= + and the demand function given by

Transcription

1 Math 8 REVIEW Part I: Problms. A firm s fid cost is $60,000; it costs th firm $77 to mak ach unit of its product; and it slls ach unit for $90. a) Find its cost, rvnu, and profit functions. b) How much dos ach unit mad and sold contribut towards profit? c) Find its brak-vn point. d) What is happning to th profit at 5,000 units? Wh? ) What is happning to th profit at,000 units? Wh?. A firm has a product whos total rvnu function is givn b R ( ) = 75 and total cost function is C ( ) = Dtrmin th possibl production runs th product can brak vn.. A firm has for a product its total cost givn b p = 500. Giv its profit function. C ( ) 0.5 0, 000 = + and th dmand function givn b 4. Th cost function for a product is givn b C ( ) = + 00 and its dmand function is p= 760, whr is th numbr of units pr month. a) Giv its rvnu and profit functions. b) Find th marginal cost, marginal rvnu, and marginal profit at 50 units. c) Intrprt our answrs to part b. d) Dtrmin th numbr of units that should b producd and sold pr month in ordr to maimiz profit. 5. A product has a dmand function givn b p = 480 and suppl function givn b p= a) Find its quilibrium point. b) Dtrmin whthr a unit pric of $400 will ild a surplus or shortag and plain wh. 6. A product has a total rvnu function givn b R ( ) 500 C ( ) = a) Giv th profit function. b) Dtrmin th numbr of units ndd to achiv maimum profit. c) What is th maimum profit? 7. If log ( ) = log log ( ), find. = and total cost function givn b 8. Dtrmin th numbr of ars it would tak to tripl th amount of principl invstd at 8.8% compoundd continuousl. 9. A radioactiv substanc has a half-lif of 6 months. Aftr 0 months, 4 grams rmain. Dtrmin th initial amount. 0 t h A= A 0. Dtrmin th futur valu of an annuit of $40 paid at th nd of ach 6-month priod for 0 ars at a 6% rat of intrst, compoundd smiannuall.. Dtrmin th futur valu of an annuit of $00 paid at th bginning of ach 6-month priod for 8 ars at a 6% intrst rat, compoundd smiannuall.. A dbt of $8,000 is to b amortizd with 8 qual smiannual pamnts at a % intrst rat, compoundd smiannuall. Dtrmin th amount of thos pamnts.

2 . Th Smiths want to hav $0,000 in two ars for a down pamnt on a hous. To mt this goal, th will mak dposits at th nd of ach month for th nt two ars into an account that pas % compoundd monthl. a) Find th siz of th monthl pamnts. b) What wr thir total arnings on this account? 4. Th Smiths in problm # abov dcid to bu a hous pricd at $00,000. With th $0,000 accumulatd from thir sinking fund account and anothr $0,000 from savings, th mak a 0% down pamnt and tak out a loan on th balanc at 8% compoundd monthl for th nt 0 ars. a) Dtrmin th amount of thir monthl pamnts. b) What would b th total intrst paid ovr th 0 ars? c) Aftr living in thir hom for 0 ars, th found it ncssar to rlocat du to Mr. Smith s job. Find th unpaid balanc of th loan at this tim. For #5-9, dtrmin th indicatd its: 5. h 0 f( + h) f( ) h if f( ) = 7. h 0 ( a+ h) a h ( + ) f( ), f( ), f( ) + if, < f( ) = , 0. If, 5 f( ) =, c, > 5 dtrmin th valu of c that will allow f( ) to b continuous. 4. Givn: f( ) = + + a) Dtrmin th intrval(s) whr f is incrasing and whr it is dcrasing. b) Dtrmin th intrval(s) whr f is concav up and whr it is concav down. c) Giv local trmas and point(s) of inflction, if an ist. d) Using th abov information, sktch its graph. ) Dtrmin th ara of th rgion boundd b th curv and th -ais and btwn = - and =.. If f() = 5 +, us th it dfinition of th drivativ to find f '( ). Find th indicatd drivativs:. f '( ), if f( ) ( ) = 5. d, if d = ln 4. d, if d + = 7 d 6., if d =

