AP Calculus resource Guide

7/29/2019 AP Calculus resource Guide

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AdvAnced PlAcement

Teacher resource Guide

AP|Cac Mike Reiners | Jan Bourg

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TAbl AnAlyss

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This workbook is a product of the:

CASIO TEACHER ADVISORY COUNCIL (CTAC)

theyre written By Teachers, For Teachers. CTAC represents an

embodiment of this philosophy and the foundation for which we

generate all CASIO support materials. For more information on CTAC,

our ACE training program or how to get involved email us at

[email protected]

2011 by Casio America, Inc.570 Mt. Pleasant Avenue

Dover, NJ 07801

The contents of this book can be used by the classroom teacher to make reproductions for student

use. All rights reserved. No part of this publication may be reproduced or utilized in any form by any

means, electronic or mechanical, including photocopying, recording, or by an information storage or

retrieval system without permission in writing from CASIO.

Printed in the United States of America.

Advanced PlacementTEACHER RESOURCE GUIDE

AP|Calculus

C d u C w k b k

A U T H O R S :

Mike Reiners, Jan Bourg

E D I T O R S :

Amber Branch, Amy Chow

AP_Calculus.indd 1 10/8/11


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THE AUTHORS

Mike ReineRs

Mike Reiners is the Key Advisory Teacher for Casio America.

He has been a classroom teacher for 15 years, serving as the

Math/Science Department Chair at CHOF School in St. Paul,

MN. Mike is on the Board of Directors of the Minnesota State

High School Mathematics League. He also leads the problem-

writing team for the MSHSML, and is the Head Coach of theMinnesota All-State Math Team. Mike has conducted numerous

workshops around the world concerning how to challenge and

engage gifted math students, the writing and implementation

of mathematically-rich problems, and proper pedagogical use of

educational technology.

Jan BouRg

Jan Bourg (BA 1983 Stephen F. Austin U., magna cum laude)

has over 20 years of teaching experience at the middle schooland high school levels. She has been math department chair at

Coppell High School in Texas since 2006, and was nominated

for campus Teacher of the Year in 2010. Jan is passionate about

innovative instruction and student discovery, especially in the

areas of Precalculus and Calculus. She is married with 3wonderful grown children and a brand-new grandson.



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Graph Analysis

P Calcls stents nee to e ale to... plot the graph of a

fnction ithin an aritrary vieing ino, an preict an explain theoserve local an gloal ehavior of a fnction.

nvestigating slope:

1. From the Main Men (p), enter the Graphmoe (5).

2. npt the fnction: y = x2 2.

(fs-2l)

3. dw the graph (u).



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4. race along the crve (q) to the pointhere x = 0.5.

(Notice: by efalt, Casio calclators se

x-nits of 0.1 for tracing.)

5. Pressl to mar the coorinates of the pointirectly on the crve.(use the right arro to move ot of the ay ofthe mare point.)

6. deactivate the race featre (q), then scrollpar (BB).

7. ctivate race (q), then type 2.25l tomove irectly to a point that ol otherisee sippe over hen tracing.



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8. Mar the point (l), then activate the Zoomn featre (we).

9. use the arro eypa to move the crosshairson top of the mare point, an pressl tozoom in at that point.

10. use the Zoom n shortct ey (+) to zoomin again.

(his atomatically ses the center of the

screen as the center of zoom.)

11. otice the local linearity exhiite y this

fnction. use race (q) anl to mar the

points (2.25, 3.0625) an (2.275, 3.1756), inthat orer.

Note: be sre to immediatelytrn o race

(q), ithot frther pressing the left or rightarros, after maring the secon point.



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12. ccess calclators n-Matrix moe

(p1).

13. Calclate the slope eteen the to

mare points. (usez

to create the

fraction template,a- to create the Y,ana+ to create the X; move to theenominator y pressingN.Pressl to perform the calclation.)

f all stents in a class calclate the slope

near a ierent point on y = x2 2, yo col

mae lists in the tatistics moe (p2)of the stents x-vales an calclate slopes,

then create a scatter plot to investigate the

relationship eteen these lists. his leas

stents toar the concept of erivative.



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Compaing nctional gowth

(an fning points o intesection):


2. npt these three fnctions:

y = x2 (fsl)y = 2x (2^)y = log

2x (iwr2$)


4. we are typically only concerne ith groth starting from some zero time, so only

the rst qarant is necessary; open the Vie

wino settings (e).



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5. et Xmin an Ymin to 0, Xmax an Ymax to10. Pressl to store a change vale, anN to sip over a vale that is staying thesame.

