Clculus AP U5 Itegrtio (AP) Nme: Big ide Clculus is etire rch of mthemtics. Clculus is uilt o two mjor complemetry ides. The first is differetil clculus, which is cocered with the istteous rte of chge. This c e illustrted y the slope of tget to fuctio's grph. The secod is itegrl clculus, which studies the res uder curve. These two processes ct iversely to ech other. Clculus llows you to fid optiml solutios to mthemticl epressios d is used i medicie, egieerig, ecoomics, computer sciece, usiess, physicl scieces, sttistics, d my more res. Feedck & Assessmet of Your Success Dte Pges Topics.5dys Atiderivtives & Idefiite -4 Itegrtio (AP) Jourl # dys Estimtig Ares Numericl 5- Itegrtio (AP) Jourl #.5dys Sigm Limits of Fiite Sums (AP) -4 Jourl # 5-7 Defiite Itegrls (AP) Jourl #4 dys FTC & Averge Vlue (AP) 8- Jourl #5 dys More FTC & MVT for Itegrls -7 (AP) Jourl #6 dys Iterpret Itegrls (AP) 8- Jourl #7 Fiished ssigmet pges? Mde correctios? Summrized otes i jourl? Added your ow epltios? How my etr prctice questios did you try i ech topic? Tettive TEST dte: Questios to sk the techer:
Clculus AP U5 Itegrtio (AP) Nme: ASSIGNMENT Atiderivtives & Idefiite Itegrtio (AP). So fr, give fuctio, we kow how to fid rte of chge usig the, ut wht if ll we kew ws how fuctio ws chgig with time, d we wted to fid out out the fuctio itself? e. you kow velocity fuctio ut wht to kow the fuctio.. Grph possile F( ) for the give grph of F ( ) A fuctio F is clled tiderivtive of f o itervl I if F ( ) = f ( ) for ll i I Theorem: If F is tiderivtive of f o itervl I, the the most geerl tiderivtive of f o I is F ( ) + C where C is ritrry costt.. Fid the tiderivtives ) f ( ) = Epli why there re ifiitely my swers: 4. Differet ottio: Fid the idefiite itegrls ) 4 d ) f ( ) = ) d 8 c) f ( ) = + + c) ( + ) d d) f ( ) = 5si + d) d 4 + 4 e) f ( ) = 7sec t + 4 6 e) + + d f) f ( ) = 8 f) 5 f ( ) = 8cos, fid f () + g) If g)
Clculus AP U5 Itegrtio (AP) Nme: I ech of the grphs elow, determie which curve is f (), d which curve is the tiderivtive F( ). 5. 6. 7. A differetil equtio is equtio eplicitly solved for derivtive of prticulr equtio. Solvig differetil equtio ivolves fidig the origil fuctio from which the derivtive cme. The solutio ivolves +C. The solutio uses to fid the specific vlue of C. A seprle differetil equtio is oe where it is possile to seprte ll the d y vriles. dy Tke Leiiz Form: = f ( ) d Chge to Differetile Form: dy = f ( ) d Perform tiderivtive or idefiite itegrtio opertio: dy = f ( ) d 8. Sometimes, it is difficult or impossile to fid the tiderivtive of fuctio, ut we c still gther ifo out it grphiclly. A directio field, which shows the slope t give poits, c e used to sketch grph of the tiderivtive of fuctio. E. If f ( ) = si ( + ) d f ( 0) =, sketch f (). Solve the differetil equtios. 9. dy Give = 4 + d y () = 6, fid the d equtio for y 0. 5 + If f ( ) = d f ( ) =, fid f ()
4 Clculus AP U5 Itegrtio (AP) Nme:. If f ( ) = si + with f ( 0) = 7 d. f ( 0) =, fid f () Fid y().. Suppose the rte of chge of cocetrtio of vitmi i the loodstrem t time t is give y dc = 0.e dt 0. t If there is iitilly mg of the vitmi i the loodstrem, the wht is the cocetrtio s fuctio of time? 4. A ll is throw upwrd with speed of 0 m/s from uildig tht is 0m tll. Fid formul descriig the height of the ll ove the groud t secods lter. 4
