Student s Printed Name:

Size: px

Start display at page:

Download "Student s Printed Name:"

Randolph Dean
5 years ago
Views:

1 Studt s Pritd Nam: Istructor: CUID: Sctio: Istructios: You ar ot prmittd to us a calculator o ay portio of this tst. You ar ot allowd to us a txtbook, ots, cll pho, computr, or ay othr tchology o ay portio of this tst. All dvics must b turd off ad stord away whil you ar i th tstig room. Durig this tst, ay kid of commuicatio with ay prso othr tha th istructor or a dsigatd proctor is udrstood to b a violatio of acadmic itgrity. No part of this tst may b rmovd from th xamiatio room. Rad ach qustio carfully. To rciv full crdit for th fr rspos portio of th tst, you must:. Show lgibl, logical, ad rlvat justificatio which supports your fial aswr.. Us complt ad corrct mathmatical otatio.. Iclud propr uits, if cssary.. Giv aswrs as xact valus whvr possibl. You hav 90 miuts to complt th tir tst. Do ot writ blow this li. Fr Rspos Problm Possibl Eard Fr Rspos Problm Possibl Eard. 5. a b. 7. a c.. b a (Scatro). b. 5 Fr Rspos 70. c. 5 Multipl Choic 0. d. 5 Tst Total 00 Vrsio A Pag of 8

2 Multipl Choic. Thr ar 0 multipl choic qustios. Each qustio is worth poits ad has o corrct aswr. Th multipl choic problms will cout 0% of th total grad. Circl your choic o your tst papr.. Exprss th followig sum usig sigma otatio. ( pts.) 87 a) c) k k ( ) ( k ) b) k k k ( ) ( k ) d) k k k0 k k ( ) ( k) k ( ) ( k ). Dtrmi whr th fuctio f( x ) is dcrasig if th first drivativ is ( pts.) x f ( x) x ( x ) ( x ). a), b) (, ) 0, c),0 (, ) d) (0,) Vrsio A Pag of 8

3 . ( pts.) Assum f ( x) g( x) dx 6 ad f ( x) dx. Fid g( x) dx. a) c) g( x) dx 9 b) g( x) dx d) g( x) dx 0 g( x) dx 9. Th figur shows th aras of thr rgios boudd by th graph of f ad th x-axis. ( pts.) Fid 5 f ( x) dx. a) c) 5 f ( x) dx b) 5 f ( x) dx 0 d) 5 f ( x) dx 60 5 f ( x) dx Vrsio A Pag of 8

4 5. Fid th x-valus of th absolut xtrma of ( pts.) h( x) x x 8 o th itrval [0, ]. a) absolut maximum at x, absolut miimum at x b) o absolut maximum, absolut miimum at x 0 c) absolut maximum at x, absolut miimum at x 0 d) o absolut xtrma 6. Lt f( x ) b twic diffrtiabl o th itrval (, ). Th graph of th drivativ of ( pts.) f( x) is show blow. Us it to dtrmi th itrval(s) o which th graph of f( x) is cocav up. a) (,) (5, ) b) (,5) c) (,) d) No such itrvals Vrsio A Pag of 8

5 7. ( pts.) Fid f( x ) if f ( x) x ad f. x x a) f ( x) x b) f ( x) x l x c) f ( x) x l( x) d) f ( x) x l x 8. Us rctagls to stimat th ara abov th x-axis ad udr th graph of ( pts.) f ( x) si x o th itrval [0, ] (s graph). Partitio th itrval ito two subitrvals of qual width ad valuat th fuctio at th midpoits of th subitrvals. a) 8 b) c) d) 6 Vrsio A Pag 5 of 8

6 Th graph of gx ( ) is show blow. It cosists of two straight lis. Us it to valuat th dfiit itgrals i problms 9 ad ( pts.) gx ( ) dx a) c) gx ( ) dx 9 b) gx ( ) 7 dx d) gx ( ) dx 7 gx ( ) dx 0. ( pts.) 6 g( x) dx a) c) 6 g( x) dx b) 6 g( x) dx d) 6 6 g( x) dx 7 g( x) dx Vrsio A Pag 6 of 8

7 (this pag ittioally lft blak) Vrsio A Pag 7 of 8

8 Fr Rspos. Th Fr Rspos qustios will cout 70% of th total grad. Rad ach qustio carfully. To rciv full crdit, you must show lgibl, logical, ad rlvat justificatio which supports your fial aswr. Giv aswrs as xact valus.. ( pts.) A right triagl whos hypotus is 8 mtrs log is rvolvd about o of its lgs to grat a right circular co (s figur). Fid th radius ad hight that maximizs th volum of th co. I your work you should: Stat th fuctio to b optimizd i trms of a sigl variabl. Stat th domai of th fuctio. Show all work dd to fid ad vrify th valus of r ad h that maximiz th volum. h r 8 Not: Th volum V of a right circular co is V r h V r h, whr h r 8 r 8 h V ( h) (8 h ) h (8 h h ) V h h h h V ( h) (8 h ) Solv V( h) 0 to fid critical poits ( ) (8 ) domai: 0, 8 or 0, 8 (8 h ) 0 8 h 0 h 6; h ( h oly solutio i domai) Vrify maximum volum at h 8 For a closd itrval: chck V (0) 0, V (), V 8 =0 (closd itrval mthod); or first drivativ tst or scod drivativ tst For a op itrval: First drivativ tst: V '( h) 0 o (0,) ad V '( h) 0 o (, 8), so thr is a maximum at h ; or apply scod drivativ tst V ( h) h 0 o ( 0, 8], so cocav dow at th critical poit h is a maximum. Solvig for r : r 8 ; r 8 Th maximum volum th co ca hav is cubic uits wh h m ad r m. Vrsio A Pag 8 of 8

