Answers to Selected ExercisesChapter 0148323517991113313141551817133 - PDF document

1 A18 Answers to Selected ExercisesChapter
A18 Answers to Selected ExercisesChapter 01.Š483.2/35.Š17.99.111.3313.1415.5/1817.13.3119.621.43/1623.025.273.36/91/81611.32217.x519.Šy x21.1 x23.x3y25.z4 y327.x6 y629.x4y6 z431.3 x433.3 4x2/335.1Š0.3x2Š6 5x37.239.1/241.4/343.2/545.747.549.Š2.66851.3/253.255.257.x+961.x3
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notasolution.) 213/34 ANSWERS TO SELECTED EXERCISESue. We can construct a table of values from any graph byreading off a set of values.alse. In a numerically speciÞedfunction, only certain values of the function are speciÞed, givingonly certain points on the graph.They are different por-tions of the graph of the associated equation is the same as the graph of Mi

2 ssing value: 11; Missing value: Missing
ssing value: 11; Missing value: Missing value: 7; (0)25.27.29.31.33.35.55.a.Undefined Compute the corresponding successive changes (3, 2) y 2 xy0 3 xy0
11xy
5)
4xy 23xy
11xy . If the answer is alwaysthe same number, then the values in the table come from a linear bx+c b=0 isundeÞned,cannot be specified as a function of . (The graph of thesulting equation would be a vertical line.)is positive then negative then hile leaving Þxed.per day, marginal cost bicycle5.a.500)9.a.11.a.; supply: 13.a.Thenumberofnewin-groundpoolsincreasedmostrapidlyduringtheperiods1996Ð1997and1998Ð1999,whenitroseby10,000newpoolsinayear.transactions.Theslopegivestheadditionalnumberofonlineshop-pingtransactionsperyear,andismeasuredin(millionsof)trans-actionsperyear.17.a.Medicarespendingispredictedtoriseatarateof$14.4billionperyear19.a.21.a.50,000$1,000,000; marginal income is100)domain [0, 405]. For proÞt, 20,000g per day for break evenIncreasing by$355,000peryear43.a.200if0200if530

3 ,000if014,000if574if1 3Š103 if1(1)It mus
,000if014,000if574if1 3Š103 if1(1)It must increase by10units each day, including the third.the number of items from the breakeven results in a proÞt:Because the slope of the revenue graph is larger than the slope ofthe cost graph, it is higher than the cost graph to the right of thepoint of intersection, and hence corresponds to a proÞt. 16314_19_ans_pA18-A42.qxd 7/17/06 11:54 PM Page 20 ANSWERS TO SELECTED EXERCISESarbitrary7,12,4(1),arbitrary19.a.possible; setting 550 packages from DufÞn House, 350 from600 packages from DufÞn House, 200 from5000 hits per day33.a.arbitrary100 book orders per dayday100 book ordersper dayNew York to OHaganBooks.com: 450 packages,New York to FantasyBooks.com: 50 packages, Illinois toOHaganBooks.com: 150 packages, Illinois to FantasyBooks.com:91522042548940167387633012398511593546249437.a.a.b.[760 800 300]39.Sales=70013002000400300500Inventory300700300060047001500
2
102
1xy [49.158.975.443.2]; 1990 dis-[50.859.785.45

4 2.8]; Net change 1980 to[1.70.8109.6] (a
2.8]; Net change 1980 to[1.70.8109.6] (all net increases)otal Bankruptcy Filings Brooklyn ilings in Newark [150250150100150] [300400300200250] [250400250200200] [7001050700500ilings in Brooklyn Fil-ings in Newark [300400300200250] [250400250200200] [50050050].The difference wasreatest in January 01, July 01, and January 02.49.a.ProcMemTubesPomII216201440Inventory = 500500010,000200200020,000Inventory30034008000100160016,00051.a.4401909509501790200204050502008Tourism420230100010001790300 4302109759751790250th entry of the sum is obtained by adding theIt would have zeros down the maintrary numbers.Answers will vary.401The associativity of matrix addition is a consequence of the associativity of additionof numbers, since we add matrices by adding the correspondingentries (which are real numbers).Answers will vary. 200420052006 $8000$7200$8800 $5600$5760$7040 $2800$3500$4000 Revenue 16314_19_ans_pA18-A42.qxd 7/17/06 11:54 PM Page 23 ANSWERS TO SELECTED EXE

5 RCISESYesYes 20Š1 2Š1 21Š1 2 19.11
RCISESYesYes 20Š1 2Š1 21Š1 2 19.11 3Š1 31Š2 3Š1 3Š11 32 3 21.01Š125.1Š01227.Š21 21 21 2Š1 229.Š2Š 2Š1 3808126500702602351692690(6,49.a.(10,(6,1,5)(0,0,0)51.a.servingsofbeans,andslicesofbreadservings of beans and 53.a.100 batches of vanilla, 50 batches of mocha, 100 batches of strawberry100 batches of vanilla, no mocha,200 batches of strawberry101batches of Vanilla, batches of strawberry$5000 in PNF,$2000 in FDMMX, $2000 in FFLIX100 APPL, 20 HPQ,[53.364.0101.6[53.564.1101.265.5]61.a.[3781408015454096295948]The inverse does not existÑthe ma-trix is singular.(If two rows of a matrix are the same, then row re-ducing it will lead to a row of zeros, and so it cannot be reducedto the identity.).)D| I] easily reduces to a matrix thathas a row of zeros on the left-hand portion, so that is singular,Conversely, if none of the are zero, then [] easily reducesto a matrix of the form [], showing that is invertible.AIAhas an inverse, then every s

6 ystem of equations has a unique solution
ystem of equations has a unique solution, namely matrix with a row of zeros, then such a system has either inÞnitelymany solutions or no solution at all. 1100Š1]5.BbAq[0]7.Strictly determined. AÕs optimal strategy is ; BÕs optimal strat-; value: 1Not strictly determinedNot strictly determinedmined0010]; e=2.2519.[100]Tor [010]T;e=1/421.R=[1/43/4],C=[3/41/4]T, e=Š1/423.R=[3/41/4],C=[3/41/4]T,e=Š5/425.FriendHTYouSweden; N Norway;ourOpponentDefendsYouIourOpponentBNSYou10000PleasantTap; T Thunder Rumble; S Strike the Gold,PTSNYou1035104033.a.CE should charge $1000 and GCS should charge $900;15% gain in market share for CECE should charge $1200(the more CE can charge for the same market, the better!)lo vs. Noto; evenly matchedshould use the northern route; 1 day41.a.24212524Ohio, leaving Bush with a 21% chance of winning the election.ou can expect to lose 39 customers.Option 2: move to47.a.About 66%es; spend the whole night 16314_19_ans_pA18-A42.qxd 7/17/06 1

