Santa Fe ISD Year Overview for AP Calculus AB /10/17

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1 St Fe ISD Yer Overview for AP Clculus AB /10/17 From the College Bord AP Clculus AB is roughly equivlet to first semester college clculus course devoted to topics i differetil d itegrl clculus The AP course covers topics i these res, icludig cocepts d skills of limits, derivtives, defiite itegrls, d the Fudmetl Theorem of Clculus The course teches studets to pproch clculus cocepts d prolems whe they re represeted grphiclly, umericlly, lyticlly, d verlly, d to mke coectios mogst these represettios Studets ler how to use techology to help solve prolems, experimet, iterpret results, d support coclusios Mthemticl Prctices for AP Clculus (MPACs) The MPACs cpture the importt spects of the work tht mthemticis egge i, t the level of competece expected of AP Clculus studets The MPACs explicitly rticulte the ehviors i which studets eed to egge i order to chieve coceptul uderstdig i the AP Clculus courses Ech cocept d topic ddressed i the course c e liked to oe or more of the MPACs The MPACs re ot iteded to e viewed s discrete items tht c e checked off list; rther, they re highly iterrelted tools tht should e utilized frequetly d i diverse cotexts MPAC 1: Resoig with defiitios d theorems Studets c: use defiitios d theorems to uild rgumets, to justify coclusios or swers, d to prove results cofirm tht hypotheses hve ee stisfied i order to pply the coclusio of theorem c pply defiitios d theorems i the process of solvig prolem d iterpret qutifiers i defiitios d theorems (eg for ll, there exists ) e develop cojectures sed o explortio with techology; d f produce exmples d couterexmples to clrify uderstdig of defiitios, to ivestigte whether coverses of theorems re true or flse, or to test cojectures MPAC 2: Coectig cocepts Studets c: relte the cocept of limit to ll spects of clculus use the coectio etwee cocepts (eg rte of chge d ccumultio) or processes (eg differetitio d its iverse process, tidifferetitio) to solve prolems; c coect cocepts to their visul represettios with d without techology d idetify commo uderlyig structure i prolems ivolvig differet cotextul situtios MPAC 3: Implemetig lgeric/computtiol processes Studets c: select pproprite mthemticl strtegies sequece lgeric/computtiol procedures logiclly c complete lgeric/computtio processes correctly d pply techology strtegiclly to solve prolems e tted to precisio grphiclly, umericlly, lyticlly, d verlly d specify uits of mesure f coect the results of lgeric/computtiol processes to the questio sked MPAC 4: Coectig multiple represettios Studets c: ssocite tles, grphs, d symolic represettios of fuctios develop cocepts usig grphicl, symolicl, verl, or umericl represettios with d without techology c idetify how mthemticl chrcteristics of fuctios re relted i differet represettios d extrct d iterpret mthemticl cotet from y presettio of fuctio (eg utilize iformtio from tle of vlues) e costruct oe represettiol form from other (eg tle from grph or grph from give iformtio); d f cosider multiple represettios (grphicl, umericl, lyticl, d verl) of fuctio to select or costruct useful represettio for solvig prolem MPAC 5: Buildig ottiol fluecy Studets c: kow d use vriety of ottios (eg f ( x), y, dy ) dx coect ottio to defiitios (eg reltig ottio for the defiite itegrl to tht of the limit of Riem sum) c coect ottio to differet represettios (grphicl, umericl, lyticl, d verl), d d ssig meig to ottio, ccurtely iterpretig the ottio i give prolem d cross differet cotexts MPAC 6: Commuictig Studets c: clerly preset methods, resoig, justifictios, d coclusios use ccurte d precise lguge d ottio c expli the meig of expressios, ottio, d results i terms of cotext (icludig uits) d expli the coectios mog cocepts e criticlly iterpret d ccurtely report iformtio provided y techology; d f lyze, evlute d compre the resoig of others St Fe ISD Mthemtics Deprtmet Pge 1 of 7