5 Clculus AP U5 Itegrtio (AP) Nme: ASSIGNMENT Estimtig Ares Numericl Itegrtio (AP). A scout moves orth i the forest for distce of 50m i 50sec, stops for 0sec, the moves south 80m i 60sec.. Sketch displcemet-time grph c. Fid the re uder the velocity-time grph. d. Wht does the slope of d-t grph represet?. Sketch velocity-time grph e. Wht does the re uder the v-t grph represet? f. Wht would the slope of v-t grph represet? g. Wht would the re uder the -t grph represet?.. Fid the verge ccelertio from t= to t=6. c. Fid the distce trvelled from t= to t=6 d. Sketch d-t grph. Fid the istteous rte of chge t t=. Wht does it represet? 5
6 Clculus AP U5 Itegrtio (AP) Nme: Clculus swers two very importt questios. The first, how to fid the istteous rte of chge, we swered with our study of the derivtive. We re ow redy to swer the secod questio: how to fid the re of irregulr regios.. Gol: To fid the re of the shded regio R tht 4. lies ove the -is, elow the grph of y = d etwee the verticl lies = 0 Right Riem Approimtio Method: RRAM d =. Left Riem Approimtio Method: LRAM R = f ( ) + f ( ) + + f ( ) = f ( k ) k = Mid Riem Approimtio Method: MRAM L = f ( ) + f ( ) + + f ( ) 0 = f ( k ) k = 0 we divide the itervl [, ] ito suitervls of equl width = d k = + k M = f k= 0 + k k + ( ) 6
7 Clculus AP U5 Itegrtio (AP) Nme: 5. For cotiuous fuctios f, Notes: If you do thik of the itegrl Defiite itegrl of f from to is is vrile s re uder curve, the keep the followig i mid: f ( ) d = lim f ( ) k = where the itervl [, ] is divided ito suitervls of equl width the let k f ( ) d = f ( t) dt f ( ) d is clled while f ( ) d is clled f ( ) d represets ccumultio over itervl [, ] ot sice re is lwys positive! 6. Velocity fuctio of prticle movig left/right o = + o [ ] horizotl lie is V ( t) t cost.5 0,5. ) Wht does egtive velocity o [ ],4.5 me for positio of prticle? ) Fid the Right Riem sum usig te suitervls. c) The sum you foud, wht does it represet? d) The ctul vlue of the re uder is -.887m wht is the % error i your swer? 7
8 Clculus AP U5 Itegrtio (AP) Nme: 7. The velocity, m/s, fuctio of projectile fired stright up ito the ir is f ( t) = 60 9.8t. ) Use the Left Riem sum with si suitervls to estimte how fr the projectile rises durig the first sec. ) How close (% error) do the sums come to the ctul vlue of 45.9 m? 8. Use the midpoit rule to fid pproimtio to d + usig = 4 9. Use the midpoit rule to pproimte + 5d usig = 8
9 Clculus AP U5 Itegrtio (AP) Nme: 9 Trpezoidl Rule )] ( ) ( )... ( ) ( ) ( [ ) ( 0 f f f f f T d f + + + + = where = d k k = + Simpso s Rule )] ( ) ( 4 ) ( )... ( 4 ) ( ) ( 4 ) ( [ ) ( 0 f f f f f f f S d f + + + + + + = where is eve d = 0. Use the trpezoidl rule to fid pproimtio to d e usig 4 =.. Use Simpso s rule to fid pproimtio to d e usig 4 =..
0 Clculus AP U5 Itegrtio (AP) Nme:. ) Assume f ( ) is cotiuous, pproimte ) Approimte f () f ( ) d usig LRAM d RRAM d TRAP 0 4. ) Assume f ( ) is cotiuous, pproimte ) Approimte f (7) 8 f ( ) d usig LRAM d RRAM d TRAP 0
Clculus AP U5 Itegrtio (AP) Nme: 5. So, we ow hve methods for pproimtig As emple, let s cosider itegrl tht we c defiite itegrls, ut the questio still remis how good re these pproimtios??? d. The ect vlue is 4 evlute ectly: Here re the ssocited errors:. Error Bouds If E T d E M re the errors i the Trpezoidl d Midpoit rules, respectively, the E T Where K( ) f '( ) K K( E M ' for ) 4 K ( ) 80 ) K For Simpso s Rule, E S 4 where K is ow such tht f (4) ( 5 6. Suppose we pproimte Rule with = 5 d usig Midpoit ) Wht is the mimum possile error? ) If we wt mimum possile error 8 of 0, wht vlue of should we use? 7. π If we pproimte cos d usig Simpso s 0 Rule with = 4, wht is the mimum possile error?
