Basic Mathematics for Economists phần 3 pdf

... 20% = 0.2 then q = 32 0 − 55(0.2)6.5 + 4(0.2)= 32 0 − 116.5 + 0.8= 30 97 .3 = 42 .33 If t = 30 % = 0 .3 then q = 32 0 − 55(0 .3) 6.5 + 4(0 .3) = 32 0 − 16.56.5 + 1.2= 30 3.57.7= 39 .42These examples ... 6720 19 30 0 6000 20 32 0 5120 21 34 0 4080 22 36 0 2880 23 380 1520 24 400 0 25 420 -1680 © 19 93, 20 03 Mike RosserADemandB0 q£9.992.2094 120 166 250MRЈ6016.0012.16MCMR8 .32 Figure ... + 3y +4z = 144 (3) 2. Solve for x, y and z when12x + 15y +5z = 158 (1)4x + 3y +4z = 50 (2)5x + 20y +2z = 148 (3) 3. Solve for A, B and C when 32 A + 14B + 82C = 664 (1)11.5A + 8B +52C = 34 9...

... 1752.55 35 31 19254.05 1884.05 36 32 21274.40 2020 .35 37 33 234 35.85 2161.45 38 34 257 43. 20 230 7 .35 39 35 28201.25 2458.05 40 36 30 814.80 26 13. 55 41 37 33 588.65 27 73. 85 42 38 36 527.60 2 938 .95 ... 158 11.5 -5 13 22.5 -700 33 .5 -4 03 6 1 117 12 - 532 23 -697 34 -37 8 7 1.5 77 12.5 -550 23. 5 -6 93 34.5 -35 2 8 2 38 13 -567 24 -688 35 -32 5 9 2.5 0 13. 5 -5 83 24.5 -682 35 .5 -297 10 3 -37 14 -598 ... 21 8.5 -37 8 19.5 -697 30 .5 - 532 41.5 117 22 9 -4 03 20 -700 31 -5 13 42 158 23 9.5 -427 20.5 -702 31 .5 -4 93 42.5 200 24 10 -450 21 -7 03 32 -472 43 2 43 25 10.5 -472 21.5 -7 03 32.5 -450 43. 5 287...

... as powers of the same number one adds all thepowers together.Example 2 .31 3 3× 3 5= (3 × 3 × 3) × (3 3 3 × 3 × 3) = 3 8= 6,561To divide numbers in terms of powers of the same base number, ... byitself 8 times gives 226 .34 , i.e. the eighth root. Thuslog 226 .34 = 2 .35 476 13 18log 226 .34 = 0.29 434 52Therefore8√(226 .34 ) = 100.29 434 52= 1.9694509.© 19 93, 20 03 Mike Rosser7.8 Perpetual ... 0.1qc© 19 93, 20 03 Mike RosserTest Yourself, Exercise 2.10Use logs to answer the following.1. 424 × 638 .724 =2. 6, 434 ÷ 29.12 = 3. 22. 43 7=4. 9.6128 .34 =5. 36 √5,200 =6. 14 3. 2× 6.24×...