3 7. (ln ) If f( ) =, find f '( ). 8. A ball is thrown vrticall upward from th ground with an initial vlocit of 8 ft/sc. If th positiv dirction of th distanc from th starting point is up, th quation of th motion is st ( ) = 6t + 8 t. a) Find th instantanous vlocit and acclration of th ball aftr sconds. b) Dtrmin how long it will tak th ball to rach its highst point. c) What is th maimum altitud attaind b th ball? d) What is th maimum vlocit attaind b th ball? Dtrmin th quation of th tangnt lin that satisfis th following conditions. 9. Tangnt to th graph of 6 5,. + = at ( ) 0. Tangnt to th curv 4 = that has slop of 7.. An fficinc stud was conductd for th postal sstm of a larg mtropolitan cit. It was found that th numbr of pics of mail sortd b th avrag workr t hours aftr starting work at 8:00 a.m. is givn b Nt ( ) = t + 5t + 0, whr 0 t 4. At what tim during th morning shift is th avrag workr prforming at pak fficinc? ( ). Using th curv-sktching procdur discussd in class, sktch th graph of =. (Domain, intrcpts, asmptots, sign analsis to dtrmin local trmas, concavit). Th total ara of a circl with radius and a squar with sid 0 is givn b th formula Dtrmin th valu of that will minimiz th total ara. A= π + (0 ). 4. A rctangular bo with a squar bas and a volum of 0 ft is rquird to ship an itm. Th matrial for th bas costs 40 cnts/ ft, th sids costs 5 cnts/ ft, and th top costs 5 cnts/ ft. a) If rprsnts th lngth of a sid of th squar bas, prss th surfac ara S of th bo as a function of. b) Eprss th total cost C of th rquird matrial as a function of. c) Giv th domain of C. d) Dtrmin th dimnsions of th bo that can b constructd for th last cost. ) Find th cost of th bo if th abov dimnsions ar to b usd. 5. If th profit of a product is prsntd b P ( ) = and if th profit is incrasing at th rat of $4000 pr month, dtrmin th approimat rat of chang of sals with rspct to tim (in units/month) whn th monthl sals ar =00 units.

4 6. Th monthl dmand function for a product is givn b p =. Th manufacturr is incrasing th 500 production of this product at th rat of 50 units pr month. a) Giv th rvnu function. b) Find th rat of chang of th rvnu with rspct to tim (in months) whn th monthl production is 50 units. 7. Us diffrntials to approimat (to four dcimal placs). 5, Th cost function for th production of a product is givn b C ( ) =. Us th diffrntial of C to 75 dtrmin th approimat chang in cost as production is incrasd from 00 to 0 units. 9. If ( ) = 4, ( + 5) us logarithmic diffrntiation to find d d. 40. Considr th dmand quation = 5 5 p, whr 0 p 5. a) Dtrmin whn th dmand will b unitar, th intrval whr dmand will b inlastic, and th intrval whr it will b lastic. b) In ach of th abov thr cass, tll what happns to th total rvnu whn th pric dcrass. Evaluat: 4. ( + ) d d ( ) 4. 4 d 46. d (ln ) d 0 d 47. If G"( ) = +, with G'(0) =, and G(0) =, find G ( ). 48. If f( ) =, comput th Rimann sum of f ovr th intrval [,4] using four subintrvals of qual lngth. Choos th rprsntativ points to b th midpoints of th subintrvals Th marginal rvnu for a product is givn b R'( ) = + 0. ( + ) Dtrmin its rvnu function R. ( ) 50. Th marginal cost function of a product is givn b approimat total cost of th first 400 units. dc d = 89, whr is th numbr of units. Dtrmin th + 5. Find th ara of th rgion boundd b f( ) =, th -ais, =, and = 5. 4

