B) 25y e. 5. Find the second partial f. 6. Find the second partials (including the mixed partials) of

Transcription

1 Sampl Final Suppos z = (, y), ( a, b ) = 0, y ( a, b ) = 0, ( a, b ) = 1, ( a, b ) = 1, and y ( a, b ) =. Thn (a, b) is h s is inconclusiv a saddl poin a rlaiv minimum a rlaiv maimum. * (Classiy criical poin using. A manuacurr wih clusiv righs o a sophisicad nw indusrial machin is planning o sll a limid numbr o h machins o boh orign and domsic irms. Th pric h manuacurr can pc o rciv or h machins will dpnd on h numbr o machins mad availabl. (For ampl, i only a w o h machins ar placd on h mark, compiiv bidding among prospciv purchasrs will nd o driv h pric up.) I is simad ha i h manuacurr supplis machins o h domsic mark and y machins o h orign mark, h machins will sll or y y 0 + housand dollars apic a hom and or 50 + housand dollars apic abroad. I h manuacurr can produc h machins a a cos o $1000 apic, how many should b supplid o ach mark o gnra h largs possibl proi? = 50, y = 1 =, y = 9 = 10, y = 50 = 85, y = 150. * (Classiy criical poin using. Th only grocry sor in a small rural communiy carris wo brands o rozn appl juic, a local brand ha i obains a h cos o cns pr can and a wll-known naional brand ha i obains a h cos o 0 cns pr can. Th grocr simas ha i h local brand is sold or cns pr can and h naional brand or y cns pr can, approimaly y cans o h local brand and y cans o h naional brand will b sold ach day. How should h grocr pric ach brand o maimiz h proi rom h sal o h juic? local brand () a 8 cns, naional brand (y) a 5 cns local brand () a 8 cns, naional brand (y) a 118 cns local brand () a 5 cns, naional brand (y) a 59 cns local brand () a cns, naional brand (y) a 59 cns. Compu or 5y 5 5y 5y 5 5 y 5 y 5y (, y) 5y =. 5. Find h scond parial y givn 5y y (1 + ) + ( 5y 0 ( + 5 y) + ( 5y (1 + ) + ( 5y 0 ( + 5 y) ( 5 (, y) = y + y ln + y. Find h scond parials (including h mid parials) o 9 y 9 y 9 y =, =, y y = 9 y + y, =, 5 = 19 y + 5, 5 = 19 y + 5, y y y 5 5 = 19 y + 5, = 0 y. Compu (ln, ln8) i 18 1 non o h abov (, y) + 5 = 19 y + 5, 5 = 19 y + 5, y y =. 5 = 19 y + 5, (, y) = 8 y + 5y = 0 y = 0 y 5 = 0 y L ( ) = 1 1ln, or > 0. Find h minimum valu o or > 0. ( 1ln()) 1( 1ln()) 0 (1 ln()) Pag 1 Pag

2 9. (Find rlaiv ras o rurn) Suppos your amily owns a rar book whos valu yars rom now will b V ( ) = 9 dollars. I h prvailing inrs ra rmains consan a % pr yar compoundd coninuously, whn will i b mos advanagous or your amily o sll h book and invs h procds? 1. yars 8.11 yars yars 5 yars 10. How quickly will mony ripl i i is invsd a % inrs compoundd coninuously? 15.9 yars 15.1 yars 15. yars 15.5 yars 11. Find d ( ) d, whr ( ) ln 8 =. 1. Th quaion o h angn lin o 1 y = 1 y = y = 8 y = 1 ( ) ln = a = is ( ) 1. Find h drivaiv o ln ( ln ) ln ln + ln ln(ln ) p Th consumr dmand or a crain commodiy is D( p) = unis pr monh whn h mark pric is p dollars pr uni. Eprss consumrs'oal monhly pndiur or h commodiy as a uncion o p and drmin h mark pric ha will rsul in h gras consumr pndiur. $1.91 $ $.1 $ Us logarihmic dirniaion o ind ( ). ( ) = 5 + ( ) = ( ) ( ) = ( ) /5 + 9 ( ) = + / ( ) = ( ) Solv or : ln ln = 18. = = 18 = = Pag Pag

3 1. A radioaciv subsanc dcays ponnially. I 00 grams wr prsn iniially and 00 grams ar prsn 100 yars lar, how many grams will b prsn ar 00 yars?.1 grams 0 grams.1 grams.91. Find h inrvals o incras and dcras or h uncion dcrasing or > and incrasing or < dcrasing or < and incrasing or > dcrasing or all incrasing or all ( ) = Solv or : log ( 1) = Drmin whr h graph o > 1 < 1 > -1 < -1 ( ) = is concav down. I $500 is invsd a prcn compoundd coninuously, wha is h balanc ar 11 yars? $115.5 $5 $599. $ Find all h criical poins o h uncion ( ) = , 1, non Find h inrvals o incras and dcras or ( ) =. + 8 incrasing on <., dcrasing on >. incrasing on <. and >. incrasing on 0.15 and on >., dcrasing on 0.15 <. incrasing on 0.15 <., dcrasing on 0.15 and on >.. A 5-yar projcion o populaion rnds suggss ha yars rom now, h populaion o a crain communiy will b P( ) = housand. 1) A wha im during h 5-yar priod will h populaion b growing mos rapidly? ) A wha im during h 5-yar priod will h populaion b growing las rapidly? ) A wha im is h ra o populaion growh changing mos rapidly? = 5 yars; = 0 yars; = 0 yars = 0 yars; = 0 yars; = yars = yars; = 1 yar; = yars = yars; = 0 yars; = yars 5. Find h absolu maimum o h uncion ( ) 5 = on h inrval. An aparmn compl has 50 unis. Whn h monhly rn or ach uni is $0, all unis ar occupid. Eprinc indicas ha or ach $1 pr monh incras in rn, unis will bcom vacan. Each rnd aparmn coss h ownr o h compl $50 pr monh o mainain. Wha monhly rn should b chargs o maimiz proi? $15.5 $51 $.5 $.5 Pag 5 Pag

