05 AB Integrals and Their Applications Presenter Notes...Copyright © 2017 National Math + Science...

Copyright © 2017 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org

AP Calculus

ABIntegralsandTheirApplications

PresenterNotes

2017‐2018 EDITION

Copyright © 2017 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 1

StudentStudySession‐PresenterNotes

ThankyouforagreeingtopresentatoneofNMSI’sSaturdayStudySessions.Wearegratefulyouaresharingyourtimeandexpertisewithourstudents.Saturdaymorningscanbea“toughsell”forstudents,soweencourageyoutoincorporatestrategiesandtechniquestoencouragestudentmovementandengagement.Suggestionsfordifferentpresentationoptionsareincludedinthisdocument.Ifyouhaveanyquestionsaboutthecontentoraboutpresentationstrategies,pleasecontactMathematicsDirectorCharlaHolzbogatcholzbog@nms.orgorAPCalculusContentSpecialistKarenMikschatkmiksch@nms.org.ThematerialprovidedcontainsmanyreleasedAPmultiplechoiceandfreeresponsequestionsaswellassomeAP‐likequestionsthatwehavecreated.ThegoalforthesessionistoletthestudentsexperienceavarietyofbothtypesofquestionstogaininsightonhowthetopicwillbepresentedontheAPexam.Itisalsobeneficialforthestudentstohearavoiceotherthantheirteacherinordertohelpclarifytheirunderstandingoftheconcepts.Suggestionsforpresenting:ThevastmajorityofthestudysessionsareonSaturdayandstudentsandteachersarecomingtobeWOWed!WewantactivitiestoengagethestudentsaswellaspreparethemfortheAPExam.Thefollowingpresenternotesincludepacingsuggestions(youonlyhave50minutes!),solutions,andrecommendedengagementstrategies.Suggestionsonhowtoprepare: Thenotes/summariesonthelastpage(s)areforreference.Wewantthestudents’timeduring

thesessiontobefocusedonthequestionsasmuchaspossibleandnottakingorreadingthenotes.Asthequestionsarepresentedduringthesession,youmaywishtoreferthestudentsbacktothosepagesasneeded.Itisnotourintentforthesessionstobeginwithalectureoverthesepages.

Asyouprepare,workthroughthequestionsinthepacketnotingthelevelofdifficultyandtopicorskillrequiredforthequestions.

Designaplanforwhatquestionsyouwouldliketocoverwiththegroupdependingontheirlevelofexpertise.Somegroupswillbereadyforthetougherquestionswhileothergroupswillneedmoreguidanceandpracticeontheeasierones.Createaneasy,medium,andhardlistingofthequestionspriortothesession.Thiswillallowyoutoadjustontheflyasyougettoknowthegroups.Inmostinstances,therewillnotbeenoughtimetocoverallthequestionsinthepacket.Useyourjudgementontheamountofquestionstocoverbasedonthestudents’interactions.Remembertoincludebothmultiplechoiceandfreeresponsetypequestions.Discussionsontesttakingstrategiesandscoringofthefreeresponsequestionsarealwaysgreattoincludeduringtheday.

Theconceptsshouldhavebeenpreviouslytaught;however,bepreparedto“teach”thetopicifyoufindoutthestudentshavenotcoveredtheconceptpriorinclass.Insessionswheremultipleschoolscometogether,youmighthaveamixtureofstudentswithandwithoutpriorknowledgeonthetopic.Youwillhavetouseyourbestjudgementinthissituation.

Considerworkingthroughsomefreeresponsequestionsbeforethemultiplechoicequestions,orflippingbackandforthbetweenthetwotypesofquestions.Sometimes,iffreeresponsequestionsaresavedforthelastpartofthesession,itispossiblestudentsonlygetpracticewithoneortwoofthemandmoststudentsneedadditionalpracticewithfreeresponsequestions.


Integrals (AB) and Their Applications Student Session Presenter Notes

This session includes a reference sheet at the back of the packet. We suggest that the presenter spends a few minutes only on the integration rules and does not spend time going over the whole reference sheet, but may point it out to students that it is available to refer to if needed. We suggest that students will work in small groups of 3 or 4 (depending on the size of the class), arrange the desks prior to the start of the session. We have intentionally included more material than can be covered in most Student Study Sessions to account for groups that are able to answer the questions at a faster rate. Use our trainer notes and your own judgment, based on the group of students, to determine the order and selection of questions to work in the session. Be sure to include a variety of types of questions (multiple choice, free response, calculator, and non-calculator) in the time allotted. Notice in the solutions guide the questions are categorized as 3, 4, or 5 indicating a typical question of the difficulty level (DL) for a student earning these qualifying scores on the AP exam.

