 1 4 y 1 ycos xy x cos xy Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 6

13. C (AP-like) DL: 4 fx e x 3 f 0 2 f e x x 2 e x 3 f 0 1 4 Tangent line equation : y 1, when, 4 x 0 2 x 1 2 y 1 8 2 2.125 14. A (AP-like) DL: 3 f x 1 3e3x f 0 x 4 e 3x 1 3e0 0 4 e 2 0 5 15. D (2008 AB3) DL: 4 f 2 2x x x 13 x 2 2 16. A (2008 AB28) DL: 4 If fx contains a,b then its inverse gx f 1 x will contain b,a. Their erivatives 1 are relate with the equation gb. f a Similarly, when f x contains then its inverse f 1 x contains 3, 6. 1 Then g3. f 6 3 1 x 2 2 factoring out the common term x 2 2 2 helps to conense the answer; f f 2 x 1 x x 2 2 x x 2 2 6x x 2 2 2 6x 2 6x x 2 2 x 2 2 2 7x 2 6x 2 6,3 g x 1 2 Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 7

17. D (AP-like) DL: 4 slope of the secant line ; m 15 3 4 1 4 h x 3x 2 2kx h 2 12 4k setting h2 4 an solving for k yiels k 2 18. D (AB Sample Question #5 from AP Calculus Course an Exam Description) DL: 4 The function therefore gx an its tangent line will contain the same point (point of tangency), g 1 2 y 1 2 4 1 2 1 3 At the point of tangency, they also have the same slope, therefore g 1 1 2 y 2 4 g 1 2 g 1 2 4 3 7 19. A (AB Sample Question #4 from the early version of AP Calculus Course an Exam Description) DL: 4 V t r 4r2 t 2 4 5 2 r t r t 2 425 1 50 A t 8r r t 8 5 1 50 4 5 20. B (AP-like) DL: 4 fx e sin x 1 0 e sin x 1 sinx ln1 0 [1,4] x is the only solution on f f x e sin x cos x e sin cos e 0 11 Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 8

21. A (2008 AB18) DL: 5 The tangent line, y xk, an the graph of the function y has the same slope, -1, at the point of tangency; y 2x 3 1 at x 2 At the point of tangency, they also have the same y-coorinate; 4 6 1 2 k k 3 22. D (1998 AB90) DL: 4 A 1 2 bh A t 1 2 b h t h b t A t 1 2 b3 h3 3 2 b h A When b h, is negative therefore A ecreases. t 23. A (2008 AB11) DL: 3 f x x3 1 2x 2 2 f 6 1 17 1 2x 3 24. B (AB Sample Question #3 from AP Calculus Course an Exam Description) DL: 4 f f x cos ln2x cos ln 2x x x 1 2x 2 Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 9

25. D (AP-like) DL: 4 A 2xy 2x9 x 2 18x 2x 3 A an changes sign from positive to negative at. 18 6x2 0 x 3 The omain is x : 0,3, at, there must be the absolute max of the area function. A x 3 2 3 9 3 2 26. B (AP-like) DL: 4 2 y y 2y 2 y x 3 12 3 at x 3 an y 1, y 12 1 2 then, 2 y 21 2 2 6 2 Free Response 27. 2003-Form B- AB6a Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 10

29. 2014 AB/BC 1a,b (a) A(30) - A(0) =-0.197(or -0.196)lbs/ay 30-0 1: answer with units (b) A (15) =-0.164(or - 0.163) The amount of grass clippings in the bin is ecreasing at a rate of 0.164 (or 0.163) lbs/ay at time t = 15 ays. 2 1: A (15) 1: interpretation 30. 2012 AB/BC 1a 31. 2001 AB/BC 2a Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 12

32. 2010 AB6a,b (a) y f 1 8 1, 2 2 1: f 1 1: answer An equation of the tangent line is y 28 x 1. (b) f 1.1 2.8 Since y f x 0 on the interval 1 x 1.1, 2 y 3 2 2 y 2 13x y 0 on this interval. 2 1: approximation 1: conclusion with explanation Therefore on the interval 1 x 1.1, the line tangent to the graph of y f x at x 1 lies below the curve an the f 1.1. approximation 2.8 is less than Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 13

33. 2002-Form B-AB6 34. 2014 AB/BC 3 () p ( x) = f ( x 2 -x)( 2x- 1) ( ) p - 1 = f (2)(- 3) = (-2)(- 3) = 6 3 2: p ( x) 1: answers 35. 2011-Form B- AB 2c r3 50 (c) The rate at which water is raining out of the tank at time t 3 hours is increasing 2 at50 liters/hour. 2 1: r 3 1: meaning of r 3 Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 14

36. 2009-FormB- AB5a 37. AP-like h0 oes not exist. Even if lim hx lim 4x 3 3 3 an lim hx lim 3e3x 3, since lim h x lim hx, x0 x0 x0 x0 x0 x0 h x is not continuous at, therefore is not ifferentiable at. x 0 hx x 0 Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 15

General Rules Definition of Derivative: f ( x) f ( x) lim h0 f (x h) f (x) h Sum an Difference Rule: f ( x) g( x) f (x) g (x) Constant Multiple Rule: k f (x) k f (x) Prouct Rule: f ( x) g( x) Quotient Rule: f ( x) g( x) Chain Rule: f ( g( x)) Power Rule: n x Particular Rules nx n1 Exponential functions: f (x)g(x) f (x) g (x) f (x)g(x) f (x) g (x) [g(x)] 2 f (g(x)) g (x) eu e u u u u Bases Other than e: ( a ) a (ln a) u Trigonometric Functions: sin(u) u cos(u) cos(u) u sin(u) tan(u) sec2 (u) u csc(u) u csc(u)cot(u) sec(u) u sec(u)tan(u) cot(u) csc2 (u) u Logarithmic Functions: ln(u) 1 u u ln u 1 1 u log a ( u) ln a ln a u Inverse Trigonometric Functions: sin1 (u) 1 1 u u 2 1 tan1 (u) 1 u u 2 Copyright 2017 National Math + Science Initiative, Dallas, Texas. All rights reserve. Visit us online at www.nms.org 15