2 St Fe ISD Yer Overview for AP Clculus AB /10/17 1 st Nie Weeks Istructiol Uits 38 Dys August 22 d Octoer 14 th Lor Dy: Septemer 5 th Fculty/Stff PD Dy: Octoer 10 th BIG IDEA 1: Limits 20 dys (The umer of dys is lwys pproximte) Edurig Uderstdig 11: The cocept of limit c e used to uderstd the ehviors of fuctios Lerig Ojectives LO 11A: Express limits symoliclly usig correct ottio LO 11A: Iterpret limits expressed symoliclly Essetil Kowledge EK 11A1: Give fuctio f, the limit of f(x) s x pproches c is rel umer if f(x) c e mde ritrrily close to y tkig x sufficietly close to c (ut ot equl to c If the limit exists d is rel umer, the the commo ottio is lim f( x) = x c EK 11A2: The cocept of limit c e exteded to iclude oe-sided limits, limits t ifiity, d ifiite limits EK 11A3: A limit might ot exist for some fuctios t prticulr vlues of x Some wys tht the limit might ot exist re if the fuctio is uouded, if the fuctio is oscilltig er this vlue, or if the limit from the left does ot equl the limit from the right Lerig Ojective LO 11B: Estimte limits of fuctios Essetil Kowledge EK 11B1: Numericl d grphicl iformtio c e used to estimte limits Lerig Ojective LO 11C: Determie limits of fuctios Essetil Kowledge EK 11C1: Limits of sums, differeces, products, quotiets, d composite fuctios c e foud usig the sic theorems of limits d lgeric rules EK 11C2: The limit of fuctio my e foud y usig lgeric mipultio, lterte forms of trigoometric fuctios, or the squeeze theorem EK 11C3: Limits of the idetermite forms 0 0 d my e evluted usig L Hospitl s Rule Lerig Ojective LO 11D: Deduce d iterpret ehvior of fuctios usig limits Essetil Kowledge EK 11D1: Asymptotic d uouded ehvior of fuctios c e explied d descried usig limits EK 11D2: Reltive mgitudes of fuctios d their rtes of chge c e compred usig limits Edurig Uderstdig 12: Cotiuity is key property of fuctios tht is defied usig limits LO 12A: Alyze fuctios for itervls of cotiuity or poits of discotiuity EK 12A1: A fuctio f is cotiuous t x = c provided tht f(c) exists, lim f( x) exists, d lim f( x) = f( c) x c x c EK 12A2: Polyomil, rtiol, power, expoetil, logrithmic, d trigoometric fuctios re cotiuous t ll poits i their domis EK 12A3: Types of discotiuities iclude removle discotiuities, jump discotiuities, d discotiuities due to verticl symptotes LO 12B: Determie the pplictio of importt clculus theorems usig cotiuity EK 12B1: Cotiuity is essetil coditio for theorems such s the Itermedite Vlue Theorem, the Extreme Vlue Theorem, d the Me Vlue Theorem St Fe ISD Mthemtics Deprtmet Pge 2 of 7

3 St Fe ISD Yer Overview for AP Clculus AB /10/17 1 st Nie Weeks Istructiol Uits cotiued BIG IDEA 2: Derivtives 16 dys (The umer of dys is lwys pproximte) Edurig Uderstdig 21: The derivtive of fuctio is defied s the limit of differece quotiet d c e determied usig vriety of strtegies LO 21A: Idetify the derivtive of fuctio s the limit of differece quotiet f( + h) f( ) f( x) f( ) EK 21A1: The differece quotiets d express the verge rte of chge of fuctio over itervl h x ( ) ( ) EK 21A2: The istteous rte of chge of fuctio t poit c e expressed y lim f + h f ( ) ( ) or lim f x f, provided tht the limit h 0 h x x exists These re commo forms of the defiitio of the derivtive d re deoted f ( ) ( ) ( ) EK 21A3: The derivtive of f is the fuctio whose vlue t x is lim f + h f provided this limit exists h 0 h dy EK 21A4: For y = f( x), ottios for the derivtive iclude, f ( x ), d y dx EK 21A5: The derivtive c e represeted grphiclly, umericlly, lyticlly, d verlly LO 21B: Estimte derivtives EK 21B1: The derivtive t poit c e estimted from iformtio give i tles or grphs LO 21C Clculte derivtives EK 21C1: Direct pplictio of the defiitio of the derivtive c e used to fid the derivtive for selected fuctios, icludig polyomil, power, sie, cosie, expoetil, d