6. xit the Vie wino settings (d),an re-ra the graph (u).

7. he graph colors match the colors of the

fnction eqations. his reveals that y = log2x

is groing mch more sloly than the other

to fnctions. we can also see that y = x2

srpasses y - 2x at aot x = 2. does y - 2x ever

tae the lea again? Choose appropriate Vie

wino settings to n ot.

8. gain, exit the Vie wino settings (d),an re-ra the graph (u).



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9. y = 2x

oes eventally overtae y = x2

; this cane seen even more clearly y sing the right

an p arros to investigate frther.

Fin the points of intersection

(yyll).

10. his is the left-most point of intersection

ithin the crrent vieale omain.Pressl to mar this point, then$ to moveto the next point of intersection, anlagain to mar it.



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Fining x-vales o points on a cve,

given a y-vale:


2. npt the fnction:

(f^3$-4)


4. nter the Graph olver (y), go to thenext page of men options (u), anchoose the X-Calclator (w).



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5. ype a y-vale of 2 into the ialog ox, anpressl.

6. he left an right arros can e se to cycle

throgh all three points here y = 2; any

point can also e mare y pressingl.



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Fining oots (zeos) gaphically:


2. npt the fnction:y = x4

5

3x3 76x2 125x + 75.

(f^4$-z5N3 f^3$-76fs-125f+75l)


4. re there really only to real roots to this4th-egree polynomial?

Lets n ot y performing an

tomatic Zoom (wy).



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5. snt the nmerical laeling of the axes nice?

t loos lie there might e another root jst to

the left of 6. use the Zoom t shortct ey

(-) to chec.

6. n fact, there ere to more real roots! Lets

n them all; se the comman ithin

the Graph olver (yq).

7. nce again, e can cycle throgh the roots

an mar their locations sing the right arro

($) an xecte (l) eys.



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Table Analysis

P Calcls stents nee to e ale to...

constrct a tale of vales to investigate iscontinosehavior an conjectre a limit.

xploing limits:

1. From the Main Men (p), enter the alemoe (7).

2. npt the fnction: y =x 1

x2

+ x 2 .

(usez to create the fraction template anf for the inepenent variale x; pressl to store the fnction hen nishe.)

3. pen the ale ettings (y). Change thetart vale y typing 5l, then pressN toleave n nchange, an then type 0.5lfor the tep vale. used hen nishe to

go ac to the ale Fnction screen.



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4. display the tale (u).

5. croll on to the x-vale of 2. otice an

is liste for the vale of theY1

fnction, sggesting a point of iscontinity.

a ne ro into the tale (ew).Note: he crrent ro creates a plicate

above itself.

6. we ant to investigate hat happens as e

approach x = 2 from the left; type 2.1l togenerate a ne y-vale.

7. dd another ro beneath the crrent ro

(e), an type 2.01l.



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8. dd another ro for 2.001. what seems toe happening to the y-vales?

9. o lets approach x = 2 from the right;

scroll on to the x-vale of 1.5, a

ro, an type 1.9l.

10. to more ros, for the x-vales of

1.99 an 1.999.

o hat is happening to the y-vales?

11. Pressd to close the tale-eiting softmen, then plot the points from the tale onto

a graph (u). he limit at x = 2 oes notexist (d).



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12. Pressd to retrn to the ale, an scrollon to the x-vale of 0.5.

dd ros (ee) an type in ne valeseteen x = 0.5 an x = 1 to investigate hat

happens as x approaches 1 from the left.

13. croll on to the x-vale of 1.5; ros

(w) an type in ne vales eteen x = 1an x = 1.5 to investigate hat happens as x

approaches 1 from the right.

14. croll p ntil vales can e seen on othsies of x = 1. what o the y-vales seem to

e approaching? Pressd, an this time plotthe points from the tale as a CCd

graph (y).

15. race the graph (q); notice that the cross-hairs only trace onto actal points from the tale.

xperiment ith tracing an zooming in at the points of iscontinity.

epeat the trace/zoom investigation sing the Graph mole (p5).

Can yo estimate x 1x

2+ x 2

limx = 1

?



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Equation Solving

P Calcls stents nee to e ale to... n the zeros of

fnctions (solve eqations nmerically).

Fining oots (zeos) nmeically:

1. From the Main Men (p), enter theqation moe (f).(hef ey can e se to access mole even ithot pressinga.)

2. we ol lie to n all locations here

x3 4x = 2; aing 2 to oth sies allos s

to se the Polynomial solver (w).

3. he polynomial is egree 3 (w).



