Clculus for Businss nd Socil Scincs - Prof D Yun Finl Em Rviw vrsion 5/9/7 Chck wbsit for ny postd typos nd updts Pls rport ny typos This rviw sht contins summris of nw topics only (This rviw sht dos hv prctic problms for ll topics) Finl Em is on Sctions to 74 Antidrivtivs/indfinit intgrls ruls n n d C for n ( f ( bg( ) d f ( d b g( d d ln C d C b b d ln b C d C b Dfinit intgrls b b f ( d f ( d (Find ntidrivtiv nd plug in b nd nd subtrct th two rsults) Applictions of dfinit intgrls b F '( d F( b) F( ) which is th nt chng in F from input to input b This nt chng cn b computd if w r givn formul for its drivtiv F '( For mpl if w hv formul for th mrginl cost thn w cn comput ny nt chng in cost Anothr mpl is th nt chng in position is th dfinit intgrl of vlocity b f ( d is th r undr y f ( nd bov th -is btwn th vrticl lins nd b b ( f ( g( ) d is th r undr th curv y f ( nd bov th curv y g( btwn th vrticl lins nd b Givn dmnd curv hving pric s function of quntity p f ( th consumrs surplus is A ( f ( f ( A)) d whr A is th quntity dmndd Givn incom strm K pr yr thn ssuming nnul intrst rt r th blnc ftr N yrs N N r( N (futur vlu) is K dt which lso hs prsnt vlu of K rt dt Lvl curvs Th lvl curv of function f ( t hight k is th curv f ( k drwn in th y -pln Th st of lvl curvs (contour mp) cn dscrib th 3-dimnsionl surfc z f ( Prtil drivtivs f is th prtil drivtiv of f with rspct to which is th rgulr drivtiv trting ll othr input vribls s constnt f is th rt of chng of f with rspct to holding ll othr input vribls constnt
f If f ( is productivity whr is lbor nd y is cpitl thn is th mrginl productivity f with rspct to lbor nd is th mrginl productivity with rspct to cpitl y f f ( h b) f ( b) ( b) h Rltiv m/min of f ( f f A cndidt for rltiv m/min must stisfy nd (Solv two qutions in two y unknowns) Us th two-input vrsion of th nd drivtiv tst t cndidt for rltiv m/min: f f f o Clcult D y y f o If D nd it is locl minimum f o If D nd it is locl mimum o If D it is sddl point (nithr m nor min) o If D it is inconclusiv Lgrng multiplirs for constrind m/min problms Th cndidts for locl m/min of n objctiv function f ( with constrint g ( cn only occur whr th function F( y ) f ( g( hs ll thr prtil drivtivs qul to Solv 3 qutions in 3 unknowns y Formuls providd on th m: S th Finl m formul sht Prctic Problms Mtch th function with th lvl curvs () f ( y (b) f ( y (c) f ( y (d) f ( y (I) (II) (III) (IV) (V) (VI)
Find ll st ordr prtil drivtivs () f ( 3 8y y (b) f ( ln(3 8y y ) (c) y (d) f ( p q) pln( q) () f ( y z) (f) y 5z 3 Find ll nd ordr prtil drivtivs f f ( / y ( y p q) p qy 3 4 y () f ( 8 y 7y (b) f ( 4 Find th cndidts for locl m/min nd thn us th scond drivtiv tst to dcid whthr th cndidts r locl m locl min sddl point or inconclusiv () f ( 3 y y 6 4y (b) f ( 6 4y 3 y 4 (c) f ( 4y y 8 4y (d) f ( y y 3y (For (d) you r givn th cndidts r ( 3/ )( )( ) ) 5 Us th mthod of Lgrng multiplirs to find th loctions of possibl m/min () Minimiz 3y subjct to constrint 5y (b) Mimiz 3y 6 subjct to constrint 4y 6 Suppos w r givn th productivity function f ( y whr is lbor nd y is cpitl () At nd y 5 find th mrginl productivity with rspct to lbor (b) At nd y 5 find th mrginl productivity with rspct to cpitl (c) At nd y 5 us th prtil drivtiv to pproimt th chng in productivity if th lbor is incrsd by (d) At nd y 5 us th prtil drivtiv to pproimt th chng in productivity if th cpitl is incrsd by 7 Mtch th surfc with th lvl curvs () (b) (c) (I) (II) (III)