logrithmic fuctios EK 21C2: Specific rules c e used to clculte derivtives for clsses of fuctios, icludig polyomil, rtiol, power, expoetil, logrithmic, trigoometric, d iverse trigoometric EK 21C3: Sums, differeces, products, d quotiets of fuctios c e differetited usig derivtive rules EK 21C4: The chi rule provides wy to differetite composite fuctios EK 21C5: The chi rule is the sis for implicit differetitio EK 21C6: The chi rule c e used to fid the derivtive of iverse fuctio, provided the derivtive of tht fuctio exists Review (1 Dy) St FE ISD District 1 st Nie-Weeks Test (1 Dy) St Fe ISD Mthemtics Deprtmet Pge 3 of 7

4 St Fe ISD Yer Overview for AP Clculus AB /10/17 2 d Nie Weeks Istructiol Uits 40 Dys Octoer 17 th Decemer 16 th TAKS Retests: Octoer 17 th -20 th Pret Cofereces: Octoer 26th d 27th Erly Relese: Octoer 26th, 27th, d 28th Thksgivig Brek: Novemer 21 st -25 th EOC Retests: Decemer 5-9 th BIG IDEA 2: Derivtives cotiued 37 dys Edurig Uderstdig 21: The derivtive of fuctio is defied s the limit of differece quotiet d c e determied usig vriety of strtegies cotiued LO 21D: Determie higher order derivtives EK 21D1: Differetitig f produces the secod derivtive f, provided the derivtive of f exists; repetig this process produces higher order derivtives of f d 2 y EK 21D2: Higher order derivtives re represeted with vriety of ottios For y = f( x), ottios for the secod derivtive iclude, f ( x ), d y dx 2 d y Higher order derivtives c e deoted or f ( ) ( x) dx Edurig Uderstdig 22: A fuctio s derivtive, which is itself fuctio, c e used to uderstd the ehvior of the fuctio LO 22A: Use the derivtives to lyze properties of fuctio EK 22A1: First d secod derivtives of fuctio c provide iformtio out the fuctio d its grph icludig itervls of icrese or decrese, locl (reltive) d glol (solute) extrem, itervls of upwrd or dowwrd cocvity, d poits of iflectio EK 22A2: Key fetures of fuctios d their derivtives c e idetified d relted to their grphicl, umericl, d lyticl represettios EK 22A3: Key fetures of the grphs of f, f, d f re relted to oe other LO 22B: Recogize the coectio etwee differetiility d cotiuity EK 22B1: A cotiuous fuctio my fil to e differetile t poit i its domi EK 22B2: If fuctio is differetile t poit, the it is cotiuous t tht poit Edurig Uderstdig 23: The derivtive hs multiple iterprettios d pplictios icludig those tht ivolve istteous rtes of chge LO 23A: Iterpret the meig of derivtive withi prolem EK 23A1: The uit for f ( x) is the uit for f divided y the uit for x EK 23A2: The derivtive of fuctio c e iterpreted s the istteous rte of chge with respect to its idepedet vrile LO 23B: Solve prolems ivolvig the slope of tget lie EK 23B1: The derivtive t poit is the slope of the lie tget to grph t tht poit o the grph EK 23B2: The tget lie is the grph of loclly lier pproximtio of the fuctio er the poit of tgecy LO 23C: Solve prolems ivolvig relted rtes, optimiztio, d rectilier motio EK 23C1: The derivtive c e used to solve rectilier motio prolems ivolvig positios, speed, velocity, d ccelertio EK 23C2: The derivtive c e used to solve relted rtes prolems, tht is, fidig rte t which oe qutity is chgig y reltig it to other qutities whose rtes of chge re kow EK 23C3: The derivtive c e used to solve optimiztio prolems, tht is fidig mximum or miimum vlue of fuctio over give itervl LO 23D: Solve prolems ivolvig rtes of chge i pplied cotexts EK 23D1: The derivtive c e used to express iformtio out rtes of chge i pplied cotexts Review (2 Dys) St FE ISD District Cumultive 1 st Semester Exm (1 Dy) St Fe ISD Mthemtics Deprtmet Pge 4 of 7