7 1:54 PM Page 25 ANSWERS TO SELECTED EXE
1:54 PM Page 25 ANSWERS TO SELECTED EXERCISES1.3.UnboundedUnbounded5.7.UnboundedUnbounded9.11.UnboundedUnbounded13.15.Corner point: (2, 0)Corner points: (2, 0), (0, 3)17.19.Bounded; Corner points:Bounded; Corner points: (5, 0), (10, 0), (10, 8), (0, 0), (5, 0), (0, 5), (0, 8), (0, 5)(2, 4), (4, 2) y 610 Solutionset5810x  y   y
332Solutionset3x
 6 3x   6 y 8Solutionset 124x
y   y 
5 5Solution setxy 443 Solutionsetx
y  114 3y setx  3y y set13x  2y 2 5xy x
2y  8 8
4Solution set x y 1052set x  y  21.23.Unbounded; Corner points: Unbounded; (0, 10), (10, 0), (2, 6), (6, 2)Corner points:25.27.Unbounded; Corner point: (0, 0)Corner point: (Corner points: quartsofCreamyVanilla,quartsofContinentalMochaCorner points: x y2x  y  500x  y  300250300500300set 1.072.33 yx  8.5y  104.1x
4.3y  4.47.5x
4.4y  5.7 4.710
16.5     y

2x
y  0x
3y  0 y Solutionset52 
3 x
 6
3x  53 y2 20x  10y  10010x  20y  10010x

8  10y  80 810 16314_19_ans_pA18-A42.qx
 10y  80 810 16314_19_ans_pA18-A42.qxd 7/17/06 11:54 PM Page 27 ANSWERS TO SELECTED EXERCISEShouldpurchase500calculustexts,nohistorytextsandnomarketingtexts.ThemaximumproÞtis$5000persemester.ThecompanycanmakeamaximumproÞtof$650bymaking100gallonsofPineOrange,200gallonsofPineKiwi,and150gallonsofOrangeKiwi.Thedepart-mentshouldoffernoAncientHistory,30sectionsofMedievalHistory,and15sectionsofModernHistory,foraproÞtof1,050,000.Therewillbe500studentswithoutclasses,butallsectionsandprofessorsareused.Plant80acresoftomatoesandleavetheother20acresunplanted.ThiswillgiveyouaproÞtof$160,000.ItcanmakeaproÞtof$10,000byselling1000servingsofGranola,500servingsofNuttyGranolaandnoNuttiestGranola.Itisleftwith2000oz.almonds.illiongalstoprocessAand45milliongalstoprocessC.An-othersolution:Allocate10milliongalstoprocessBand40mil-liongalstoprocessC.Use15servingsofRiboForceHPandnoneoftheothersforamaximumof75gcreatine.iswrong;youshouldbuy125sharesofIBMandnoothers.Allocate$2,250

9 ,000toautomobileloans,$500,000tosigna-tu
,000toautomobileloans,$500,000tosigna-tureloans,and$2,250,000toanycombinationoffurnitureloansandothersecuredloans.Invest$75,000inUniversal,noneintherest.Anotheroptimalsolutionis:Invest$18,750inUniver-sal,and$75,000inEMI.ucsontoHonolulu:290boards;ucsontoVeniceBeach:330boards;TorontotoHonolulu:0boards;TorontotoVeniceBeach:200boards,giving820boardsFly10peoplefromChicagotoLosAngeles,5peoplefromChicagotoNewYork,and10peoplefromDenvertoNewYork.es;thegivenproblemcanbestatedas:Maxi-subjecttoThe graphical method applies only to LP problems in two un-knowns, whereas the simplex method can be used to solve LPproblems with any number of unknowns.She is correct.Because there are only two constraints, there can only be two ac-tive variables, giving two or fewer nonzero values for the un-knowns at each stage.equations is a solution in which all the non-pivotal variables aretaken to be zero; that is, all variables whose values are arbitraryare assigned the value zero. To o

10 btain a basic solution for a givensystem
btain a basic solution for a givensystem of linear equations, one can row reduce the associatedaugmented matrix, write down the general solution, and then setall the parameters (variables with ÒarbitraryÓ values) equal toNo. Let us assume for the sake of simplicity that allthe pivots are 1Õs. (They may certainly be changed to 1Õs withoutaffecting the value of any of the variables.) Because the entry atthe bottom of the pivot column is negative, the bottom row getsreplaced by itself plus a positive multiple of the pivot row. Thealue of the objective function (bottom-right entry) is thus re-placed by itself plus a positive multiple of the nonnegative right-most entry of the pivot row. Therefore, it cannot decrease.200,tomatoes and no other crops. This will give you a proÞt of$200,000. (You will be using all 100 acres of your farm.)10 mailings to the East Coast, none to the Midwest, 10 to theest Coast. Cost: $900. Another solution resulting in the samecost is

11 no mailings to the East Coast, 15 to th
no mailings to the East Coast, 15 to the Midwest, none tothe West Coast.10,000 quarts of orange juice and 2000quarts of orange concentraterap CDs, and 5000 classical CDs for a maximum retail value ofOne serving of cereal, one serving of juice, andno dessert!15 bundles from Nadir, 5 from Sonny, and nonefrom Blunt. Cost: $70,000. Another solution resulting in the samecost is 10bundles from Nadir, none from Sonny, and 10 fromMix 6 servings of Riboforce HP and 10 servings ofCreatine Transport for a cost of $15.60.37.a.Build 1 conven-tion-style hotel, 4 vacation-style hotels and 2 small motels. The$37.6 million, you will still be covered by the subsidy.to Honolulu: 500 boards/week; Tucson to Venice Beach:120boards/week; Toronto to Honolulu: 0 boards/week; Toronto toenice Beach: 410 boards/week. Minimum weekly cost is $9700.$2500 from Congressional Integrity Bank, $0 from CitizensÕust, $7500 from Checks R Us.Fly 10 people fromChicago to LA, 5 from Chicago to Ne

12 w York, none from Denverto LA, 10 from D
w York, none from Denverto LA, 10 from Denver to NY at a total cost of $4520.no more cardiologists, 12 rehabilitation specialists, and 5 infec-,... ,represented by the initial tableau may not be feasible. In phase I use pivoting to arrive at a basic solution that is feasible.The basic solution corresponding to the initial tableau has allthe unknowns equal to zero, and this is not a feasible solution be-cause it does not satisfy the given inequality.ers may vary. Examples are Exercises 1 and 2.Answersmay vary. A simple example is: Maximize xample is: Maximize /52/5], C=[2/53/50]]/403/4], C=[1/201/2]T, e=1/227.R=[031131155/1102111 16314_19_ans_pA18-A42.qxd 7/17/06 11:54 PM Page 29 A30Answers to Selected Exercises ANSWERS TO SELECTED EXERCISES 4 ounces each of Þsh and cornmeal, for a total cost of 40¢ per¢ per gram of protein, ¢ per gram of fat.100 oz of grain and no chicken, for a total cost of $1; per gram of protein, 0¢ per gram of fat.One serving

13 of ce-real, one serving of juice, and no
of ce-real, one serving of juice, and no dessert! for a total cost of 37¢;¢ per % U.S. RDA of Vitamin C.10 mailings to the East coast, none to the Midwest, 10 to theest Coast. Cost: $900; 20¢ per Democrat and 40¢ per Republi-can. OR 15 mailings to the Midwest and no mailing to the coasts.Cost: $900; 20¢ per Democrat and 40¢ per Republican.costing 360,000 pico-shirleys of energy OR 288073% of the days in Littleville, 27% in Metropolis, and skip Ur-bantown. T. L. Down should spend about 91% of the days in Lit-tleville, 9% in Metropolis, and skip Urbantown. The expectedoutcome is that T. L. Down will lose about 227 votes per day ofEach player should show one Þnger with, two Þngers with probability . The expected outcome is that player Apoint per round, on average.Write moves as) where represents the number of regiments sent to the ÞrstColonel Blotto should play (0, 4) with probability several optimal strategies, one of which is to play (0, 3) with. T