5 5. Find th ara of th rgion boundd b f( ) = + and f ( ) =. 5. A product has a suppl function givn b p= + and a dmand function givn b p = Find a) its consumrs surplus and b) its producrs surplus if production is st at markt quilibrium Suppos gold is bing producd at a rat of = t mtric tons pr ar t, whr t = 0 rprsnts th prsnt. Find th ara undr this function from t = to t = 4 and intrprt our rsult. (Answrs to Part I). a) C ( ) = ,000 ; R ( ) = 90; P ( ) = 60,000 b) $ c) (0,000,,800,000) d) Sinc 5,000<0,000, thr will b a loss. [ Also, P (5, 000) = 65, 000] ) Sinc,000>0,000, thr will b a profit. [ Also, P (, 000) = 6, 000]. 0 units; 0 units. P ( ) = , a) R ( ) = 760 ; P ( ) = b) C'(50) = 00; R'(50) = 460; P'(50) = 60 c) incras in cost of $00 on production of 5 st unit; incras in rvnu on sal of 5 st unit; incras in profit of $60 on sal of 5 st unit d) 90 units 5. a) ( 0,40 ) b) shortag; dmand > suppl units ars grams 0. $ $45.. $88.9. a) Sinking fund; $74.46 b) $, a) $98.6 b) $44,7.60 c) $6, a , 8, DNE. a) incrasing on ( 0, + ) ; dcrasing on (,0) b) concav upward on (, ) and (, + ) ; concav downward on (, ) c) no local maimum local minimum at (0,) points of inflction at (, ) and (, ) 47 ) sq. units ( ) ( ) d) ( ) ( ) 8 ( + )

6 8. a) v() ft/sc; () - ft/sc = a = b) 4 sconds c) (4) 56 ft s = d) 8 ft/sc = 0. = : 0 a.m π +. - Vrtical asmptot at = 0 Horizontal asmptot at = Crossovr at (, ) Critical valu at = Local minimum at (, 0) a) S ( ) = + b) C ( ) = c) > 0 d) 5 ) $ incrasing units/month 6. a) R ( ) = b) $,0, d ( )( + 4) $ = 5 d ( + 5) 40. a) unitar whn p = $4.4; inlastic whn 0 p < $4.4; lastic whn $4.4 < p $5 b) unitar: rvnu dos not chang; inlastic: rvnu dcrass; lastic: rvnu incrass ( ) C 4. ( 4) ( 4) 4 C C ( ) C G ( ) = R ( ) = + 05 ( + ) 50. $ ln sq. units 5. ( + ) d =.06 sq. units 6 5. a) $4. b) $ mtric tons; total production from t = to t = 4 ars 6

7 Part II: Multipl Choic. Th graph of f( ) = + 7 has (A) On local maimum, on local minimum, on point of inflction. (B) On local minimum, no local maimum, on point of inflction. (C) On local minimum, no local maimum, on point of inflction (D) On point of inflction onl (E) Non of ths. Th diffrntial of p for th dmand function p ( ) = 80 7 is 7d d 7d (A) (B) (C) d (D) 80 7 (E) non of ths. If, for all valus of, f '( ) < 0 and f "( ) > 0, which of th following curvs could b part of th graph of f? (A) (B) (C) (D) (E) 4. For th function f dfind, at which point is f "( ) not positiv? (A) A (B) B (C) D (D) F (E) non of ths D E B C A F 5. If a commodit s lasticit of dmand at pric p is givn b E( p) and E( p ) =.5, thn which of th following is tru? (A) Incrasing th pric b a small amount will rsult in a dcras in rvnu. (B) Incrasing th pric b a small amount will rsult in an incras in rvnu. (C) Incrasing or dcrasing th pric b a small amount will lav rvnu approimatl constant. (D) Cannot b dtrmind. (E) Non of ths (Answrs to Part II). D. C. E 4. B 5. A 7