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4. npt the coecients of the polynomial,pressingl to store each vale.(o sip over the x2 term, either type0l,or simply press$.)

5. LV the polynomial (q); all three valesfor hich x3 4x = 2 are shon in rone

form in a soltion matrix. Longer ecimal

approximations of each vale can e seen in

the loer right corner, an cycle throgh

sing the p an on arros.



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olving any eqation

nmeically:

1. From the Main Men (p),enter the qation moe (f).

2. elect the olver (e).

3. ype the eqation x3 4x = 2.

(f^3$-4fL.

n2l)Note: Yor isplaye x-vale may e ierent

than the one shon here.

his x-vale is notnecessarily a soltion to the

eqation; it is simply the calclators crrent

store vale for the variale x.

ometimes, the P Calcls exam ill

provie a graph for stent se, sch as the

one shon elo:

eeing this graph, stents shol no that

there is one x-vale eteen 0 an 1 for hich

x3 4x = 2, an another sch vale eteen 1

an 2. hese ons can e se to ientify

particlar soltions sing the olver.



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4. Move on one line (N), an enter a loer

on of 0 (0l), then enter an pperon of 1 (1l).

5. LV the eqation (u). he soltion thatexists eteen x = 0 an x = 1 is shon, along

ith verication that oth the left an rightsies of the eqation evalate to 2.

6. Pressd to retrn to the eqation entry

screen, an this time enter a loer on of 1an an pper on of 2.

7. LV the eqation (u); e no see thesoltion that exists eteen x = 1 an x = 2.



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Derivatives

P Calcls stents nee to e ale to...

present the erivative graphically, nmerically, an analytically.

Conceptalizing the limit o the

ieence qotient:

1. From the Main Men (p),enter the Geometry moe an press (l).

2. f the initial elcome screen, shon elo,

isplays, clear it (d).

3. Presse, then$ to enter the dra pecialmen. croll on an select item #7 (l) to

inpt a fnction in f(x) format.



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4. npt the fnction f(x) = 2 sin 3

x.

(2h((Lcz3)f)l)

5. cess the uP screen (Lp

).

6. croll on to the Gri setting an change

it to Line (e).croll on to the xes setting an change

it to cale (e).

7. Pressd to retrn to the Geometry ino.



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5. he tangent line at x = 0 is isplaye in green;

se the left an right arros to trace along

the sine crve, oserving ho the tangent line

changes.

6. sefl time-saving shortct hile tracing:

simply type in a esire x-vale!

7. t any point hile tracing, pressl to marthe point an leave a more permanent setch

of the tangent line at that point. n this ay,

stents can see the locs of tangent lines as

a onary that enes the crve.

8. f necessary, open the etch men (r) anclear the setch (q).



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Calclating the slope o a cve

at a point:

1. From the Main Men (p),enter the n-Matrix moe (1).

2. pen the soft-men of MH templates (r).

3. Create a rst erivative template (r).

4. npt the fnction expression: 2x + 3 .

(Ls2f+3)



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5. Move over to the small

entry el ($$) an inpt 1

4

for the x-vale.

(nz1N4)

6. Pressl to evalate

the erivative of f(x) = 2x + 3 at x = 14

.

he erivatives exact vale is

isplaye in rationalize form.

7. hex ey toggles the reslt eteen exactform an a ecimal approximation.



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Compaing chaacteistics o

the gaphs o f, f , an f :

1. From the Main Men (p), enter the Graphmole (5).

2. npt the fnction y = x4 + 5x3 3x2 20x 4.(f^4$+5f^3$-3fs-20f-4l)


4. t seems lie e nee to zoom ot, t the

x-vales for or ino are ne; its the range

of or y-vales that nee to change.

pen the Zoom men (w),an choose the FC setting (w).



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13. pen the etch men (r), go to the nextpage of options (u), an select the Verticalline tool (r).

14. use the left an right arros to move to an

x-vale here the rst erivative (re graph)

eqals zero; pressl to setch a vertical lineat that vale.

15. Highlighting all of these vales helps to sho

the relationship eteen the zeros of the rst

erivative an the maxima/minima of f(x).

16.d to the fnction entry screen an openthe ption men again (i).



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21. unfortnately, the efalt color for theY3fnction (green) is the same as the efalt

color se y the etch men. o change the

color of theY3 graph, se the FM

setting (L5).

22. heY1 graph (le) egins lining yello.

ote the hint in the stats ar at the top ofthe screen, hich sggests e can se the p

an on arros to cycle eteen the three

graphs; pressN tice to highlightY3.

