Direction: You may use computer or graphing calculator to answer the following questions. Use separate paper to show all work.

Transcription

1 / MATH 76 Test Review Coputer Part Direction: You ay use coputer or graphing calculator to answer the following questions. Use separate paper to show all work.. Starbucks profit is growing at a rate of P( ) 5..8 billion dollars per year, where is the nuber of years after. If Starbucks had a $ billion profit in 5, what is the epected profit in?. Marginal cost, in hundred dollars, is C( ).6, where is the nuber of units. If the fied costs are $5, find the cost of producing 5 units.. Maria takes a cake out of a 75 F oven and places it on a counter. The rate at which.7t the cake cool is given by T( t).5e degrees per inute. If the cake needs to be not ore than 5 F before it can be iced, how long will Maria have to wait before icing the cake?.5t. A bacteria culture grows at a rate of P( t) 5.8e bacteria per hour. If there were initially 8 bacteria to the culture, how any will there be after day? 5. Annual per capita spending on health care in the United States since 5 has grown at a rate of S( ) 5.8(.67) dollars per year, where is the nuber of years after 5. If annual per capita spending in is $87, how uch was spending in? 6. The nuber of new cases in a flu epideic during the first week grows at a rate of.6 N( ).e new cases per day. How any cases were reported between the end of the third day and the end of the seventh day? 5 7. Sales of a new sailboat have grown at a rate of S( ) 9 boats per onth. How any boats were sold fro the end of the second onth to the end of the sith onth? 8. Estiate the area of the function between f ( ) and the -ais on [, ] a) Sketch the graph of the function. Use window: -in:, a: yin: ya: 6 b) Use trapezoid geoetry to estiate the area. c) Draw four rectangles to estiate the area. Show work. d) Find d e) Are answers fro b), c) and d) above the sae? Different? Why or why not? Eplain clearly.

2 / 9. Estiate the area of the function f ( ) e and the -ais on [, ] a) Sketch the graph of the function. Use window: -in:, a: yin: ya: b) Draw four rectangles to estiate the area using the idpoint of each of the four rectangles. Show work. c) What is the final estiate for the area of this function?. Estiate the area of the function f ( ) and the -ais on [, ] a) Sketch the graph of the function. Use window: -in: 5, a:, yin: 5, ya: 5 b) Draw three rectangles to estiate the area using the idpoint of each of the three rectangles. Show work. c) Find d d) Does answer fro c) above the sae as fro b) above? Why or why not? Eplain clearly.. Find the area of the region enclosed between the two curves f ( ) and f ( ) on [, ]. Find the area of the region enclosed between two curves f ( ) and f ( ) on [, ]. Money is being transferred into an account at a rate of $5 a year with $5 increases per year. If the account pays 5.5% copounded continuously, what is the future value of the account after four years?. The owners of a sall airline are aking big plans. They hope to be able to buy a large airline years fro now by investing an account pays 9.% continuous copounding. The owners have deterined that they can afford to invest $. illion each year. How uch will these investents be worth ten years fro now? 5. Last year, profit for the HiTech Corporation was $7. illion and the profits increase continuously at a rate of $. illion per year. The account earns % annual interest, copounded continuously. What is the value of the account after five years? 6. Find the consuer surplus for the deand curve D( q).5q.5 dollars at the arket price of P $8 7. Find the producer surplus for the supply curve S( q).q.75 dollars at the arket price of P $

3 / 8. Supply for a product is given by S( q) 5 q and deand for a product is given 6 by Dq ( ) q. a) Find the equilibriu point. b) Find the arket surplus up to equilibriu using the integral definition. c) Find the consuer surplus. d) Find the producer surplus. 9. The supply and deand functions for a new video gae are given by.8q S( q) q and D( q) e, where q. a) Sketch the supply and deand curves on the sae set of aes. b) Find the equilibriu point. c) Find the arket surplus up to equilibriu.

4 / FORMULAS FOR TEST MATH 76 Gini Inde (Optional) Consuer Surplus Producer Surplus Market Surplus G L d q CS D( q) P dq where P is the arket price that consuers will deand q quantity units. q PS P S q dq where P is the arket price that producers are willing to supply q quantity units. Market Surplus = Consuer Surplus + Producer Surplus q MS = CS + PS = ( ) ( ) D q S q dq Equilibriu Point Continuous Incoe Flow D(q) = S(q) Future Value T rt rt FV e f () t e dt dollars

5 / Price p Consuers surplus p D(p) q q quantity p price p Producers surplus q q quantity P price Consuers surplus D(q) S(q) Producers surplus p q, p q q quantity 5