14 he expected out-points on average.The du
he expected out-points on average.The dual of a standard minimization problem satisfying thenonnegative objective condition is a standard maximization prob-lem, which can be solved using the standard simplex algorithm,thus avoiding the need to do Phase I.Answers will vary. An. This problem can be solved using theBuild 1 convention-style hotel, 4Answers willvary.1.3.UnboundedBounded; Corner points: (0, 0), x y 3x
2y  30x
2y  20 10101520 Solution set416 2x  3y  12 xy /21/20],[011/27727],,/2752714from DufÞn House, and 375 from Higgins Press for a minimum90,000600,Billy Sean should take the following combination: Sciences:24 credits, Fine Arts: no credits, Liberal Arts: 48 credits, Mathe-Books should choose between Ò2 for 1Ó and Ò3 for 2Ó with prob-abilities 20% and 80%, respectively. OÕHaganBooks shouldchoose between Ò3 for 1Ó and ÒFinite MathÓ with probabilities60% and 40%, respectively. OÕHaganBooks expects to gain12

15 ,000 customers from FantasyBooks.$120,$5
,000 customers from FantasyBooks.$120,$505,$20,705$250,$14,457In 2 years85.28% if you sold in February,No. Simple interest increase is linear. The graph isvisibly not linear in that time period.time in years since 1950) Graph:Graph (A) is the only possible choice, because the equationPVrtgives the future value as aWrong. In simple interest growth,the change each year is a Þxed percentage of the and not the preceding yearÕs value. (Also see Exercise 42.)Simple interest is always calculated on a constant amount,. If interest is paid into your account, then the amount on which$55,526.45 per yearone earning 11.9% compounded monthlyes. The invest-ment will have grown to about $150,281 millionThe Ecuadorian investmentis better: it is worth 1.01614 units of currency (in constant units) per 194019802000 16314_19_ans_pA18-A42.qxd 7/19/06 3:03 PM Page 30 ANSWERS TO SELECTED EXERCISESunit invested as opposed to 1.01262 units for Chile.51.90% if you sold in Febru

16 ary, 2005interest increase is exponentia
ary, 2005interest increase is exponential. The graph looks roughly exponen-tialinthat period, but to really tell we can compare interest rates betweenmarked points to see if the rate remained roughlyconstant:From December 1997 to August 1999 the rate was(16or 161.79%, while from August1999 to March 2000 the rate was (33or 251.40%. These rates are quite different.31 years;about$26,1002.3 years65.a., but an exponential function. Thus, its graph is not aWrong. Its growth is exponential and can bemodeled by 01(1The graphs are the same becausethe formulas give the same function of estment behaves as though it was being compounded once a yearat the effective rate.The effective rate exceeds the nominalrate when the interest is compounded more than once a year becausethen interest is being paid on interest accumulated during each year,resulting in a larger effective rate. Conversely, if the interest is com-pounded less often than once a year, the effective rate is

17 less than theCompare their future value
less than theCompare their future values in constant dollars.The investment with the larger future value is the better investment.The graphs are approaching a particular curve as gets larger,approximately the curve given by the largest two values of ou should take the loan from Solid Savings & Loan: it willhave payments of $248.85 per month. The payments on the otherloan would be more than $300 per month.Answers usingcorrectly rounded intermediate results:irst Þve years: $402.62/month; last 25 years: $601.73Original monthly payments were $824.79. The new monthlypayments will be $613.46. You will save $36,481.77 in interest.13 years4.5 years24 yearswill be earned by your annuityÑboth while it is increasing andhile it is decreasing. Your payments will be considerably smaller(depending on the interest earned).He is not correct.Forinstance, the payments on a $100,000 10-year mortgage at 12%are $1434.71, while for a 20-year mortgage at the same rate,they are $

18 1101.09, which is a lot more than half t
1101.09, which is a lot more than half the 10-yearmortgage payment. ii)Šn=PMT1Ši)Šn 14.0 years10.8 years7.0 years earInterestPrincipal $3934.98$1798.98 $3785.69$1948.27 $3623.97$2109.99 $3448.84$2285.12 $3259.19$2474.77 $3053.77$2680.19 $2831.32$2902.64 $2590.39$3143.57 $2329.48$3404.48 10$2046.91$3687.05 11$1740.88$3993.08 12$1409.47$4324.49 13$1050.54$4683.42 14$661.81$5072.15 15$240.84$5491.80 20002001200220032004 Revenue$180,000$216,000$259,200$311,040$373,248 spring,summer,fall,winter1,2,3,4,5,61,2,32,4,6,89.a.(H,H)(H,T),(T,H),(T,T)(H,H),(H,T),(T,T)(1,5),(2,4),(3,3),(4,2),(5,1)(1,5),(2,4),(3,3) GoogleHotmaileBayC OHaganBooks 16314_19_ans_pA18-A42.qxd 7/17/06 11:54 PM Page 31 ANSWERS TO SELECTED EXERCISES (1,1)(1,2)(1,3)(1,4)(1,5)(1,6)(2,1)(2,2)(2,3)(2,4)(2,5)(2,6)(3,1)(3,2)(3,3)(3,4)(3,5)(3,6)(4,1)(4,2)(4,3)(4,4)(4,5)(4,6)(5,1)(5,2)(5,3)(5,4)(5,5)(5,6)(6,1)(6,2)(6,3)(6,4)(6,5)(6,6)(1,4),(2,3),(3,2),(4,1)(1,1)(1,2)(1,3)(1,4)(1,5)(1,6)(2,2)(2,3)(2,4)

19 (2,5)(2,6)(3,3)(3,4)(3,5)(3,6)(4,4)(4,5)
(2,5)(2,6)(3,3)(3,4)(3,5)(3,6)(4,4)(4,5)(4,6)(5,5)(5,6)(6,6)(1,3), (2,2)as in Exercise 7; (2, 2),(2, 3),(2, 5),(3, 3),(3, 5),(5, 5)m,o,z,a,r,to,a(s, o), (s, r), (s, e), (o, s), (o, r), (o, e), (r, s), (r, o), (r, e), (e, s), (e, o), (e, r)(o,s),(o,r), (o,e), (e,s), (e,o), (e,r)domestic car, imported car, van,antique car, antique truckan, antique truck19.a.all sets of 4 gummy candies chosen from the packet of 12y candiesin which two are strawberry andtwo are blackcurrant21.a.from20all lists of 14people chosen from 20, in whichColin Powell heÞrstpositionNew England,acific, Middle Atis the event that youchoose a region that saw an increase in housing prices of 15%ormore or is on the east coast. aciÞc, New England,Middle Atlantic, South Atlanticis the event that youchoose a region that saw an increase in housing prices of 15%ormore and is on the east coast. New England,43.a.Mutually exclusiveNot mutuallyxclusiveis the event that an author is successfuland new.