9 Fids a volum fuctio i trms of a sigl variabl (two poits to rcogiz poits h r 8 ) Domai of volum fuctio poit Fids th critical valu poits Vrifis maximum poits Givs th volum maximizig valus for radius ad hight poit Subtract ½ poit for missig or icorrct drivativ otatio. Subtract ½ poit for ot showig wh takig squar root. Subtract poit for ot givig valus for both r ad h. Subtract ½ poit for ot icludig uits. Los poits for drawig a umbr li with + ad - ovr parts but o justificatio. Los poit if thy justify but do't say aythig about that maig it is a max. Vrsio A Pag 9 of 8

10 . (0 pts.) At a Fourth of July clbratio, a bottl rockt is fird straight upward from a picic tabl o mtr high. Th acclratio fuctio for th rockt is giv blow. Fid th vlocity ad positio fuctios for th rockt. t a( t) cos t m/s, whr t is tim sic th rockt lauchd t a( t) cost t v( t) si t C Apply v(0) si(0) C 0 0 C 0 C t v( t) si t m/s t s( t) cost t C Apply s(0) cos(0) 0 C 0 C C t s( t) cost t m Work o Problm Fids vt () up to costat poits Solvs for costat i vt () poit Stats complt vt () with uits poit Fids st () up to costat poits Solvs for costat i st () poits Stats complt st () with uits poit OK if calculat vt () ad st () without showig itgratio symbol ½ poit dductio for icorrct otatio with a maximum palty of poit for all otatio rrors Vrsio A Pag 0 of 8

11 . (0 pts.) Evaluat th limits. Us of L Hôpital s Rul must b idicatd ach tim it is usd, ithr symbolically or i words. No crdit will b awardd without supportig work. x0 x0 x0 bx b bl( bx) a. (5 pts.) lim x0 x bx b b l( bx) 0 lim x 0 L L lim lim bx b b( b) bx 0 x 0 bx b b bx b b(0) b b b b b b b ( ) ( ) ( (0)) ( ) ( ) ( ) (costat b>0) Rcogizs idtrmiat form (xplicitly or / poit implicitly) Applis L Hopital s Rul corrctly two tims poits (two poits ach applicatio) Substituts to gt fial aswr / poit Subtract ½ poit for failig to idicat us of L Hopital s Rul Subtract ½ poit for ach otatio rror with a maximum of o poit total for all otatio rrors (xcludig rrors idicatig us of L Hopital s Rul) Subtract ½ poit for statmt: aythig = a idtrmiat form Subtract ½ poit for wrog idtrmiat form b b b b Vrsio A Pag of 8

12 b. (5 pts.) lim x x ta x ta x lim x x L lim x lim x l ta x x x x ta l x x lim ta l 0 x x x l 0 lim x cot 0 lim x x csc x csc Rcogizs idtrmiat form (xplicitly) / poit Rwrits usig ad atural log fuctio / poit Limit i xpot / poit Uss log proprty to mov xpot / poit Divids by rciprocal of tagt fuctio (OK if / poit covrtd to si ad cosi) Applis L Hopital s Rul corrctly ( poit for poits umrator ad poit for domiator) Substituts to gt fial aswr / poit Subtract ½ poit for failig to idicat us of L Hopital s Rul Subtract ½ poit for ach otatio rror with a maximum of o poit total for all otatio rrors (xcludig rrors idicatig us of L Hopital s Rul) Subtract ½ poit for statmt: aythig = a idtrmiat form Subtract ½ poit for wrog idtrmiat form Max of /5 if divids by th wrog rciprocal Othr tchiqus OK o Dfi y as fuctio, atural log of both sids o Dfi y as limit i xpot, fid valu, th xpotiat at th d Vrsio A Pag of 8

13 . (0 pts.) Lt f( x) x a) (5 pts.) Dtrmi th quatio(s) of ay horizotal asymptots o th graph of f( x ). lim 0 x x y 0 lim x x 0 y Fids limit as x Equatio of horizotal asymptot as x Fids limit as x Equatio of horizotal asymptot as x poits / poit poits / poit b) (5 pts.) Dtrmi th itrvals o which f( x) is icrasig or dcrasig. B sur to show th calculatio of th first drivativ. Put your fial aswrs i th appropriat spacs blow. f( x) f( x) x x f( x) x ( ) x ( ) x x ( ) ( ) Calculats first drivativ Icrasig itrval Dcrasig itrval (OK if lft blak) poits poit poit f( x) 0 has o solutio f ( x) is dfid for all x o critical valus f ( x) 0 for all x Icrasig: (, ) Dcrasig: o itrvals Vrsio A Pag of 8