8 Find th following intgrls Not which r indfinit intgrls nd which r dfinit intgrls () 6 6 (6 6 d ) (b) 3 ( 3) d (c) d (d) 7 3 d () 8 d (f) 3 d 9 Givn th mrginl cost function C' ( find th nt chng in cost from to Find th r btwn th two curvs y nd y 4 Suppos you r givn th dmnd curv p 4 Find th consumrs surplus if th dmnd is An incom strm gnrts $5 yr continuously for th nt yrs Suppos w continuously invst this mony in somthing tht rns 4% nnully continuously () Wht will th blnc b ftr yrs? (b) Find th prsnt vlu of this yr incom strm 3 Givn th following grph of f ( y ssum th domin is only [8] () Giv th intrvls of incrsing nd of dcrsing (b) Stt ny locl mim or minim (c) Giv th intrvls of concv up nd of concv down (d) Stt ny inflction points () Which is grtr f '(4) or f '(5)? 4 Find th following limits 4 3 () lim (b) lim 3 5 Us drivtiv ruls to find th drivtiv of th following functions () 5 4 y 7 3 (b) g ( u) (c) h ( 3 3 (5u ) 3t 7 (d) y ln(3 7) () f ( t (f) f ( ln( t (g) q p p 7 p (h) g ( (i) f ( p) ( ) ln(5 ) 6 Find th tngnt lin t th point () to th following grphs 3 3 3 3 () y (b) y (c) y y 7 7 Suppos dmnd nd pric p r rltd by p 5p 65 () At th momnt whn th dmnd is nd th pric is $3 if th pric incrss t rt of $ pr month how is th dmnd chnging? (b) At th momnt whn th dmnd is nd th pric is $3 if th dmnd incrss t rt of pr month how is th pric chnging? 35
8 Suppos dmnd nd pric p r rltd by p 45 () At th momnt whn th dmnd is nd th pric is $5 if th pric dcrss t rt of $ pr month how is th dmnd chnging? (b) At th momnt whn th dmnd is nd th pric is $5 if th dmnd dcrss t rt of pr month how is th pric chnging? 9 Suppos w wish to mk 3 sid by sid idnticl rctngulr pig pns nt to n isting wll so tht ll 3 pns r touching th wll No fnc is ndd ginst th wll () W hv ft of fnc Find th dimnsions tht would mimiz th totl r Suppos w wish to mk 3 sid by sid idnticl rctngulr pig pns nt to n isting wll so tht ll 3 pns r touching th wll No fnc is ndd ginst th wll Ech pig pn should b 48 squr ft Find th dimnsions tht would minimiz th lngth of fnc usd Suppos w sll 6 units during yr Suppos th dlivry chrg is $6 pr ordr nd th crrying cost pr itm pr yr is $5 And suppos w modl th crrying cost bsd on th vrg numbr of units in th invntory how mny units pr ordr would minimiz th totl invntory cost? Suppos th cost of producing units is C( 5 () If th pric pr unit is fid t find tht would mimiz profit (b) If th pric is bsd on dmnd by th dmnd qution p 5 (b) find tht would mimiz rvnu nd (b) find tht would mimiz profit 3 A quntity of rdioctiv lmnt dcys t rt proportionl to its siz Lt A ( dnot th mss whr t is in yrs Suppos w strt with grms nd ftr yr 9 grms rmin () Find formul for A ( (b) How long will it tk for th quntity to dcy down to grm? (c) Wht is th mss ftr 5 yrs? (d) Writ th diffrntil qution for A ( () How fst is th quntity dcying whn th mss is grms? 4 Bctri grow t rt proportionl to its siz Lt A ( dnot th popultion of bctri whr t is in hours Suppos you r givn tht A' ( 5 A( Th popultion strtd t () Find formul for A ( (b) How long will it tk for th popultion to rch? (c) Wht is th popultion ftr 5 hours? (d) How fst is th popultion growing whn th popultion is 8? () Wht is th popultion whn th popultion is growing t 3 pr hour? 5 An invstmnt rturns t 75% nnul intrst rt nd th initil invstmnt is $ Lt A ( dnot th blnc t tim t in yrs () Writ th diffrntil qution for A ( (b) Find th formul for A ( (c) How fst is th blnc growing whn th blnc is $6? (d) Whn will th blnc rch $3? () Wht is th blnc ftr yrs? 6 Assum w hv ccss to n invstmnt tht rturns t 75% nnul intrst rt How much should w invst now so tht w hv $ in yrs? 7 Suppos w hv $ to invst Wht intrst rt do w nd to find so tht w hv t lst $ in yrs? A (t dnot th popultion of popl who hv hrd th nws ftr t dys Suppos you r givn tht A ( stisfis th diffrntil qution A' ( 4( A( ) Assum tht no on initilly hrd th nws 8 Lt )