20 is the event that an author is either s
is the event that an author is either successful ornew; is the event that an author is successful but not a new author.80; The number of stocksthat were either not pharmaceutical stocks, or were unchanged inalue after a year (or both).Internet stocks that increased in value61.a.63.a.65.a.The dogÕs ÒÞghtÓ drive is weakest.The dogÕs ÒÞghtÓand ÒßightÓ drives are either both strongest or both weakest.Either the dogÕs ÒÞghtÓ drive is strongest, or its ÒßightÓ drive(6,4)(1,1)(5,3)69.a.(7,3)is the event that Celerawins and Electoral College is in second or third place. In otherords, it is the set of all lists of three horses in which Celera isÞrst and Electoral College is second or third. (8,3)(4,1)(2,1)(2,1)ue; Consider the following experiment: Select an elementAnswers may vary. Cast a die andrecord the remainder when the number facing up is divided by 2.es. For instance, (2,5),(5,1)(4,3)two such events.Section 7.2 HHHTTHTT Probability.275.2375.3.1875 123456Pr

21 obability 1,2,3 Probability154152151 Pro
obability 1,2,3 Probability154152151 Probability.1.4.4.1 LowMiddleHighProbability.5.3.2 UCRProbability.2.64.16 slightly biased in favor ofheads, because heads comes up approximately 58% of the time.On the other hand, it is conceivable that the coin is fair and thatheads came up 58% of the time purely by chance. Deciding whichconclusion is more reasonable requires some knowledge of infer-37.a.39.a.41.a.Dial-up: .63, Cable Modem: .21, DSL: .15, 16314_19_ans_pA18-A42.qxd 7/17/06 11:54 PM Page 34 ANSWERS TO SELECTED EXERCISES 2·1 2=1 45 18·1 2 =1 9 36·5 18 =2 (giving up | used Brand X ) larger than (giving up) Probability Rain todayNo rain today (small and breaksdown)  .042P(small and does notbreak down)  .258P(luxury and breaksdown)  .0192P(luxury and does notbreak down)  .2208P(budget and breaksdown)  .184P(budget and does notbreak down)  .276.30.24.46.86.14.92.08.6.4 .6.7.3P(sell 2-door)  .12P(sell 4-door)  .28P(No sale)  .6 SaleNo sale (I
C) 

22 .14P(J
C)  .56P(G
B)  .01P(H
B)  .09P
.14P(J
C)  .56P(G
B)  .01P(H
B)  .09P(E
A)  .1P(F
A)  .1 .5.2.8.1.9.2.1.7 The claim is correct.The probability that an unemployed person has a high schooldiploma only is .35, while the corresponding Þgure for anemployed person is .30.75.a.It was ineffective.77.a.(Internet user | )(Internet user | 83.a.Answers will vary. Here is a simple one: head, : the second toss is a head, The probability you seek is xample, you were going to place a wager on whether ornot, it is crucial to know that the sample space has been re-(you know that did occur). If you base your wagerou will misjudge your likelihood of winning.our friend is correct. If mutually exclusive then . On the other hand, if . If a product is 0 then one of the factors must be. Thus, it cannot be true that are mutually exclusive, have nonzero probabilities, and are nonzero probabilities, and areP(A)+P(B)ŠP(AB)]=1Š[P(A)+P(B)ŠP(A)P(B)]=(1ŠP(A))(1ŠP(B))=P(A )P(B ).Section 7.623.a.19.81% of single hom

23 eowners have pools.Thus they shouldgo af
eowners have pools.Thus they shouldgo after the single homeowners.: Airbag deployed; ed; .31(.25)+.75]=.3035.Show him an examplelike Example 1 of this section, where butSuppose that the steroid test gives 10%alse negatives and that only 0.1% of the tested population usessteroids. Then the probability that an athlete uses steroids, giventhat he or she has tested positive, is 9)(001) 9)(001)01)(999)Draw a tree in which the Þrst branching shows which of occurred, and thesecond branching shows which of then occurred. Thereare three Þnal outcomes in which . In only one of these, theoccur. Thus, P(T)=P(T|R1)P(R1) P(T|R1)P(R1)+P(T|R2)P(R2)+P(T|R3)P(R3) 16314_19_ans_pA18-A42.qxd 7/17/06 11:54 PM Page 36 A38 ANSWERS TO SELECTED EXERCISESBrandA:65339,BrandB:127Here are three: (1) it is possible forsomeone to be a customer at two different enterprises; (2) somecustomers may stop using all three of the companies; (3) new2,3,. . . ,120,1,(negative proÞts indicate

24 loss)an assume any value between 0 and 6
loss)an assume any value between 0 and 60.0,1,11.a.is the rule that assigns to each outcome the number of tails.13.a.(1,1),(1,2),. . . , (1,6),(2,1),(2,2),. . . ,(6,6)is the rule that assigns to each outcome the sum of the two15.a.(4,0),(3,1),(2,2)(listed in order (red, green))is the rule that assigns to each outcome the number of redmarbles.17.a.the set of students in the study groupis the rule that assigns to each student his or her Þnal examThe values of , in the order given, are 89%, 85%, 95%,63%, 92%, 80%.19.a.29.a.5000)The random variable is count on a given day xP(X  x) 0.50.30.20321 2000300040005000600070008000 .2.1.1.1.2.0.3 750225037505250 162161162 HHHTTHTT0112 (1,1)(1,2)(1,3)(6,6)234...12 (4, 0)(3, 1)(2, 2)432 123456 (X=x)1 61 61 61 61 61 6 P(X  x) 0.50.30.20321 0149 (X=x)1 83 83 81 8 P (X  x) 049 23456789101112 (X=x)1 362 363 364 365 366 365 364 363 362 361 36 (X  x) 2810 123456 (X=x)1 363 365 367 369 3611 36 16314_19_ans_pA18-A42.qxd 7/1

25 7/06 11:54 PM Page 38 A40 ANSWERS TO S
7/06 11:54 PM Page 38 A40 ANSWERS TO SELECTED EXERCISES33.a.6.5; There were an average of 6.5 checkout lanes in asupermarket that was surveyed., and is thus larger. Most supermarkets havemore than the average number of checkout lanes.The average age of a school goer in 1998 was 14.3.Large cars49.a.2 defective airbagsastForward: 3.97%; SolidState: 5.51%; SolidState gives thehigher expected return.is wrong; for example, the collection 0, 0, 300 has mean 100 andNo. The expected number of times you will hitthe dart-board is the average number of times you will hit theullÕs eye per 50 shots; the average of a set of whole numbers neednot be a whole number.only a small fraction of people in the class scored better than youbut received exceptionally high scores that raised the class aver-our received 100%, you received 80%, and the rest received70%. Then the class average is 83%, 5 people have lower scoresthan you, but only four have higher scores.a very large s

26 ample is only an The means of larger and
ample is only an The means of larger and larger samples approachWrong. The statement at-tributed to President Bush asserts that the mean tax refund wouldbe $1000, whereas the statements referred to as ÒThe TruthÓ sug-tax refund would be close to $100 (and thatthe 31st percentile would be zero).Select a U.S. householdbe the income of that household. The ex-pected value of is then the population mean of all U.S. house- 51015202535 .17.33.21.19.03.07 3210 .0625.6875.125.125 3210 .1.4.4.1 1234 (X=x)4 3518 3512 351 35 Expected value 3.5, variance standard deviation Expectedvalue5,standarddeviationExpectedvalue47,variancestandarddeviationExpectedvalue36,standarddeviationExpectedvalue2,variancestandarddeviation25.a.a.We must assume that the population distribution is bell-shapedand symmetric.27.a.29.a.5000,31.a.a.c.100%; Empirical rule37.a.old,78yearsAt most6.25%43.a.45.a.780,47.a.[2.5, 10.5]; 3 checkout56%,78%; The empirical rule predicts 68%. Theassociated pro

