CALCULUS WORKSHEET ON INTEGRATION WITH DATA Work th following on notbook papr. Giv dcimal answrs corrct to thr dcimal placs.. A tank contains gallons of oil at tim t = hours. Oil is bing pumpd into th tank at a rat, whr is masurd in gallons pr hour and t is masurd in hours. Slctd Rt valus of Rt Rt ar givn in th tabl blow. t (hours) 5 9 Rt (gallons pr hour) 8.9 6.8 6. 5.9 5.7 (a) Estimat th numbr of gallons of oil in th tank at t = hours by using a trapzoidal approimation with four subintrvals and valus from th tabl. Show th computations that lad to your answr. (b) A modl for th rat at which oil is bing pumpd into th tank is givn by th function Gt, whr ln t Gt is is masurd in gallons pr hour and t is masurd in hours. Us th modl to find th numbr of gallons of oil in th tank at t = hours.. A hot cup of coff is takn into a classroom and st on a dsk to cool. Th tabl shows th rat at which tmpratur of th coff is dropping at various tims ovr an ight minut Rt priod, whr Rt is masurd in dgrs Fahrnhit pr minut and t is masurd in minuts. Whn t =, th tmpratur of th coff is F. t (minuts) 5 8 ( F/min.) 5.5.7.6.8 Rt (a) Estimat th tmpratur of th coff at t = 8 minuts by using a lft Rimann sum with thr subintrvals and valus from th tabl. Show th computations that lad to your answr. (b) Us valus from th tabl to stimat th avrag rat of chang of Rt ovr th ight minut priod. Show th computations that lad to your answr. (c) A modl for th rat at which th tmpratur of th coff is dropping is givn by th.t function yt 7 is masurd in dgrs Fahrnhit pr minut and t is y t, whr masurd in minuts. Us th modl to find th tmpratur of th coff at t = 8 minuts. (d) Us th modl givn in (b) to find th avrag rat at which th tmpratur of th coff is dropping ovr th ight minut priod.. (Modification of AB / BC ) Th tmpratur, in dgrs Clsius ( C), of th watr in a pond is a diffrntiabl function W of tim t. Th tabl blow shows th watr tmpratur as rcordd vry days ovr a 5-day priod. t (days) 6 9 5 W t ( C) 8 (a) Approimat th avrag tmpratur, in dgrs Clsius, of th watr ovr th tim intrval t 5days by using a trapzoidal approimation with subintrvals of lngth t days and valus from th tabl. Show th computations that lad to your answr. (b) A studnt proposs th function P, givn by P t t t, as a modl for th tmpratur of th watr in th pond at tim t, whr t is masurd in days and Pt is masurd in dgrs Clsius. Us th function P to find th avrag valu, in dgrs Clsius, of Pt ovr th tim intrval t 5 days.
. (Modification of Form B AB / BC ) A tst plan flis in a straight lin with positiv vlocity minuts, whr v is a diffrntiabl function of t. Slctd valus of vt, in mils pr minut at tim t for vt vt ar shown in th tabl blow. t (min) 5 5 5 5 (mpm) 7. 9. 9.5 7..5... 7. vt (a) Us a midpoint Rimann sum with four subintrvals of qual lngth and valus from th tabl to approimat v t dt plain th maning of. Show th computations that lad to your answr. Using corrct units, v t dt in trms of th plan s flight. t 7t (b) Th function f, dfind by f t 6 cos sin, is usd to modl th vlocity of th plan, in mils pr minut, for. According to this modl, what is th avrag vlocity of th plan, in mils pr minut ovr th tim intrval vt? t 5. (Modification of 5 AB / BC ) A mtal wir of lngth 8 cntimtrs is hatd at on nd. Th tabl blow givs slctd valus of th tmpratur, in dgrs Clsius, of th wir cm from th hatd nd. (a) Estimat T Distanc (cm) 5 6 8 Tmpratur T ( C) 9 7 6 55 T 7. Show th work that lads to your answr. Indicat units of masur. T for th avrag tmpratur of th wir. Estimat th avrag tmpratur of th wir using a trapzoidal sum with th four subintrvals indicatd (b) Writ an intgral prssion in trms of by th data in th tabl. Indicat units of masur. 8, and indicat units of masur. Eplain th maning of (c) Find T d th tmpratur of th wir. 6. (Modification of 6 AB / BC ) Rockt A has positiv vlocity vt 8 T d in trms of aftr bing launchd upward from an initial hight of ft at tim t = sconds. Th vlocity of th rockt is rcordd for slctd valus of t ovr th intrval t 8 sconds, as shown in th tabl blow. t (sconds) 5 6 7 8 vt (ft pr scond) 5 9 5 7 9 (a) Using corrct units, plain th maning of 7 v t dt in trms of th rockt s flight. Us a midpoint Rimann sum with subintrvals of qual lngth to approimat v t dt. (b) Rockt B is launchd upward with an acclration of at ft pr scond pr scond. t At tim t = sconds, th initial hight of th rockt is ft, and th initial vlocity is ft pr scond. Which of th two rockts is travling fastr at t = 8 sconds? Eplain your answr. 7