5 St Fe ISD Yer Overview for AP Clculus AB /10/17 3 rd Nie Weeks Istructiol Uits 46 Dys Jury 5 th Mrch 10 th Jury 5 Mrch 11 Techer Workdy: Jury 3 rd Fculty/Stff PD Dys: Jury 16 th d Ferury 20 th Sprig Brek: Mrch 13 th -17 th BIG IDEA 2: Derivtives cotiued 12 dys Edurig Uderstdig 23: The derivtive hs multiple iterprettios d pplictios icludig those tht ivolve istteous rtes of chge LO 23E: Verify solutios to differetil equtios EK 23E1: Solutios to differetil equtios re fuctios or fmilies of fuctios EK 23E2: Derivtives c e used to verify tht fuctio is solutio to give differetil equtio LO 23F: Estimte solutios to differetil equtios EK 23F1: Slope fields provide visul clues to the ehvior of solutios to first order differetil equtios Edurig Uderstdig 24: The Me Vlue Thereom coects the ehvior of differetile fuctio over itervl to the ehvior of the derivtive of tht fuctios t prticulr poit i the itervl LO 24A: Apply the Me Vlue Theorem to descrie the ehvior of fuctio over itervl EK 24A1: If fuctio f is cotiuous over the itervl [, ] d differetile over the itervl (, ), the Me Vlue Theorem gurtees poit withi tht ope itervl where the istteous rte of chge equls the verge rte of chge over the itervl BIG IDEA 3: Itegrls d the Fudmetl Theorem of Clculus 32 Dys Edurig Uderstdig 31: Atidifferetitio is the iverse process of differetitio LO 31A: Recogize tiderivtives of sic fuctios EK 31A1: A tiderivtive of fuctio f is fuctio g whose derivtive is f EK 31A2: Differetitio rules provide the foudtio for fidig tiderivtives Edurig Uderstdig 32: The defiite itegrl of fuctios over itervl is the limit of Riem sum over tht itervl d c e clculted usig vriety of strtegies LO 32A(): Iterpret the defiite itegrl s the limit of Riem sum LO 32A(): Express the limit of Riem sum i itegrl ottio EK 32A1: A Riem sum, which requires prtitio of itervl, I, is the sum of products, ech of which is the vlue of the fuctio t poit i suitervl multiplied y the legth of tht suitervl of the prtitio EK 32A2: The defiite itegrl of cotiuous fuctio f over the itervl [, ], deoted y f ( x ) dx, is the limit of Riem sums s the widths of the suitervls pproch 0 Tht is ( ) lim ( i ) x i is vlue i the ith suitervl, f x dx = f x x i where x i is the width of the ith suitervl, is mx xi 0i= 1 the umer of suitervls, d mx x i is the width of the lrgest suitervl Aother form of the defiitio is ( ) lim f x dx = f ( x ) x i i where i=1 x i = d x is vlue i the ith suitervl i EK 32A3: The iformtio i defiite itegrl c e trslted ito the limit of relted Riem sum, d the limit of Riem sum c e writte s defiite itegrl LO 32B: Approximte defiite itegrl EK 32B1: Defiite itegrls c e pproximted for fuctios tht re represeted grphiclly, umericlly, lgericlly, d verlly EK 32B2: Defiite itegrls c e pproximted usig left Riem sum, right Riem sum, midpoit Riem sum, or trpezoid sum; pproximtios c e computed usig either uiform or ouiform prtitios LO 32C: Clculte defiite itegrl usig res d properties of defiite itegrls EK 32C1: I some cses, defiite itegrl c e evluted y usig geometry d the coectio etwee the defiite itegrl d re EK 32C2: Properties of defiite itegrls iclude the itegrl of costt times fuctio, the itegrl of the sum of two fuctios, reversl of limits of itegrtio, d the itegrl of fuctio over djcet itervls EK 32C3: The defiitio of the defiite itegrl my e exteded to fuctio with removle or jump discotiuities d x EK 33A2: If f is cotiuous fuctio o the itervl [, ], the f () t dt = f ( x), where x is etwee d dx x EK 33A3: Grphicl, umericl, lyticl d verl represettios of fuctio f provide iformtio out the fuctio g defied s g( x) = f () t dt St Fe ISD Mthemtics Deprtmet Pge 5 of 7