27 bability distribution is roughly bell-sh
bability distribution is roughly bell-shaped but not96%. ChebyshevÕs rule is valid, since it predictsThe sample standard deviation is bigger; the formula forsample standard deviation involves division by the smaller term, which makes the resulting number larger.The grades in the Þrst class were clustered fairly close to 75.By ChebyshevÕs inequality, at least 88% of the class had grades inthe range 60Ð90. On the other hand, the grades in the secondclass were widely dispersed. The second class had a much widerThe variable musttake on only the value 10, with probability 1.The U.S.sler.Because this test has a smaller standard deviation, a greaterpercentage of scores fall within 20 points of the mean.surprising, because the time between failures was more than 5 stan-darddeviations away from the mean, which happens with an ex-tremely small probability.that a person will say Goode is normalNeither. They are equal.A normal distribution with standard deviation 0.5,

28 because itis narrower near the mean, bu
because itis narrower near the mean, but must enclose the same amount ofarea as the standard curve, and so it must be higher. 16314_19_ans_pA18-A42.qxd 7/17/06 11:54 PM Page 40 5000 pounds. The model is not trustworthy for vehicleeights larger than 5000 pounds, because it predicts increasingfuel economy with increasing weight, and 5000 is close totheupper limit of the domain of the function.Maximum revenue when Maximum revenue with 70 houses, 29.a.per log-on; 33.a.; $120.2 billion, which agreeswith the actual value to the nearest $1 billion.37.a.Mathematical regression cannot reliably beused to make predictions about sales. (Answers will vary.)-coordinate of the vertex represents the unit price thatleads to the maximum revenue, the -coordinate of the vertexgives the maximum possible revenue, the -intercepts give theunit prices that result in zero revenue, and the -intercept gives therevenue resulting from zero unit price (which is obviously zero).Graph

29 the data to see whether the points sugg
the data to see whether the points suggest a curverather than a straight line. If the curve suggested by the graph isconcave up or concave down, then a quadratic model would be alikely candidate.), then the rev-enue is given by . This is the equation of a, and so is concave down. Thus the ver-tex is thehighest point on the parabola, showing that there is asinglehighest value for , namely, the -coordinate of the vertex., the demand must be given by p=Š50p2+60p pp+60.Section 9.21. 4006008001000 051015202530 ANSWERS TO SELECTED EXERCISES 0123 641 161 1416 0123 2Š3 4Š3 8 0123 41 24816 012327931 31 91 27 0123 8Š3 4Š1 0137 0123 161 81 41 124 13.15.17.5(32(12(24(0
1
20
30
10yx 
3(2
x) xy48121 1y  2(2x) 3020
y  3
xx -3-2-10123403.454.607.38910.13530.018320.002479 -3-2-101234662280.016.821.010.060660.0036430.0002188 -3-2-101239.78116.8529.025086.13148.4255.6 not2^x-1not2/1-2^-4*x;not2/1-2^(-4*x)not(3+x)^(3*x)/x+1;not(3+x^(3*x))/(x+1)or2*EXP((1+x)/x);not2*e^

30 1+x/x;not2*e^(1+x)/x;not2*EXP(1+x)/x 163
1+x/x;not2*e^(1+x)/x;not2*EXP(1+x)/x 16314_19_ans_pA18-A42.qxd 7/17/06 11:54 PM Page 42 base to a power never results in a negative number, so there can beno such real number as the logarithm of a negative number. logAny logarithmic curve logwill eventually surpass 100%, and hence not besuitable as a long-term predictor of market share.Time isincreasing logarithmically with population; Solving gives logloglog, which is of the formlog7/(1+6*2^-x)10/(1+4*0.3^-x) 19(2 1+2Šx11.y=7.2 4(1 2(0942)21.a.20% per year23.a.33(110,000 9(1(7) 29(2days29.a. 8(1; 6300 articles 5200 articles31.a. 8(7predicts that book sales will level off at around 82.8 millionbooks per year. Not consistent; 15% of the market is repre-sented by more than double the predicted value. This shows the 020 60 80100 020 80100 0 8 0 5 4 2 812 810 0 8 difÞculty in making long-term predictions from regression 080(1056)diseases are communicated via the spread of a pathogen (suchasavirus), new te

31 chnology is communicated via the spread
chnology is communicated via the spread ofinformation (such as advertising and publicity). Further, just as thespread of a disease is ultimately limited by the number of suscep-tible individuals, so the spread of a new technology is ultimatelylimited by the size of the potential market.predict where the sales of a new commodity might level off.5(1,or5(25.7.5(9 33x19.Š1 log log110.2 years10.8 years 8(1 3(037.a.erage of approximately 2100 hits per dayThe fact that , the coefÞcient of negative.per novel,nthly revenue 41.a. 51015202530 101520
3
2
(x)g(x) 10123 y
1,
4)
3x ANSWERS TO SELECTED EXERCISES 16314_19_ans_pA43-A69.qxd 7/17/06 11:57 PM Page 44 $1.33 per monthin unemployment per 1 percentage point increase in the deÞcit19.21.Answers may varyThe index was increasing at an average rate of 300 points perday.$0.08 per year. The value of the euro in U.S. dollars waswing at an average rate of about $0.08 per year over the period43.a.45.a.million per year;

32 Over the period 1997Ð1999, annual adver
Over the period 1997Ð1999, annual advertisingrevenues increased at an average rate of $305 million per year.$590 million per year; The model projects annual adver-tising revenues to increase by $590 million per year in 2000.47.a.Forhousehold incomes between $40,000 and $40,500, the poverty ratedecreases at an average rate of 0.69 percentage points per $1000Theaveragerateofchangeoferaninterval[canbede-terminednumerically,usingatableofvalues;graphically,bymea-suringtheslopeofthecorrespondinglinesegmentthroughtwopointsonthegraph;oralgebraically,usinganalgebraicformulaforthefunction.Ofthese,theleastpreciseisthegraphicalmethod,becauseitreliesonreadingcoordinatesofpointsonagraph.Answers will vary.6 units of quantity A per unit of quantity Ces. Here is an example: y 1991Year1999 2.3 ANSWERS TO SELECTED EXERCISES25.a.million people per year. During the period2000Ð2004, employment in the U.S. decreased at an average rate of0.25 million people per year.Zero people

33 per year. During theperiod 1999Ð2002 the
per year. During theperiod 1999Ð2002 the average rate of change of employment in the.S. was zero people per year.27.a.1998Ð2000. The number ofcompanies that invested in venture capital each year was increasingmost rapidly during the period 1998Ð2000, when it grew at anerage rate of 650 companies per year. 1999Ð2001. The num-ber of companies that invested in venture capital each year wasdecreasing most rapidly during the period 1999Ð2001, when itdecreased at an average rate of 50 companies per year.29.a.a.Š0.25thousand articles per year. During the period1993Ð1995,the number of articles authored by U.S. researchersdecreased atan average rate of 250 articles per year., Average ratethousand articles/year.Over the period 1993Ð2003, the number of articles authored by.S. researchers decreased atanaverage rate of 90 per year, repre-31.a.33.a.transactions per year,Ð150 million transactions per year, 50 million transactions per year.Over the period January 2000ÐJan