14 c) (5 pts.) Th scod drivativ of f( x ) is show blow. Us it to dtrmi th itrvals o which f( x) is cocav up or cocav dow. Put you fial aswrs i th appropriat spacs blow. f x x ( ) ''( x) x ( ) x x ( ) f( x) x ( ) f( x) 0 x x ( ) 0 x ( ) x x 0 x 0 Solvs f( x) 0 poits Cocav up itrval poit Cocav dow itrval poit f ( x) 0 for all x 0 f ( x) 0 for all x 0 Cocav Up: (,0) Cocav Dow: (0, ) d) (5 pts.) Sktch f( x) x. Show th ordrd pair (x, y) at ay poit whr f has a local xtrm or a iflctio poit. Labl all axis itrcpts. Show th quatio of ay horizotal asymptots o th graph. Horizotal asymptots (OK if is ot labld) y 0 y-itrcpt Basic shap (cocavity, icrasig, o-liar, o x- itrcpt) poits poit poits Vrsio A Pag of 8

15 5. ( pts.) Cosidr th limit blow. * lim (xi ) x, whr i th itrval [0, 5] is partitiod ito subitrvals of width * xi is th right dpoit of th i th subitrval 5 x a. ( pts.) Exprss th limit as a dfiit itgral. 5 * lim ( i ) ( ) i 0 x x x dx Itgrad Limits of itgratio (/ ach) poit poit b. (7 pts.) Usig th summatio formulas blow as dd, valuat th limit. ( ) ( )( ) ( ) c c, i, i, i 6 i i i i 5i 5 x x x x * * lim ( i ), i, i 5i 5 lim i 50i 5 lim i 50i 5 lim lim i i 50 5 lim lim i i i 50 ( ) 5 lim lim 5lim lim 5 5() 5 0 dtrmis Substituts * x i x ito f( x ) * i Summad formula i trms of i ad Uss summatio formulas to gt a xprssio o oly Evaluats limit (OK if o work show to rsolv idtrmiat form ) Fial aswr poit poit poit poits poit poit Vrsio A Pag 5 of 8

16 c. ( pts.) Evaluat th dfiit itgral by usig basic ara formulas. Iclud a sktch Or (x ) dx Ara of rctagl + ara of triagl (5)() + (5)(0) (x ) dx Ara of trapzoid Sktchs rgio Fial aswr with supportig work: rctagl + triagl OR ara of a trapzoid poit poit Vrsio A Pag 6 of 8

17 6. (5 pts.) Th air forc dcids to tst a xprimtal jt by flyig it from o military bas to aothr military bas 00 mils away. Th jt maks this trip i xactly 0 miuts. Popl complai of harig a soic boom, but th air forc dis thir jt vr brok th soud barrir of 768 mils pr hour. Us calculus to show that th jt must hav brok th soud barrir at som tim durig its flight. B sur to stat ay rlvat thorms from calculus you us i makig your coclusio. Lt st () b th distac (mils) travld by th jt at tim t (hours), whr t [0,0.5]. Th avrag vlocity for th trip is s(0.5) s(0) MPH Sic th positio fuctio is cssarily cotiuous ad diffrtiabl, th Ma Valu Thorm applis. Calculats avrag vlocity Mtios cotiuous ad diffrtiabl (/ poit ach) Coclusio that mtios th Ma Valu Thorm poits poit poits Do t d to dfi a positio fuctio; OK if just calculat avrag vlocity Do t d a complt stc fial aswr, but th argumt should b clar By th Ma Valu Thorm, thr must b som tim c (0,0.5) such that s( c) v( c) Thrfor, th jt must hav brok th soud barrir at som tim durig its flight. Vrsio A Pag 7 of 8

18 Scatro ( pt.) My Scatro: Chck to mak sur your Scatro form mts th followig critria. If ay of th itms ar NOT satisfid wh your Scatro is hadd i ad/or wh your Scatro is procssd o poit will b subtractd from your tst total. is bubbld with firm marks so that th form ca b machi rad; is ot damagd ad has o stray marks (th form ca b machi rad); has 0 bubbld i aswrs; has MATH 060 ad my sctio umbr writt at th top; has my istructor s last am writt at th top; has Tst No. writt at th top; has th corrct tst vrsio writt at th top ad bubbld i blow my XID; shows my corrct XID both writt ad bubbld i; Bubbl a zro for th ladig C i your XID. Plas rad ad sig th hoor pldg blow. O my hoor, I hav ithr giv or rcivd iappropriat or uauthorizd iformatio at ay tim bfor or durig this tst. Studt s Sigatur: Vrsio A Pag 8 of 8

1985 AP Calculus BC: Section I

1985 AP Calculus BC: Section I 985 AP Calculus BC: Sctio I 9 Miuts No Calculator Nots: () I this amiatio, l dots th atural logarithm of (that is, logarithm to th bas ). () Ulss othrwis spcifid, th domai of a fuctio f is assumd to b