Answrs to Intgration using Data 5 8 8.8 8 9. 8 8. 58.8gallons E. (a) (b) 5 88 9.8 89. 8.9 88.9 6.7 5. gallons (c) 57.6 gallons (d) 8.6 gallons pr hour 8.9 6.8 6.8 6. 6. 5.9 5.9 5.7 98.75 gal (b) 97.975 gal ln t 5.5.7.6 86. F. (a) Amt.. (a) Tmp. (b) Av. rat of chang = 8.t (c) 7 dt 9.78 F.8 5.5.588F / min 8 8.t (d) Th tmpratur is dropping at a rat of 7 dt.65 F / min. 8. aav.tmp. 8 8 5 5.C/day 5 t (b) t dt 5.757C/day 5. (a) v t dt 9. 7... 9 mi. Th intgral givs th total distanc in mils that th plan flis btwn t = and t = min. t 7t (b) Av. vlocity 6 cos sin dt 5.96 mils pr minut. 55 6 7 T 7 C/cm 8 6 8 (b) Av. tmp. T d 8 Av. tmp. 9 97 7 6 6 55 75.688C 8 5. (a) 8 (c) T d T T 8 55 5C. Th tmpratur drops 5 C from th hatd nd of th wir to th othr nd of th wir. 6. (a) Sinc th vlocity is positiv, th intgral rprsnts th distanc, in ft, travld by rockt A from t = sconds to t = 7 sconds. 7 v t dt 5 ft (b) Lt v t b th vlocity of rockt B at tim t. B vb t dt 6 t C. vb 6 C. vb t 6 t. t 8 5 9 v 8 so Rockt B is travling fastr at tim t = 8 sconds. vb
CALCULUS WORKSHEET ON SECOND FUNDAMENTAL THEOREM AND FUNCTIONS DEFINED BY INTEGRALS. Evaluat. d sin t d (a) dt t d cos d (b) dt t d (c) dt d t d (d) ln t dt d d () cos d 7 t dt d (f) sin t dt 5 d tan. Th graph of a function f consists of a smicircl and two y lin sgmnts as shown. Lt g b th function givn by g f tdt. (a) Find g, g, g,and g 5. (b) Find all valus of on th opn intrval,5 which g has a rlativ maimum. Justify your answrs. (c) Find th absolut minimum valu of g on th closd intrval and th valu of at which it occurs. Justify your answr. Graph of f (d) Writ an quation for th lin tangnt to th graph of g at =. () Find th -coordinat of ach point of inflction of th graph of g on th opn intrval [,5] at y,,5. Justify your answr. (f) Find th rang of g.. Lt g f t dt, whr f is th function whos graph is shown. g, g, g, g, and g 7. (a) Evaluat (b) Writ an quation for th lin tangnt to th graph of g at =. (c) On what intrvals is g incrasing? Dcrasing? Justify your answr. (d) Find all valus of on th opn intrval < < 7 at which g has a rlativ maimum. Justify your answr. () Whr dos g hav its absolut maimum valu? Graph of f What is th maimum valu? Justify your answr. (f) Whr dos g hav its absolut minimum valu? What is th minimum valu? Justify your answr.. Lt g f t dt, whr f is th function whos graph is shown. (a) On what intrvals is g dcrasing? Justify. (b) For what valu(s) of dos g hav a rlativ maimum? Justify. (c) On what intrvals is g concav down? Justify. (d) At what valus of dos g hav an inflction point? Justify. Graph of f TURN->>>
5. Th graph of th function f, consisting of thr lin sgmnts, is shown on th right. Lt g f tdt, (a) Find g g g (b) Find g and g.,,. (c) Find th instantanous rat of chang of g with rspct to at =.. Justify. Graph of f (d) Find th absolut maimum valu of g on th intrval, () Th scond drivativ of g is not dfind at = and at =. Which of ths valus ar -coordinats of points of inflction of th graph of g? Justify. y
Answrs to Worksht on Scond Fund. Th. & Functions Dfind by Intgrals. (a) sin (b) 6 () cos (f) sc sin tan. (a),,, (b) g has a rl. ma. at = bcaus g f ln (c) - tan (d) changs from positiv to ngativ thr. (c) Abs. min. = at = - (Justify with Candidats Tst.) (d) y () g has an I.P at = bcaus g changs from incrasing to dcrasing thr. g has an I.P at = bcaus g changs from dcrasing to incrasing thr. (f) [, ]. (a),, 5, 7, (b) y 6 (c) g is incrasing on bcaus g f bcaus g f g g is dcrasing on 7 (d) g has a rlativ maimum at = bcaus, which quals, is positiv thr., which quals, is ngativ thr. changs from positiv to ngativ thr. (Justify with Candidats Tst.) () and (f) Ma. valu = 7 at = and Min. valu = at = (Justify with Candidats Tst.). (a) g is dcrasing on and 5 bcaus g f is ngativ thr. (b) g has a rl. ma. at = and at = bcaus g f changs from positiv to ngativ thr. (c) g is concav down on and 5bcaus g f is dcrasing thr. (d) g has an I.P at,, and bcaus g changs from incrasing to dcrasing or vic vrsa thr. 5. (a),, (b), (c) 9 (d) Abs. ma = 5 at = (Justify with Candidats Tst.) () g has an inflction point at = bcaus g f changs from incrasing to dcrasing thr. g dos not hav an inflction point at = bcaus g f is dcrasing for < < and continus to dcras on < <.