34 uary 2001, the (annual) numberof online
uary 2001, the (annual) numberof online shopping transactions in the U.S. increased at anaveragerate of 250 million per year. From January 2001 to January 2002,this number decreased at an average rate of 150 million per year.rom January 2000 to January 2002, this number increased at anerage rate of50 million per year.The average rate of changeer [0, 2] is the average of the rates of change over [0, 1]er [0, 1]35.a.Approximately (to two signiÞcant digits) billiondollars per year, (per year). This is much less than the(positive) slope of the regression line, dollars per year, ($13,000,000 per year). Rate of Change 0.18.1 0.018.01 0.0018.0010.00018.0001 Ave. of Change 0.10.2 0.010.02 0.0010.0020.00010.0002 Ave. of Change Š0.1667 0.1Š0.2381 0.01Š0.2488 0.001Š0.24990.0001Š0.24999 2000200120022003Revenue ($ billion)1020305 0.10.01Ave.3939.939.99 upees per day 16314_19_ans_pA43-A69.qxd 7/17/06 11:57 PM Page 46 21.g(x)xŠ3+3xŠ223.g(x) x2+2 x325.h(x).8 x1.427.

35 h(x) x3Š6 x429.r(x) 3x2+0.1 2x1.131.r
h(x) x3Š6 x429.r(x) 3x2+0.1 2x1.131.r(x)=2 3Š0.1 2x0.9Š4.4 3x2.133.t(x|x|xŠ1x235.s(x)=1 2 xŠ1 2x 2(1 t2.1Š0.3 t0.451.4
r253.355.Š257.Š559.y=3x+261.y=3 4x+163.y=1 No such values73.75.77.a.(1)Not differentiable at 079.a.differentiable at 1Not differentiable at 0No; 3es; Diverges toes; Diverges to 93.a.52teams/yearincreasingatarateof2.6percentagepointsperyear99.a.128ft/sec5 seconds; downward at160ft/sec101.a.. In January 2004thevalue of the euro was increasing at a rate of $0.188 per year.103.a.(8)boats. At a level of 800,000 boats, the number of manatee deaths 48
4

8
4
4

14 40 x y4x 1 y4 x 105.a.measures the combined marketshare of the other three providers (Comcast, Earthlink, and AOL);measures the rate of change of the combined marketshare of the other three providers.3.72% per year. In 1992, the combined market share of theother three providers was increasing at a rate of about 3.72 per-centage points per year.After graphing the curve draw t

36 he line passing through 1,3)equal to the
he line passing through 1,3)equal to the right-hand side. The equal to the right-hand side, so your friend should have written The derivative of a constant times a function is the constant times the derivative of the(2)(2our enemy mistakenlyof the constant times the derivative of thefunction. (The derivative of a product of two functions is not theproduct of the derivative of the two functions. The rule for takingthe derivative of a product is discussed in the next chapter.).Answers may vary.1000)100). Thus, at a production level of 2000,the proÞt is stationary (neither increasing nor decreasing) withrespect to the production level. This may indicate a maximumproÞt at a production level of 2000.7.a.9.a.. The cost is going up at a rate of$2,249,840 per television commercial. The exact cost of airingthe Þfth television commercial is (5)(4)(4)vision commercial. The average cost of airing the Þrst four televi-11.a.Revenue: $450, ProÞt: $80,Marginal revenue: $0

37 .90, Marginal proÞt: $0.20. The totalenu
.90, Marginal proÞt: $0.20. The totalenue from the sale of 500 copies is $450. The proÞt from theproduction and sale of 500 copies is $80. Approximate revenuefrom the sale of the 501st copy is 90¢. Approximate loss fromthe sale of the 501st copy is 20¢.maximum when you produce and sell 400 copies.proÞt on the sale of 1000 DVDs is $3000, and is decreasing at arate of $3per additional DVD sold.. Your current proÞt is $257.07 per month, and this17.a. xy ANSWERS TO SELECTED EXERCISES 16314_19_ans_pA43-A69.qxd 7/17/06 11:57 PM Page 49 3(42(23)(2) 1)(34)(5)(18(21)(3)(73(11)(1 2 x x+1 x2+( x+1 2 xŠ2 2(33(2 3(2 4)( x2+x+=2Š(x2+x+41.23xŠ0.77Š5.7xŠ2.9)Š2.9xŠ3.9(x0.23Š5.7x) xŠ2.9)243.1 2xŠ1/2(x1/2Š1 2xŠ1/2(x1/2+ x1/2Š=Š1  x( (x+1)+(x+3)](33)( (x+1)(3)(3)(1)](33)(1)( (5)(5)(5)900,000(revenue is increasing at a rate of $900,000 perDecreasing at a rate of $1 per daya rate of approximately $0.10 per month3000(3600 (10)7670mpgThis means that, at a speed of 10 m

38 ph, the fuel economy is(60)mpg/mph. This
ph, the fuel economy is(60)mpg/mph. This means that, at aspeed of 60 mph, the fuel economy is neither increasing nor(70)means that, at 70 mph, the fuel economy is decreasing at a rate ofthe most fuel-efÞcient speed for the car. (In the next chapter weshall discuss how to locate largest values in general.)Increasing at a rate of about $3420 million per year. (4)organisms per hour, per 1000 organisms. This means that thereproduction rate of organisms in a culture containing 4000organisms is declining at a rate of 2500 organisms per hour,per1000 additional organisms.decreasing at a rate of 1600 milliliters per day. This is due to theact that the number of eggs is decreasing, because (25)itive.85.a.represents the combined market shareof the other three providers (Comcast, Earthlink, and AOL).represents MSNÕs market share as a fraction of the four providers considered. m(t) percentage points) per year. In June, 2003, MSNÕsmarket share as a fraction of the four

39 providers considered wasear.The analysis
providers considered wasear.The analysis is suspect, because it seems to be assert-ing that the annual increase in revenue, which we can think of asin sales. However, because , the product rule im- q+p· Answers will vary; is one example.twice as fast as yours. The rate of change of revenue is given bydoes not depend4(22(24(45(23)(5(02)(04(2 (x+1)(1)]1)(2(32[(66.8(62(53)(13[(1[8(12(1  2x+1Š2x  54(135(7 ANSWERS TO SELECTED EXERCISES 16314_19_ans_pA43-A69.qxd 7/17/06 11:57 PM Page 51 Section 11.41.Š2/33.x5.(yŠx)7.Šy9.Šy xye31.a.33.a.35.a.37.a.39.a.41.a. 4xŠ2 2 2x+1Š4 4xŠ245.+ 4xŠ 6 3x+1Š1 xŠ6 2xŠ147.Š/3(xŠ8 3(8 xŠ149.(x3+x) x3+2 3x2+1 x3+x+1 23x2 per worker. The monthly budgetto maintain production at the Þxed level is decreasing by ap-proximately $3000 per additional worker at an employment levelof 100 workers and a monthly operating budget of $200,000.-shirts per dollar; when the price is set at $5, the de-mand is dropping by 125 T-shirts per $1