Clculus AP U5 Itegrtio (AP) Nme: ASSIGNMENT Sigm Limits of Fiite Sums (AP) Review gr mth:. 4 7 0... 5. ) Numer of terms ) Sum c) Sigm ottio. + 6 + 8 + 4 + +... ) Term formul ) Sum formul c) Sigm ottio Simplify the followig sums. 4. 5. 6. Defiite itegrl of f from to is f ( ) d = lim f ( ) k = For cotiuous fuctios f, where the itervl [, ] is divided ito suitervls of equl width = d k = + k E. Epress 5 k lim k k = cos k defiite itegrl o 0, π 4. k s +
Clculus AP U5 Itegrtio (AP) Nme: 7. Ares Uder Curve versus Itegrls 8. Set up two seprte itegrl epressios tht Cot simply evlute the defiite itegrl! Are would give the ctul re of the regio ouded is lwys! So whe usig re to y the fuctio f ( ) = 4 d the -is o fid defiite itegrls, we re resposile for the itervl [,5] ssigig the regios the correct. This mes you must fid where the grph the split up our itervl, mully mkig egtive regios c positive: f ( ) d + f ( ) d If you re usig clcultor just eter: c Note: plcemet of solute vlue mtters! 9. Fid 4 ( ) d usig sums. Does the swer represet re? 0. Fid the re uder d = usig sums. f ( ) = etwee = 0
4 Clculus AP U5 Itegrtio (AP) Nme:. 4 check your swer grphiclly usig geometric Fid ( 6) d re formuls usig sums. Fid ( 5 + ) d usig geometric re. 5 Fid d usig geometric re 4
5 Clculus AP U5 Itegrtio (AP) Nme: ASSIGNMENT Defiite Itegrls (AP). f ( + h) f ( ) Just like lim ws defied to e the h 0 h the lim f ( k ) k = is defied to e the. Defiite itegrl of f from to is f ( ) d = lim f ( ) k = For cotiuous fuctios f, divide the itervl [, ] ito suitervls of equl width = d k = + k ( = ),,,, ( = ) ie. 0 k Itegrility of Cotiuous Fuctios Theorem If fuctio f is cotiuous over the itervl [, ], the the defiite itegrl f ( ) d eists d f i itegrle over [, ] Actully the theorem is lso true for f tht hs t most my discotiuities Itegrtio Properties If f d g re itegrle o give itervls. f ( ) d = f ( ) d,. f ( ) d = 0,. cd = c( ), c is y costt f ( ) ± g( ) d = f ( ) d ± g( ) d 4. [ ] 5. cf ( ) d = c f ( ) d, c is y costt 6. f ( ) d + f ( ) d = f ( ) d. 4 6 c f ( ) d = 8 d f ( ) d = 0 4 6 4. f ( ) d =. f ( ) d = 0 6 4 4 c. 4 f ( ) d = d. f ( ) d = 0 6 c. Give the followig iformtio: 8 f ( ) d = 7, g ( ) d =, f ( ) d = 9 evlute these defiite itegrls: 8 ) [ f ( ) + 4g( )] d = 8 [6 ) g ( ) + 5] d = 0 d c) f ( ) = d) 8 8 f ( ) d = 8 0 5
6 Clculus AP U5 Itegrtio (AP) Nme: 4. 5. 6
7 Clculus AP U5 Itegrtio (AP) Nme: 6. 7. Suppose we wt to pproimte the re uder f ( ) = etwee = 0 d = 6. ) Fid epressio for this re s limit. ) Do you kow how to evlute this sum? c) Rewrite s defiite itegrl o [0, 6], why does this o loger represet re? d) Do you kow how to fid the idefiite itegrl for this fuctio? 8. Suppose we wt to pproimte the re uder π f ( ) = si etwee = 0d =. ) Fid epressio for this re s limit. ) Do you kow how to evlute this sum? c) Rewrite this limit s defiite itegrl d) Do you kow how to fid the idefiite itegrl for this fuctio? 9. For the ove questios we eed the to coect the cocepts c) d d) Without it we c ler more tedious sum simplifictios or use fiite to pproimte the swer 7