I. 10 Minutes Introductory Activity

When students enter the room, ask them to complete the reference sheet about the

integration in the back of their packet as they wait a few minutes for the start of the session.

When the session starts, display the answers and let the students first mark and then correct any mistakes.

Let the students to work in their small groups on problems #1, 3, 18, 19, 24e and 26 while you actively walk around to monitor their progress, answer questions and gauge the level of knowledge of the group. Check the answers at the end of 10 minutes.

II. 10 minutes U-Substitution Practice

Model FR 24d, FR 29d. Let the students to work on problems 11, 15 and 17 in their groups. Play “pass the pen”

to discuss the answers. Pass the Pen: Ask for a student volunteer to begin pass the pen where the questions are projected on the board and the student explains how to arrive at the correct answer. The student then selects another student to discuss the next question and so on.

Extra practice questions on u-substitution: 2, 10, and 23. You may work on some extra problems in class or advise the students to work on these problems on their own time when they use this packet to study for their AP exam.

III. 15 minutes “Integrating the Rate”, “Meaning of a Definite Integral”, “FTC”,

“Real World Application of Integration” Problems Practice

Model problems FR27 and FR31.


Let the students to work on problems 8, 12, 20 and 28 in their groups. Play “pass the pen” to discuss the answers.

Extra practice questions: #25, 30.

IV. 10 minutes Algebraic Techniques of Integration (other than u-substitution) Practice

Model problems 6 and 7. Make sure you cover these two problems during the session

since these are emphasized in the most recent AP Cal AB/BC Course and Exam Description.

Let the students work in their groups on problems 14 and 16. Check answers.

V. 5 minutes Integral as a Limit of a Riemann Sum Practice

Work out the problems 4 and 9 together with the group. At the end of the session advise students to work on the extra practice problems 5 and 32

(integrating piecewise functions), 21 and 22 (average value of a function) on their own time.

We know this is a long packet and want to encourage our presenters to “pick and choose” the best questions for the group of students in your session. Encourage the students and teachers present to be sure to revisit the questions that you didn’t have time to cover in the classroom.


Integrals and Their Applications Solutions Multiple Choice: 1. C (AP-Like) DL: 4

f (x) 2dx 2

2

f (x)dx 2dx 2

2

2

2

(1 2)

22 2x

x2

x2 3 8 11

2. D (AP-Like) DL: 4

u 3x 1 when x 1, u 4 f (3x 1)dx 1

31

4

f (u)du4

13

du 3dx x 4, u 13

dx 1

3du

3. D (2008 AB2) DL: 3

1

x2 dx x2 dx x1

1C

1

xC

4. A (AP-Like) DL: 5

f (x) (1 2x)2 on [0,1] has x 1 0

n

1

n .

Then, ck 0 1

nk

k

n, f (ck ) 1

2k

n

2

and f (ck )x 12k

n

21

n

5. B (AP Cal AB/BC Course and Exam Description, AB sample 9) DL: 4

Algebraically, x dx + 1

2

3 dx = 1

2

x2

2x1

x2

3xx1

x2 4

2

1

2

12 6 15

2

Graphically, 1

2

2 6 15

2

6. C (AP-Like) DL: 4

The integrand can be simplified using long division; x2 4x 5

x 3 (x 1)

2

x 3

x 12

x 30

3

dx x2

2 x 2 ln x 3

x0

x3

9

2 3 2ln6 2 ln 3

15

2 2 ln2


7. A (AP-Like) DL: 4 The integrand can be rewritten using the algebraic technique completing the square.

1

x2 6x 13

1

(x2 6x 9)13 9

1

4 x 3 2

Then u x 3 and du dx yields

1

x2 6x 13dx

1

4 x 3 2 dx 1

22 x 3 2 dx 1

2arctan

x 3

2

C

8. C (AP Cal AB/BC Course and Exam Description, AB sample 11) DL: 4

5e0.2t 4t dt 0

10

5e0.2t

0.2 2t 2

t0

t10

25e2 200 25 0 25e2 175

9. D (AP-Like) DL: 5

Creating a right Riemann sum with n terms from rectangles of width x 7 3

n

4

n and

height determined by the function x3evaluated at the right endpoints (the x-values at 3 kx

for k from 1 to n) of the n intervals yields 3 kx 3 xk1

n

where x 4

n and its limit as

n is x3

3

7

dx .