4 . Find wo non-ngaiv numbrs whos sum is 8 i i is rquird ha h produc o hir squars is o b as larg as possibl. and 1 and 0 and 8 and 8. A commur's rain carris 00 passngrs ach day rom a suburb o a ciy. I now coss $1 pr prson o rid h rain. A sudy shows ha 50 addiional popl will rid h rain or ach 5 cn rducion in ar. Wha ar should b chargd in ordr o maimiz oal rvnu? 8 cns 9 cns 80 cns 85 cns 1. Th ownr o a novly sor can obain joy buzzrs rom h manuacurr or 0 cns apic. H simas h can sll 80 buzzrs whn h chargs $1 apic or hm and ha h will b abl o sll 8 mor buzzrs or vry 10 cn dcras in pric. Wha pric should h charg in ordr o maimiz proi? $1 $1.1 $0.9 $1.. 1 Th drivaiv o ( ) = is 1 9. Find h lasiciy n o h dmand uncion q =. 1 + p n = 1 + p p n = 1 + p n = p n = p 0. A Florida cirus growr simas ha i 0 orang rs ar pland, h avrag yild pr r will b 00 orangs. Th avrag yild will dcras by orangs pr r or ach addiional r pland on h sam acrag. How many rs should h growr plan o maimiz h oal yild? 0 rs 0 rs 0 rs 5 rs. Th graph o ( ) = 5 + has a maimum a = 10 a minimum a = 10 a maimum a = 0 a minimum a = 0. Dirnia: ( ) = Pag Pag 8

5 5. Find h scond drivaiv o h givn uncion and simpliy your answr: ( ) = ( ) = ( + 9) 18 ( ) = ( + 9) ( ) = + 9 ( ) = Evalua h limi: lim Evalua 8 d. 8 / / / 9. Evalua + d + 5 C 0. In a crain scion o h counry, h pric o chickn is currnly $ pr kilogram. I is simad ha wks rom now h pric will b incrasing a a ra o + 1 cns pr kilogram, pr wk. How much will chickn cos wks rom now? $. $0. $. $. 1. Us h undamnal horm o calculus o ind h ara o h rgion undr h lin y = abov h inrval An animal populaion incrass a h ra o pr yar. Wha is h approima incras in h animal populaion rom h nd o yar o h nd o yar 5? An objc is moving so ha is spd ar minus is mrs pr minu. How ar dos h objc ravl bwn h nd o minu and h nd o minu? 158 mrs 9 mrs 18 mrs mrs. Rcords indica ha hours pas midnigh, h mpraur a h local airpor was ( ) = 0. + k + 5 dgrs Clsius. I h avrag mpraur bwn 11 A.M. and 1 P.M. is 50 dgrs C, wha is k? Pag 9 Pag 10

6 . Mony is ransrrd coninuously ino an accoun a h consan ra o $100 pr yar. Th accoun arns inrs a h annual ra o % compoundd coninuously. How much will b in h accoun a h nd o yars? $5. $ $1. $ I is simad ha days rom now a armrs crop will b incrasing a h ra o bushls pr day. By how much will h valu o h crop incras during h n days i h mark pric rmains id a $ pr bushl? $8 $1 $ $1. Mony is ransrrd coninuously ino an accoun a h consan ra o $1100 pr yar. Assum h accoun arns inrs a h annual ra o % compoundd coninuously. Compu h uur valu o h incom sram ovr a 1 yar priod Dmographic sudis conducd in a crain ciy indica ha h racion o h rsidns ha will rmain in h ciy or a las yars is ( ) =. Th currn populaion o h ciy is 500,000, and i is simad ha nw rsidns will b arriving a h ra o 1,000 popl pr yar. Assuming his sima is corrc, giv an prssion or h populaion as a uncion o. 50, ,000 0, ,000 / 50 0, ,000 50, ,000 Answr Ky 1. A. C. D. A 5. B. A. A 8. D 9. B 10. A 11. A 1. A 1. A 1. C 15. B 1. A 1. A 18. A 19. C 0. A 1. B. B. B. A 5. A. B. D 8. C 9. B 0. A 1. B. A. D. A 5. A. C. B 8. A 9. A 0. A 1. C. B. D. C Pag 11 Pag 1

7 5. B. D. D Pag 1