CALCULUS WORKSHEET ON FUNCTIONS DEFINED BY INTEGRALS Work th following on notbook papr.. Find th quation of th tangnt lin to th curv y F whr 7 at th F t dt point on th curv whr =.. Suppos that 5 f tdt. c (a) What is f? (b) Find th valu of c.. If F t t dt, for what valus of is F dcrasing? Justify your answr.. Th function F is dfind for all by (a) Find (b) Find F. F. (c) Find (d) Find 8. F t dt. F F. 5. Th function F is dfind for all by F f t dt, whr f is th function graphd in th figur. Th graph of f is mad up of straight lins and a smicircl. (a) For what valus of is F dcrasing? Justify your answr. (b) For what valus of dos F hav a local maimum? A local minimum? Justify your answr. (c) Evaluat F, F, and F. (d) Writ an quation of th lin tangnt to th graph of F at =. Graph of f () For what valus of dos F hav an inflction point? Justify your answr. 6. Th graph of a function f consists of a smicircl and two lin sgmnts as shown on th right. Lt g f t dt. (a) Find g, g, g. (b) On what intrval(s) of is g dcrasing? Justify your answr. (c) Find all valus of on th opn intrval, at which g has a rlativ minimum. Justify your answr. (d) Find th absolut maimum valu of g on th intrval, and th valu of at which it occurs. Justify your answr. () On what intrval(s) of is g concav up? Justify your answr. Graph of f (f) For what valu(s) of dos th graph of g hav an inflction point? Justify your answr. (g) Writ an quation for th lin tangnt to th graph of g at. y TURN->>>
7. Th graph of th vlocity vt, in ft/sc, of a car travling on a straight road, for t 5, is shown in th figur. ft / sc (a) Find th avrag acclration of th car, in, ovr th intrval t 5. (b) Find an approimation for th acclration of th car, in ft / sc (c) Approimat, at t =. Show your computations. 5 5 v t dt with a Rimann sum, using th midpoints of thr subintrvals of qual lngth. Eplain th maning of this intgral. y
Answrs to Worksht on Functions Dfind by Intgrals.. (a) y 5 (b) thr.. F is dcrasing on < bcaus F. (a) 8 (b) 6 (c) 8 (d) 8 5. (a) F is dcrasing on 5.5 and 5 thr. (b) F has a local minimum at 7, which is, is ngativ bcaus F f bcaus F, which is f, changs from ngativ to positiv thr. F has a local maimum at = bcaus F f.5 changs from positiv to ngativ thr. (c),, (d) y () F has an inflction point at,,, and changs from incrasing to dcrasing or vic vrsa thr. 6. (a), -, (b) g is dcrasing on, which is, bcaus F f bcaus g f is ngativ thr. (c) g has a rlativ minimum at = bcaus g f changs from ngativ to positiv thr. (d) Abs. ma. = at = (Justify with Candidats Tst.) () g is concav up on and (f) g has an inflction point at bcaus g f and = bcaus g f incrasing to dcrasing or vic vrsa thr. y (g), which is, is incrasing thr. changs from 6 7. (a) ft / sc 7 (b) ft / sc (using (, ) and (5, ) to stimat th slop) (c) ()() + ()() + ()() = 9 ft. This intgral rprsnts th approimat distanc in ft that th car has travld from t = 5 sconds to t = 5 sconds.
CALCULUS BC WORKSHEET ON 5. Work th following on notbook papr. Do not us your calculator.. 5 d 7. cos d sin. sin cos d d 8.. 5 sin d cos. ln d. d 9. d 5. d d.. sin d 9 6. cos d 5. d. d 7. 6 d 6. 5 tan sc d. d 8. d ln CALCULUS BC WORKSHEET ON 5. Work th following on notbook papr. Do not us your calculator.. 5 d 7. cos d sin. sin cos d d 8.. 5 sin d cos. ln d. d 9. d 5. d d.. sin d 9 6. cos d 5. d. d 7. 6 d 6. 5 tan sc d. d 8. d ln