40 increase in price. carpenters per electr
increase in price. carpenters per electrician. This means that, for a $200,000 house whose construction employs 15 electri-adding one more electrician would cost as much as approx-imately 0.307 additional carpenters. In other words, one electri-cian is worth approximately 0.307 carpenters.59.a.22.93 hours. (The other root is rejected because it is larger 4tŠ20x 0.4tŠ4x; rade point. This means that, for a 3.0 student who scores 80 onthe examination, 1 grade point is worth approximately 11.2 hours. 2r y, so 2r y the chain rule. y f(x) f(x)+g(x) g(x), yf(x) f(x)+g(x) g(x)=f(x)g(x)f(x) f(x)+g(x) as an explicit function of This can be regarded as an equation giving . The procedure of Þnding implicit differentia-tion is then the same as Þnding the derivative of as an explicit: we take Differentiate , giving 1 f(x)=1 (xŠ1)ln3ln2) 2y19.Šy/x21.Š+ 1)(3 2xŠ1+3 3x+4Š1 x+1Š9 (0)(0)(0)(0)(0)1)(1000)20(200)per week (rising)gives $110 per year 553 books p

41 er week616.8 hits per day per week, rela
er week616.8 hits per day per week, relative max: 1,1), relative min:(3,3,0)1,2)Relative min:1,1), relative max: 2,2)relative min: (1, 0), absolute max: (3, 3)Relative max:3,0), stationary non-extreme point:(2,, relative(3,2,16)(2,(2,10), relative max: 1,1), relative min:Relative max: 1,5)(3,(2,1), absolute min at (1, 0)Relative maximum at, relative minimum at Relative min: 2,5, relative max:(0,, relative min: (2, Relative max: (0, 0); absolute Relative max: (0, 0), absolute min:(1,No relative extremaRelative max: 1,1, absolute min: (0,1), absoluteRelative max: , relative min: e/, absolute min:(Š1/  Relative min at (2.45, 8.22), relative max at (1.40, 0.29)5,700), relative max at (3.10, 28.19) and (6, 40), absolute minand relative min at (5,0).StationaryStationary minima at , stationary maximum at , stationary non-extreme point at Stationary, singular non-extreme points at , stationary maximum at ANSWERS TO SELECTED EXERCISES 16314_19_ans_pA43-A69.qxd

42 7/17/06 11:57 PM Page 53 Relative maxi
7/17/06 11:57 PM Page 53 Relative maximum atNo extrema; points of; relative inßection at (0, 0), and (3,9/4)No relative extrema; point1,1)(1,relative max at 1,1Relative min at 46,0; relative max at (0.91, 1.73);Accelerating by65.a.36 ml/dayml/day67.a.o years into the epidemico years into the epidemic69.a.Concave up for concave down for The percentage of articles written by researchers in the U.S. wasdecreasing most rapidly at around 73.a.75.a.Because the graph is concave up, the derivative ofdecreaseof SAT scores with in-creasing numbers of prisoners is diminishing. In other words, theapparent effect of more prisoners is diminishing. 2
5
105xy 2
12xy 0.50.5 55xy 02046y –4 yx24–4–2 yx12–1 77.a. . Thus, for a Þrm with annual sales of $3 million, the rate at which new patents are produced de-creases with increasing Þrm size. This means that the returns (asmeasured in the number of new patents per increase of $1 millionin sales) are diminishing as the Þrm si

43 ze increases. . Thus, for a Þrm with ann
ze increases. . Thus, for a Þrm with annual sales of $7 million, the rate at which new patents are produced increases withincreasing Þrm size by 13.474 new patents per $1 million increaseThere is a point of inßection when so that in a Þrm with sales of $5,458,700 per year, the number ofnew patents produced per additional $1 million in sales is aConcave down; (C). Graph:About $570 per year, after about 12 yearsmost rapidly in 17.64 years, decreasing most rapidly now (at NonnegativeDaily sales were decreasing most rapidlyAt a point of inßection, the graph of a function changes eitherfrom concave up to concave down, or vice versa. If it changesfrom concave up to concave down, then the derivative changesfrom increasing to decreasing, and hence has a relative maximum.Similarly, if it changes from concave down to concave up, thederivative has a relative minimum.10,000 be the annual revenue of company, and let . Find 5.pair of shoes, and let 5. be the average

44 global temperature, andbe the number of
global temperature, andbe the number of Bermuda shorts sold per year. . Find 9.a. 24ft/min 1PtCompany A$ million Company B1PtCompany A$ million 0 2025 0 0.0001 ANSWERS TO SELECTED EXERCISES 16314_19_ans_pA43-A69.qxd 7/17/06 11:57 PM Page 55 Relative max at (0,0), absolute min at (1,$24 per copyrevenue, the company should charge $22.14 per copy. At thishile the revenue is not decreasing (its derivative is zero). Thus, 10,500; If weekly sales continue aspredicted by the model, they will level off at around 10,500 booksper week in the long-term.39.aÐd. 21.x6/6+C3.6x+C5.x2/2+C7.x3/3Šx2/2+C9.x+x2/2+C11.ŠxŠ4/4+C13.x3.3/3.3ŠxŠ0.3/0.3+C15.u3/3Šln|u|+C17.2x3/2 2ln x+1 2x2+C25.3x1.1/1.1Šx5.3/5.3Š4.1x+C27.x0.9 0.3+40 2ln 5ln ln2 ln3100(1 ln(149.a.320 ft/s downwards53.a.zenithat feet, 4 feet above the top of thetower.1280)59.a. times as fast(13)65.a.billion dollars per year Approximately 43,000 million gallons.They differ by aAntiderivative, mar-represents the to

45 tal cost of manufacturing 123 x y items.
tal cost of manufacturing 123 x y items. The units of is, by deÞnition, anantiderivative of be an antiderivative ofbe an antiderivative of . Then, because thederivative of (by the rule for sumsof derivatives), this means that is an antiderivativeindeÞnite integrals.xdxxdx, which is not, no matter what values we choose for theIf you take the deÞnite integral, you obtain hand, if you take the indeÞnite integral 6(32(33(x+1)5/2/5Š(x+1)3/2/3]+C39.20ln5)ln 3ln(1 3ln273.a.35ln(580,000 articles75.a. 1990)1990)2300(1900)2003)43,000simply replacesthroughout by the letter , and thus does notchange theintegral at all. For instance, the integral if we substitute The purpose ofsubstitution is to introduce a new variable that is deÞned in termsof the variable of integration. One cannot say is not a new variable. Instead, deÞne (or anyother letter different from Theintegralbesolvedbythesubstitution udu 4u2+C=(x2+ 4+C. ANSWERS TO SELECTED EXERCISES 16314_19_ans_pA43-A69

46 .qxd 7/17/06 11:57 PM Page 57 (lnlogl
.qxd 7/17/06 11:57 PM Page 57 (lnlogln32)ln24ln228,800,000(11(x+1)]Answers will vary.2ln28ln4112.5. This represents your totalproÞt for the week, $112.50.27.a.cumulated U.S. trade deÞcit with China (total excess value of im-ports over exports) for the 8-year period 1996Ð2004..S. accumulated a $640 billion trade deÞcit with China over the29.a.the region between the graphs of The area between the export and import curves representsCanadaÕs accumulated trade surplus (that is, the total excess ofxports over imports) from January, 1997 to January, 2001.The claim is wrong because the area under a curvecan only represent income if the curve is a graph of income . The value of a stock price is not income per unit timeÑthe income can only be realized when the stock is sold, and itamounts to the current market price. The total net income (pershare) from the given investment would be the stock price on the 0.25010y x 0.50.75 0
21y x 24 2010y x 46 Moving average:Mo