10. A (AP Cal AB/BC Course and Exam Description, AB sample 10) DL: 3

Usingu‐substitutionwithu ex 1,du exdx yields, ex cos(ex 1)dx cos u du sin u C sin ex 1 C

11. C (AP-Like) DL: 4

Usingu‐substitutionwithu x ,du 1

2 xdx anddx 2 x du

when x 1, u 1

x 9, u 3

yieldssin x

x

1

9

dx 2 sin u du 1

3

12. D (AP Cal AB/BC Course and Exam Description, AB sample 12) DL: 4 At any distance x from the highway, the area of a sample rectangle with a width dx is

4 x dx . Population of the same rectangular region is dP x 1 4 x dx and the total

population is x 1 4 x dx0

4

.


13. A (AP-Like) DL: 4

1 , 1

( ) 11 , 1

xx

f x xx

can be written without the absolute value as f (x) 1 , x 1

1 , x 1

therefore, f (x)dx 1

4

f (x)dx f (x)dx 1

4

1

1

2 3 1.

Alternatively,sketchandusegeometry.14. C (AP-Like) DL: 4

The integrand can be simplified using long division; x2 6x 6

x 1 (x 5)

1

x 1

x 5 1

x 11

2

dx x2

2 5x ln x 1

x1

x2

2 10 ln 3 1

2 5 ln2

6.5 ln

3

2

15. C (2008 AB15) DL: 3

Using u-substitution with u x2 4 , du 2xdx and dx 1

2xdx yields

1

2

1

u du 1

2ln u C

1

2ln x2 4 C

16. E (1998AB/BC7) DL: 4 The integrand can be simplified using division,

x 1

x1

e

dx x2

2 ln x

x1

xe

e2

2 lne

1

2 ln1

e2

2

3

2

17. B (AP-Like) DL: 4

Usingu‐substitutionwithu arctan x ,du 1

1 x2 dx anddx 1 x2 du

when x 0, u 0 and x 1, u 4

yields u0

4

du u2

2u0

u4

1

2

2

42 0

2

32

18. D (AP Cal AB/BC Curriculum Framework, Fall 2014, AB sample 15) DL: 4

h 2 h 1 r t 1

2

dt 0.75 4t 3e1.5t

1

2

dt 0.75 1.672 2.111


19. B (AP-Like) DL: 3

If 1

210

5

g x dx 16, then g x dx5

10

32.

g x dx2

10

g x dx2

5

g x dx5

10

63 g x dx2

5

+ 32

g x dx2

5

31 and 2g x dx2

5

62

20. D (2003 AP84 modified) DL: 4

( ) ( ) ( )b

af x dx f b f a

21. C (2008, AB91) DL: 3

1

3 (1)

cos x

x2 x 21

3

dx 0.183

22. C (2003 AB88) DL: 4

If 4

2

11

4 2f t dt

, then f t dt 22

4

. Answer choice C is the only answer with this

property. 23. E (1998 AB82) DL: 4

Since F is the antiderivative of f , f (2x)dx 1

3

1

2F(2x)

x1

x3

1

2F(6) F(2)


Free Response 24. 2014 AB5-d, e (AP-Like)

(e)

11

22

2 3 2 3

2 1 3 1 2 2 3 2

x

xf x dx f x x

f f

2 : Fundamental Theorem of Calculus

31 : answer

2 8 3 2 12 6 5

25. 2013AB1-b,d

26. AP-Like (2015AB5-a modified)

2

0

2

0

14 cos

2 0 4

1 44 sin

2 4

8 8sin sin 0

2

x

x

x dx

x

2 : Fundamental Theorem of Calculus

31 : answer


27. 2003B AB2-a,c, d

28. 2014 AB1c


3

29. 2012AB4-d

30. 2010 AB3-a,d

(d) 0

0 700 800tr s ds t

1: 800t 1: integral1: answer

31. 2003AB3-d, e (AP-Like)

(e) R(90) R(0)

90 0is the average rate of change

of the rate of fuel consumption, in gal / min2, over the time interval [0, 90] min.


32. 2011AB6-c


Basic Integration

( ) ( )k f u du k f u du

[ ( ) ( )] ( ) ( )f u g u du f u du g u du

du u C

1

1

nn u

u du Cn

lndu

u Cu

1

lnu ua du a C

a

u ue du e C

Inverse Trigonometric

2 2arcsin

du uC

aa u

2 2

1arctan

du uC

a u a a

Trigonometric Functions:

sin( ) cosu du u C

cos( ) sinu du u C

2sec ( ) tanu du u C

2csc ( ) cotu du u C

sec( ) tan secu u du u C

csc( )cot cscu u du u C

HelpfultoKnow:

tan( ) ln cosu du u C

cot( ) ln sinu du u C

05 AB Integrals and Their Applications Presenter Notes...Copyright © 2017 National Math + Science...

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