6 / Test Review Non Coputer or Graphing Calculator Part Direction: No graphing calculator or coputer is allowed for this part of the test. Fro probles 8, evaluate each integral.. d d e d.5.65 d d d.8 d d..5 d.... e d d d 5 d d 5 d 8. d.5e.8 d d 9. Find the specific antiderivative of f () where f ( ) where F() 5. Find the specific antiderivative of f() where f ( ) e where F() 6

7 /... P( ).6.8 C P since. (5).6(5).8(5) C C. P( ).6.8. P(). billion In, Starbucks epects to have profit $. billion. C( ). C hundred dollars Fi cost: C().5 C( )..5 C(5).(5) (5) To produce 5 units, it will cost $8.5 hundred dollars or $, t T () t e C.7.7t T ( t) 5e C F degree When the cake is out of the oven, the teperature is 75 F. This eans when t =, T () = 75..7() T () 5e C 75 5 C 75 C 7.7t T( t) 5e 7 F degree Find t =? when the teperature is 5F.7t 5e 7 5 t 9. Will wait 9. inutes before reach 5F to icing. MATH 76 Test Review Coputer Part Answer Key..5t P( t).e C bacteria P() 8 P e C Since C 6.8 (). 8 P t.5t ( ).e 6.8 P(8 hours) 65.9 After days, which is 8 hours, there would be approiate,6,5 bacteria. 7

8 / 5. S ( ) C dollars after 5 S(5) 87 Since there is $87 in, 5 S(5) C 87 C S ( ) dollars after 5 8 S(8) For, S(8).8 The aount spent in was approiate $, N ( ). e d e.567 Between the end of the third day and the end of the seventh day, there were approiate flu cases reported. 6 ( ) 5 5ln( ) S,675 boats were sold fro the end of the second onth to the end of the sith onth. 8. f()= + b) = a = f()= + ()= = b = f() = + () = 5 h = Area = ½ h (a + b) = ½ () ( + 5) = 6 c) Area = f(.5) + f(.5) + f(.5) + f(.5) Area = ( f(.5) + f(.5) + f(.5) + f(.5)) = ( ) = 6 d) d = = 6 e) The answers fro b, c, and d are sae. They are very accurate to estiate the area under the curve. 8

9 / 9. b) f ( ) e rectangle length height area I.5 f (.5) = II.5 f (.75) = III.5 f (.5) = IV.5 f (.75) = Total area: c).888. rectangle length height area I f (.5) = II f (.5) = III f (.5) = Total area: 9.5 c) d = 9 d) No, they are not the sae. Area is always positive. The integration is negative since it is below the -ais. d 9.5 show work. y = ^+ y = -. d. f ( t) 5 5t r = 5.5% =.55 T = 9

10 / T rt rt FV e f () t e dt.55.55t FV e 5 5t e dt FV The total aount will be $,55.7 after years.9().9 t). R(t) =., r = 9.% =.9, T = FV e. e dt FV 5.77 The investent will be worth $5.77 illion years fro now. 5.(5).t FV e 5. P( t) 7..t, r =., T = 5 (7.. t) e dt FV 7.9 The investent for revenue will be $7.9 illion in 5 years. 6..5q.5 8 q 85.7 y = CS D( q) 8 dq CS.75q 6.5 CS.5q.5 8 dq CS.5q 6.5 dq 85.7 CS 6.57 The consuers gain approiate $6.57 buying at $8 rather than at the price they would have been willing to pay. y = 8 7..q.75 q 75 y = y =.+.75

11 / PS S( q) dq PS.q.75 dq PS.5.q dq PS.5q.5q 75 PS The producers gain about $5.88 supplying at the price of $ rather than the price they would have be willing to supply at. 8 a) To find the equilibriu point, 6 5 q q ; q = 99.9 p (99.9, 6.8) y = 6/ b) 5 MS q dq q MS 8.6 The arket surplus is approiate $,8.6. y = 6 y = 5sqrt() c) CS 6.8 dq.9 q The consuer surplus is approiate $, d) PS q dq 68.7 The producer surplus is approiate $,68.7. The arket surplus = consuer surplus + producer surplus.

12 / 9 a).8q e q q9.79, p 8. (9, 96.9) y = e^(-.8) 9.79 MS 9..8q MS e q dq y = sqrt()+ The arket surplus is appro. $9..

13 / MATH 76 Test Review Non Coputer or Graphing Calculator Part Answer Key. d d ln C d C e d e ln C ln d C.75.8 d d 8 C d d 8. 8 d () d...5.ln.5.ln5.5.ln.67 e d e.7. d ln ln ln.86

14 / d d d d d () 8. d C e d e C e C 8. d ln ln ln F( ) d ln C ln C 5 C.5 F( ) ln.5. f ( ) e where F() F( ) e ln C F e ln C C 7.8 F( ) e ln 7.8