47 ving average:(1525)[ 5(e0.5xŠe0.Š17.¯f(x
ving average:(1525)[ 5(e0.5xŠe0.Š17.¯f(x)=2 19.21.Some changes are larger, and others are smaller.33.a. 198019841988199219962000 Spending Moving Average
24xy 224 002xy xy 412xy 1010xy ANSWERS TO SELECTED EXERCISES 12345 35103256766510313 12345 267111510235811129 1995199619971998199920002001200220032004Employment117120123126129132132130130131 Moving average122125128130131131131 16314_19_ans_pA43-A69.qxd 7/17/06 11:57 PM Page 59 33.S=2/1999 35.a.It will take about 27 months to saturate the market. Graph:A general solution gives all possible solutions to the equa-tion, using at least one arbitrary constant. A particular solution isone speciÞc function that satisÞes the equation. We obtain a par-ticular solution by substituting speciÞc values for any arbitrary (integrate twice).ln23(3 2·21/3Š1 213.2 2 315.Š117.eŠ219.3xŠ221.3 14[x7/3Š(xŠ/3]23.$160025.$250027.y x3+C29.y= 2ln31.a.Approximately1.a.3.a.5.a.7.a.Does not existxyzz(x+h)2+(y+k)2+(z+l)2]9.a.19.a.

48 046 02030 annual sales of Y. The model i
046 02030 annual sales of Y. The model is 233. (2,The marginal cost of cars is $6000 per car.Themarginalcost of trucks is $4000 per truck.43.a.ercentage points per year45.a.47.a.(11,(10,. This means that, if yourcompany now has 10 copies of Macro Publish and 10 copiesofTurbo Publish, then the purchase of one additional copyofMacro Publish will result in a productivity increase ofapproximately 5.75 pages per day.53.a.Answers will vary.(3,14,1,3,1 Theresultingellipsoidisaspherewithradius57.a.1,000)1,000,000)1,000)10,000)loglog59.a.The total weight of sulfates in theEarthÕs atmosphere61.a.The value of ould be doubled.erage lifetime of an intelligent civilizationTakelogarithm of both sides, since this would yield the linear63.a.b.c.They are reciprocals of each other.or example,or example, xyzor example, take . Then setting gives ,3). This can be viewed as a function of thesingle variable . Choosing other values for gives other func-gives. The slope is indepen

49 dent of the choice of 1
111yx k  1k 
dent of the choice of 1
111yx k  1k  2k  0 1
212yxk  0k  3k  3k  0
xk  1k  0k  11
1k  2k  2k  0 ANSWERS TO SELECTED EXERCISES 1020304052107162217 94194294394 136281426571178368558748 y 16314_19_ans_pA43-A69.qxd 7/17/06 11:57 PM Page 61 critical point that is not a relative extremum(0,0,Relative maximum at 1,0,Relative minimum at ,3(2Relative1,4)None; the relative maximum at (0, 0, 0) is notabsolute. (look at, say, (10, 10)).3,2. Thus, at least of all Mazda ownersould choose another new Mazda, and this lowest loyalty occursof Chrysler and Ford owners remain loyal to theirIt should remove 2.5 pounds of sulfur and 1 poundof lead per day.ou should charge $580.81 for the Ultra18in18in36in43.45.must be positive.relative maximum at should yield a curve with a relative maximum at happen that a slice by another vertical plane through ) does not yield a curve with a relative. [An example is =0, y=0and y=x¯Cx= xC x+y=(x+y)CxŠC (x+y)2zero, then

50 x+y)Cx=Cx=C Similarly, if . This is rea
x+y)Cx=Cx=C Similarly, if . This is reasonable because if the average, then the average cost isreater than the marginal cost . Similarly, if the average cost, then the average cost is less thanthe marginal cost . Thus, if the average cost is stationary with, then the average cost equals the marginal cost relative extremum at Substituting these into the equation of the tangent plane gives, a constant. But the graph of 10,310,1Minimumvalue Continues up indefinitelyContinues down indefinitelyFunction not defined on circle xz 10,310,1Minimumvalue50sq 3,1 3,1 (Š1/ 1/ 3,1  1/ 1/ (Š1/ 3,1 1/ (0,19,59,2518in18in36in, where lightweight cardboard and cost of heavy-duty cardboardpersquare footMethod 1: Solvefor one of the variables and substitute in. Then Þnd the maximum value of the resultingfunction of 2 variables. Advantage (Answers may vary): We canuse the second derivative test to check whether the resultingthese. Disadvantage (Answers may vary): We may

51 not be able tosolve for one of the varia
not be able tosolve for one of the variables. Method 2: Use themethod of Lagrange Multipliers. Advantage (Answers mayy): We do not need to solve the constraint equation for oneofthe variables. Disadvantage (Answers may vary): The methoddoes not tell us whether the critical points obtained are maxima,If the onlyconstraint is an equality constraint, and if it is impossible toeliminate one of the variables in the objective function bysubstitution (solving the constraint equation for a variable orAnswers may vary: Maximizees. There maybe relative extrema at points on the boundary of the domain ofthe function. The partial derivatives of the function need not be 0If the solution were located in the interiorof one of the line segments making up the boundary of the, then the derivative of a certain function would be 0.This function is obtained by substituting the linear equationin the linear objective function. But because theresult would again be a linear functio

52 n, it is either constant, or itsderivati
n, it is either constant, or itsderivative is a nonzero constant. In either event, extrema lie onthe boundary of that line segment; that is, at one of the corners1)(dydx dydx y
1y  x2
1 1 y11y  1
x ANSWERS TO SELECTED EXERCISES 16314_19_ans_pA43-A69.qxd 7/17/06 11:57 PM Page 65 3lnsec(2tan(22)sin(ln(2) cos(4[cos(1)] 4ln|4x)41.043.2
45.Šxcosx+sinx+C47. x2 2Š1 sin(2 cos(2 2eŠxcosxŠ1 DivergesConverges to
cos
(
/26(tŠ13)]67.a.eragevoltageover[0,1iszero;60cyclespersecond.116.673 volts.It is always zero.
(xŠ1)]]
(x+1)]cos(cos[2)]cos[2)]cos(sin(2tan(2 cos(1500sin[(21500sin(014159) 0.040.080.12 012 ative sentenceue (we hope!) statementillis isis a good teacher, or she is not.illisÕstudents both hate andIt is not true that either Carla is a good ANSWERS TO SELECTED EXERCISES qp TTFFTFTTFTFFFFTF TFTTFTFF TTFFFTFFTFFTTFFFFTTT pqrp TTTTTTTFTFTFTFFTFFFFFTTFFFTFFFFFTFFFFFFF pqrqrp TTTTTTTFTTTFTTTTFFFFFTTTFFTFTFFFTTFFFFFF 16314_19_ans_pA43-A69.qxd 7/17/06