8 Clculus AP U5 Itegrtio (AP) Nme: ASSIGNMENT FTC & Averge Vlue of Fuctio (AP). Fudmetl Theorem of Clculus (FTC): Suppose f is cotiuous o [, ]. Defie F s: p. F( ) = f ( t) dt, the F is cotiuous o [,] d differetile d o (,), d F ( ) = f ( t) dt f ( ) d = p. f ) d = F( ) F( ) (, where F is y tiderivtive of f. First prt sttes tht of gives ck the fuctio. Note: limit must e. Secod prt helps you to evlute defiite itegrls without, it lso gives you Ide ehid prt Proof of prt Nottio epltio of prt 8
9 Clculus AP U5 Itegrtio (AP) Nme: Review idefiite itegrls.. 7 + 4si 6e + + d + 7 + + csc ( 5) d Prctice FTC prt 4. ( ) d 5. ( 6 e ) d 6. 5 d 7. 0 d 64 6 8. π 0 (si + ) d 9. π 0 sec 5 d 0. Does this method for evlutig defiite itegrls lwys work? Cosider the followig emple d = d. 9
0 Clculus AP U5 Itegrtio (AP) Nme:.. f ( ) d where ( + ) d 0, < f ( ) = +, 4. Use Sums, wht other method(s) would work? 5. Use Geometric Shpes, wht other method(s) would work? ( + ) d + 6. Use FTC prt, wht other method(s) would work? 7. d see the eed for sustitutio method to e lered i the et uit 0
Clculus AP U5 Itegrtio (AP) Nme: Fid re ouded y: Fid re ouded y: 8. 9. 0. Use the symmetry to fid the re ouded y - is d the give fuctio o [-, ]. Fid the verge vlue of ( ) si( ) f = o [ 0, π ]. 5 7 + 5 4 d use symmetry too. The temperture of 5m log metl rod is give y 4 f ( ) = e t distce of metres from oe ed of the rod. Wht is the verge temperture of the rod?
Clculus AP U5 Itegrtio (AP) Nme: ASSIGNMENT More FTC & MVT for Itegrls (AP)... Fid the itervl o which the curve ( y = t + t + ) dt is cocve up. Justify your 0 swer. 4.
Clculus AP U5 Itegrtio (AP) Nme: Prctice FTC prt 5. 6. d d t si t dt 7. d 0 d + t dt 8. d d 7 e t t t dt 9. 0... d d t + dt 5
4 Clculus AP U5 Itegrtio (AP) Nme:. The Me Vlue Theorem for Itegrls: If f is Demostrte why theorem would rek dow for cotiuous o [, ], the there eists umer o cotiuous fuctios. c i [, ] such tht f ( ) d = f ( c)( ) 4. Fid the vlue of c gurteed y the MVT for itegrls for f ( ) = + o [-,]. Iterpret the result grphiclly. 5. Fid the vlue of c gurteed y the MVT for derivtives for f ( ) = + o [-,]. Iterpret the result grphiclly. 6. Fid the vlue of c gurteed y the MVT for itegrls for 7. Fid the equtio of the tget lie to g( ) = cos tdt t ( π,) 4
5 Clculus AP U5 Itegrtio (AP) Nme: The Net Chge Theorem: The itegrl of rte of 4. If wter flows from tk t rte of chge is the et chge: r( t) = 00 t litres per miute for 0 t 50, fid the mout of wter tht flows from the tk F ( ) d = F( ) F( ) i the first 0 miutes. Or Accumultio: F( ) F( ) F ( ) d = + wht I hve ow = wht I strted with + wht I've ccumulted sice the strt 5. Durig 4 weeks i the 000-00 flu seso, the rte of reported ifluez per 00,000 people i Ireld could e pproimted y 0.049 t I ( t) =.89e, where I is the totl umer of people per 00,000 who hve cotrcted ifluez d t is time mesured i weeks. Approimtely how my people per 00,000 cotrcted ifluez durig the whole 4 weeks? 6. At the strt of Christms Brek, t t = 0 dys, m weighed 80 pouds. If the m gied weight durig the rek t rte modeled y the fuctio πt W ( t) = 0si pouds per dy, wht ws 8 the m s weight (i pouds) t the ed of the rek, 4 dys lter? 7. π If f ( ) = 4cos 6 d f () = π ) Fid f ( ) use symmetry ) Fid f ( ) 5