6 St Fe ISD Yer Overview for AP Clculus AB /10/17 3 rd Nie Weeks Istructiol Uits cotiued BIG IDEA 3: Itegrls d the Fudmetl Theorem of Clculus cotiued Edurig Uderstdig 33: The Fudmetl Thereom of Clculus, which hs two distict formultios, coects differetitio d itegrtio LO 33A: Alyze fuctios defied y itegrl x 2 EK 33A1: The defiite itegrl c e used to defie ew fuctios, for exmple, f ( x) = e t dt 0 LO 33B(): Clculte tiderivtives LO 33B(): Evlute defiite itegrls EK 33B1: The fuctio defied y f iis tiderivtive of f EK 33B2: If f is cotiuous o the itervl [, ] d F is tiderivtive of f, the f ( x ) dx = F ( ) F ( ) EK 33B3: The ottio f ( x ) dx = F ( x ) + C mes tht F( x) = f( x) d f ( x) dx is clled the idefiite itegrl of the fuctio f EK 33B4: My fuctios do ot hve closed form tidervtives EK 33B5: Techiques for fidig tiderivtives iclude lgeric mipultio such s log divisio d completig the squre d sustitutio of vriles Edurig Uderstdig 34: The defiite itegrl of fuctio over itervl is mthemticl tool with my iterprettios d pplictios ivolvig cculumtio LO 34A: Iterpret the meig of defiite itegrl withi prolem EK 34A1: A fuctio defied s itegrl represets ccumultio of rte of chge EK 34A2: The defiite itegrl of the rte of chge of qutity over itervl gives the et chge of tht qutity over tht itervl EK 34A3: The limit of pproximtig Riem sum c e iterpreted s defiite itegrl LO 34B: Apply defiite itegrls to prolems ivolvig the verge vlue of fuctio 1 EK 34B1: The verge vlue of fuctio f over itervl [, ] is f ( x ) dx LO 34C: Apply defiite itegrls to prolems ivolvig motio EK 34C1: For prticle i rectilier motio over itervl of time, the defiite itegrl of velocity represets the prticle s displcemet over the itervl of time, d the defiite itegrl of speed represets the prticle s totl distce trveled over the itervl of time Review (1 Dy) St FE ISD District 3 rd Nie-Weeks Test (1 Dy) St Fe ISD Mthemtics Deprtmet Pge 6 of 7

7 St Fe ISD Yer Overview for AP Clculus AB /10/17 4 th Nie Weeks Istructiol Uits 48 Dys Mrch 20th My 26 th Erly Relese: Thursdy, April 13 th District Holidy: Fridy, April 14 th EOC Week: My 1 st -5 th AP Clculus AB Exm: My 9 th Erly Relese: Fridy, My 26 th Memoril Dy District Holidy: Mody, My 29 th Techer Work Dy: Tuesdy, My 30 th Fculty/Stff PD Dy: Wedesdy, My 31 st Techer PD Exchge Dy: Thursdy, Jue 1st BIG IDEA 3: Itegrls d the Fudmetl Theorem of Clculus cotiued 17 Dys Edurig Uderstdig 34: The defiite itegrl of fuctio over itervl is mthemticl tool with my iterprettios d pplictios ivolvig cculumtio cotiued LO 34D: Apply defiite itegrls to prolems ivolvig re d volume EK 34D1: Ares of certi regios i the ple c e clculted with defiite itegrls EK 34D2: Volumes of solids with kow cross sectios, icludig discs d wshers, c e clculted with defiite itegrls LO 34E: Use the defiite itegrl to solve prolems i vrious cotexts EK 34E1: The defiite itegrl c e used to express iformtio out ccumultio d et chge i my pplied cotexts Edurig Uderstdig 35: Atidifferetitio is uderlyig cocept ivolved i solvig seprle differetil equtios Solvig seprle differetil equtios ivolves determiig fuctio or reltio give its rte of chge LO 35A: Alyze differetil equtios to oti geerl d specific prolems EK 35A1: Atidifferetitio c e used to fid specific solutios to differetil equtios with give iitil coditios, icludig pplictios to motio log lie d expoetil growth d decy EK 35A2: Some differetil equtios c e solved y seprtio of vriles EK 35A3: Solutios to differetil equtios c e solved y seprtio restrictios dy EK 35A4: The fuctio F defied y F ( x) = c + f () t dt is geerl solutio to the differetil equtio = f ( x ), d F ( x) = y0 + f () t dt is prticulr dx dy solutio to the differetil equtio f ( x ) dx = stisfyig F( ) = y 0 LO 35B: Iterpret, crete, d solve differetil equtios from prolems i cotext EK 35B1: The model for expoetil growth d decy tht rises from the sttemet The rte of chge of qutity is proportiol to the size of the qutity is dy ky dx = Free Respose Prctice Multiple Choice Prctice AP Exm Preprtio 15 Dys (The umer of dys is lwys pproximte) AP Clculus AB Exm Tuesdy, My 9, 2017, 8 m After the AP Exm Suggestios Project to icorporte this yer s lerig A look t college mth requiremets d expecttios, icludig plcemet exms Advced itegrtio techiques 9 Dys (The umer of dys is lwys pproximte) Review (2 Dy) St FE ISD District Cumultive 2 d Semester Test (1 Dy) St Fe ISD Mthemtics Deprtmet Pge 7 of 7