() Find th formul for A ( (b) How long will it tk for to hr th nws? (c) How mny popl hv hrd th nws ftr 5 dys? (d) How fst is th nws sprding (in popl pr d whn popl hv hrd th nws? () How mny popl hv hrd th nws whn th nws is sprding t 5 popl pr dy? f ( p) At th pric of p wht is th lsticity of dmnd nd how would th rvnu chng if th pric ws dcrsd? y f ( of continuous function tht hs f '( for ; f '( ) f '( for 9 Lt th dmnd function (quntity in trms of pric) b p 3 Sktch th grph ) 3 Sktch th grph f ( y of continuous function tht hs f ''( for ; f ''( ) f ''( for () I (b) II (c) VI (d) V f f () 6 8y 8 y y f 6 8y f 8 y (b) 3 8y y y 3 8y y y y (c) f / y / f y y / y f f p (d) ln q p q q 3 () Solutions to Prctic Problms f f ( y 5z) ( () y 5z y (y 5z) f f f f (f) p qy y y p q (b) f 4 6 6y f y y y 4 y f 3 64y y 4 () Locl min t 8 /8 /) f y f z y ( ( 5) (y 5z) 96 y f y 3 y y y f 4 y ( (b) Locl m t ( ) (c) Sddl point (nithr m nor min) t ( ) (d) Locl mins t both ( ) nd ( ) ; sddl point t ( 3 / ) 5 () ( y ) (5 /45 /4 5 / 7) ( y ) ( 4 / 5 3/5 / 4 (b) ) nd ( y ) (4 / 53/ 5 / 4) f y f 5 5 6 () ( 5) y 4 4 (b) f y y f ( 5) y 4
f 5 5 (c) f ( 5) f (5) (5)() () 4 f (d) f ( 7) f (5) (5)() () y 7 () III (b) I (c) II 3 3 8 () 9 / 3 3 6ln 6 C (b) 3 7 (d) 3 C () 36 (f) 5( ) 9 ( ) d 5 (4 ) d 3 / 3 (( 4 ) 6) d 4( 4 () 5 dt 5( ) 6478 4t 4 (b) 5 dt 5( ) 4 3 () Incrsing on C (c) 4 nd on 6 8 ; dcrsing on 6 (b) Locl m t ; locl min t 6 (c) Concv up on 4 8; concv down on 4 (d) Inflction point t 4 () f '(4) f '(5) 4 () /3 (b) /5 7 ( 3)5u 3 3 5 4 3 5 () y ' (b) g '( u) (c) '( ) 4 h 3 3 4 5 7 3 (5 u ) 3t 3t 3 3 ( t (t ) (d) y ' ln(3 7) () f '( (f) f '( 3 7 ( t 3 5 7 p (g) q ' p p (h) g '( (i) f '( p) 7 (5 )(ln(5 )) 5 7 7 7 y 3 6 () y 5( ) (b) y 8( ) (c) y ( ) [Not: y' ] 3y 7 () Dmnd dcrss t 3/month (b) Pric dcrss t $/3 pr month 8 () Dmnd incrss t 5/month (b) Pric incrss t $8/month 9 Objctiv A 3y with constrint 3 4y whr is th dimnsion prlll to th wll M r t ft by y 5 ft Objctiv L 3 4y with constrint y 48 whr is th dimnsion prlll to th wll Minimum lngth t 8 ft by y 6 ft 5 Objctiv C 6r with constrint r 6 Minimum cost t () 35 (b) (b) kt 53t 3 () A( whr k (ln 9) / 53 so A( (b) Solv 53t to gt t (ln) /(53) 8543 hours
k (c) A (5) 5 5 967 grms (d) A' ( 53 A( () Us diff q from (d) to gt A '( 53() 6 grms pr yr 5t 4 () A( (b) Solv 5t to gt t (ln) /(5) 5 35 hours k (c) A (5) 5 7 (d) Us givn diff q to gt A '( 5(8) bctri pr hour () Us givn diff q solv 3 5 A( to gt th popultion is 5 () A' ( 75 A( 75t (b) A( (c) Us diff q () to gt A '( 75(6) 45 $/yr (d) Solv 3 75t to gt t (ln 3) /(75) 4 65 yrs 75() 75 () A () 7 6 W wnt th prsnt vlu of $ t 75% intrst rt yrs from now 75() P 47367 7 Th unknown is r nd w wnt th prsnt vlu of $ in yrs to b $ now Solv () r to gt r 69% 4 A( ( t 8 () ) 4 (b) Solv ( t ) to gt t ln( 9) /( 4) 634 dys 4(5) (c) A ( ( ) 869 (d) Us givn diffrntil qution A '( 4( ) 3 popl pr dy () Solv 5 4( A( ) to gt 875 / f '() ( 5) f '( p) 5 p ; E ( ) Th lsticity of dmnd is so f () R '() nd thus if th pric ws dcrsd thn th rvnu would dcrs 9 3 (Othr solutions r possibl) 3 (Othr solutions r possibl)