23. Pressl to open the FM men fortheY3 graph.

24.elect the Line Color setting (2),an change the color to Magenta (4).



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25. Pressd to conrm the ne settingan re-ra the graphs.

26. o e can se the Vertical line tool from

the etch men to emonstrate ho the

zeros of the secon erivative relate to the

maxima/minima of f (x) an the inection

points of f(x).



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xploing elationships etween

the ehavio o f(x) an the signs

o f an f :

Note: this activity is a contination of

the xploring Characteristics activity

27. From the Main Men (p),enter the ale moe (7).

28. pen the ale ettings (y).

29. Change the tart vale to 6, the n vale

to 6, an the tep vale to 0.5; selto store each vale.

30.etrn to the ale entry screen (d) ancreate the bL (u).



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31. crolling throgh the tale, e can see thatY1(the original fnction) is ecreasing hileY2

(the rst erivative) is negative, an thatY1 is

increasing hileY2 is positive.

32. n aition, e can relate the sign ofY3 (thesecon erivative) to the concavity ofY1; note

the sign ofY3 hile 1 x 2.5.

33. Qicly chec the connecte graph (y);

note the concavity ofY1 (the le graph) onthe interval 1 x 2.5, an compare it to the

sign ofY3. Many other important calcls

relationships can e emphasize y

comparing the vales in the tale an the

visal evience in the graph.



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Integrals

P Calcls stents nee to e ale to... evalate enite integrals

an analyze their graphical representations.

Calclate an isplay the aea

ne a cve:


2. npt the fnction y = 3. (3l)




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4. nter the Graph olver (y), go to the nextpage of men options (u), an open theintegral men (e).

5. elect the enite integral featre (q).

6. Pressl to inicate a loer on of x = 0,then type4l to inicate an pper onof x = 4.

he area ner y = 3 is shonas a shae rectangle, representing

4

x = 0

3.

tents can easily verify this area

sing geometry.



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11. etrn to the graph entry screen (d), scroll

p to highlightY1, an dL the fnction(wq).

12. npt the fnction: y = x3

.

(L()


14. his time, calclate the enite integral from

x = 1 to x = 3. (yueq1l4l)

tents cannot calclate the area sing

geometry any more, t they can

conceptally grasp the vale of the integral y

estimating the nmer of shae gri sqares.



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15. etrn to the graph entry screen (d), scrollp to highlightY1, an dL the fnction

(wq).

16. npt the fnction y = sin x. (h)


18. nter the Graph olver (y), go to the nextpage of men options (u), an open theintegral men (e), t no select the featre (w).



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19. Press$ to move to the root near 3, anesignate that root as the loer on of the

enite integral (l).

20.Move to the root near +3, an esignate that

root as the pper on.

he vale of the integral is 0, shoing that

the to shae regions are eqal in area t

opposite in irecte area. he total area is 4,

so each region has area = 2!



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valate a efnite integal:

1. From the Main Men (p),enter the n-Matrix moe (1).

2. pen the soft-men of MH templates (r).

3. Go to the next page of options (u),an select dx (q).

4. pen the ption men (i), go to thenext page (u), an select the uMCs-men (r).



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5. Create the solte vale

groping symols (q).

6. npt the expression 2 sin (ex4

) + 1.(2h(LG

zfN4$$)+1)

7. Press the right arro tice to move to theloer on entry el.

nter 0 for the loer on, then scroll p an

enter 6 for the pper on.

Pressl to evalate the integral.

his vale ol, for example, represent the

total istance travele y a particle from time

t = 0 to t = 6, if its velocity ere given y v(t) = 2 sin (ex

4) + 1.



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Calclate an isplay the aea

etween two cves:


2. npt these to fnctions:y = tan x (kfl)y = 2 x (2Lsfl)

3. dw the graphs (u).

4. o focs on the region eteen the tocrves near the origin, open the Zoom men

(w) an select bX (q).



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5. use the arro eypa to move the crosshairson an to the left of the origin. Mar the

loer left corner of the zoom ox (l), thenmove aove an to the right of the other in-

tersection point. Complete the ox (l).

6. nter the Graph olver (y), go to the nextpage of men options (u), an open theintegral men (e).

7. elect the ntersection featre (e).

8. Pressl to esignate the left intersectionpoint as the loer on, then move to the

right ($) an esignate the right intersec-tion point as the pper on (l).



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he area eteen the crves is isplaye in

the loer right corner of the screen. his

reslt can e se to emonstrate the

stractive properties of integrals, or to egin

an investigation here this region is rotate

aot an axis.

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AP Calculus resource Guide