6 Clculus AP U5 Itegrtio (AP) Nme: Solve for the idicted vrile 8. 9. 0. Evlute 0 0 6 d. Fid the umer(s) such tht the verge vlue of f ( ) = + 6 o the itervl [0,] is equl to.. f ( t) Suppose tht + dt = Fid f() t 0 6
7 Clculus AP U5 Itegrtio (AP) Nme:. 4. Popultio is give y P( t) = l(t + ) thousd of isects where t is umer of yers sice J 000. ) Fid verge popultio from t=5 to t=9 ) Fid verge chge i popultio from t=5 to t=9 7
8 Clculus AP U5 Itegrtio (AP) Nme: ASSIGNMENT Iterpret the Itegrl (AP).. If h (t) is the rte of chge of child s height mesured i iches per yer, wht does the 0 itegrl h ( t) dt represet, d wht re its uits? 0 The grph represets perso out for wlk. The perso ws m to the left of ok tree t t=0 sec. ) Wht directio is the perso trvellig? Whe did she tur roud? ) Wht does the itegrl represet? c) Fid the positio t t=0sec d) Fid the verge velocity o [0,0] usig oth formuls give i the jourl e) Fid verge ccelertio o [0,0]. Popultio is growig with rte dp = 00t + 5 people/yer where t is time dt sice 000 i yers. ) Record itegrl to e used to predict the popultio i 00 if i 000 popultio ws 0000 people. ) C you solve this itegrl with techiques we kow so fr? 4. Suppose 5.6mg of dye ws ijected ito the loodstrem. ) Record itegrl tht would represet the re of uder this curve. Wht does it represet? ) Fid the efficiecy with which the dye is eig processed i L/sec. 8
9 Clculus AP U5 Itegrtio (AP) Nme: 5. 9
0 Clculus AP U5 Itegrtio (AP) Nme: 6. (d) Fid g(-4) 0
Clculus AP U5 Itegrtio (AP) Nme: 7.
Clculus AP U5 Itegrtio (AP) Nme: 8. (d) Sketch the displcemet time grph
Clculus AP U5 Itegrtio (AP) Nme: 9. The verge vlue of homes i vlley is growig 0. If oil is lekig from ruptured tker t rte of t rte of t + 4 thousd dollrs per yer 0.t f ( t) = 0e gllos/hr where t is mesured i where t represets the umer of yers sice hours sice the tker ws dmged. Evlute d 005. If the verge vlue of home i this re epli the meig of: ws $90 000 i 007 determie whe the verge 5 ) vlue of home will e $0 000 f ( t) dt 0 ) 0 f ( t) dt. A prticle moves i stright lie so tht its velocity t time t is give y v( t) = t t m/s. ) Fid the displcemet of the prticle durig the first secods. ) Fid the distce trvelled y the prticle durig the first secods.. WITH CALC The rte t which people eter musemet prk o y 5600 give dy is E( t) = people/hr, d the t 4t + 60 9890 rte t which people re levig is L( t) = t 8t + 70 where t is mesured i hours fter midight. The prk is ope from 9m to pm. ) How my people hve etered the prk y 5pm? ) How my people re t the prk t 5pm? c) How does the umer of people t the prk chgig t 5pm?