... factor.Example 3. 33 x2+ 2xx + 2=x(x + 2)x + 2= xExample 3. 34x2+ 5x +6x + 3 =(x + 3) (x + 2)x + 3 = x + 2Example 3. 35x2+ 5x +6x2+ x − 2=(x + 2)(x + 3) (x + 2)(x − 1)=x + 3 x − 1Test ... Rosser8.Yousell900sharesviayourbrokerwhochargesaflatrateofcommissionof£20onalltransactionsoflessthan£1,000.Yourbankaccountiscreditedwith 34 0fromthesharesale.Whatpricewereyoursharessoldat?9.Yournetmonthlysalaryis£1,950.YouknowthatNationalInsuranceandpensioncontributionstake15%ofyourgrosssalaryandthatincometaxisleviedatarateof25%ongrossannualearningsabovea£5,400exemptionlimit.Whatisyourgrossmonthlysalary?10.Youhave64squarepavingstonesandwishtolaythemtoformasquarepatioinyourgarden.Ifeachpavingstoneis0.5metressquare,whatwillthelengthofasideofyourpatiobe?11.AfirmfacesthemarginalrevenuescheduleMR=80−2qandthemarginalcostscheduleMC=15+0.5qwhereqisquantityproduced.YouknowthatafirmmaximizesprofitwhenMC=MR.Whatwilltheprofit-maximizingoutputbe? 3. 8ThesummationsignThesummationsigncanbeusedincertaincircumstancesasashorthandmeansofexpress-ingthesumofanumberofdifferenttermsaddedtogether.(isaGreekletter,pronounced‘sigma’.)Therearetwowaysinwhichitcanbeused.Thefirstiswhenonevariableincreasesitsvalueby1ineachsuccessiveterm,astheexamplebelowillustrates.Example3. 43 Anewfirmsells30unitsinthefirstweekofbusiness.Salesthenincreaseattherateof30unitsperweek.Ifitcontinuesinbusinessfor5weeks,itstotalcumulativesaleswillthereforebe (30 ×1)+ (30 ×2)+ (30 3) + (30 ×4)+ (30 ×5)Youcanseethatthenumberrepresentingtheweekisincreasedby1ineachsuccessiveterm.Thisisratheracumbersomeexpressiontoworkwith.Wecaninsteadwritesalesrevenue=5i=1 30 iThismeansthatoneissummingalltheterms30iforvaluesofifrom1to5.Ifthenumberofweeksofbusinessnwasnotknownwecouldinsteadwritesalesrevenue=ni=1 30 iToevaluateanexpressioncontainingasummationsign,onemaystillhavetocalculatethevalueofeachtermseparatelyandthenaddup.However,spreadsheetscanbeusedtodotediouscalculationsandinsomecasesshort-cutformulaemaybeused(seeChapter7).© ... Exercise 3. 71. Solve for x when 16x = 2x + 56.2. Solve for x when14 =6 +4x5x 3. Solve for x when 45 = 24 + 3x.4. Solve for x if 5x2+ 20 = 1,000.5. If q = 560 − 3p solve for p when q = 31 4.6....

... 38 6.81 8 16.00% 1.24% 38 7.64 9 16.25% 1.26% 38 8.47 10 16.50% 1.28% 38 9 .30 11 16.75% 1 .30 % 39 0. 13 12 17.00% 1 .32 % 39 0.95 13 17.25% 1 .33 % 39 1.78 14 17.50% 1 .35 % 39 2.61 15 17.75% 1 .37 % 39 3. 43 ... 81 .38 4 11 2.25% 81.6 13 12 2 .30 % 81.844 13 2 .35 % 82.075 14 2.40% 82 .30 7 15 2.45% 82.540 16 2.50% 82.7 73 17 2.55% 83. 008 18 2.60% 83. 2 43 19 2.65% 83. 479 20 2.70% 83. 715 21 2.75% 83. 9 53 ... 1 .37 % 39 3. 43 16 18.00% 1 .39 % 39 4.26 17 18.25% 1.41% 39 5.08 18 18.50% 1.42% 39 5.90 19 18.75% 1.44% 39 6. 73 20 19.00% 1.46% 39 7.55 21 19.25% 1.48% 39 8 .37 22 19.50% 1.50% 39 9.19 23 19.75%...

... −8qMC =dTCdq= 3q2− 42q + 160To maximize profits MC = MR. Therefore,3q2− 42q + 160 = 184 −8q3q2− 34 q − 24 = 0(q − 12)(3q + 2) = 0q − 12 = 0or3q + 2 = 0q = 12 or q =−2 3 One cannot produce ... 65 +q1.5what is MC when q = 25?2. Derive a formula for MC if TC = 4q 3 − 20q2+ 60q + 40. 3. If TC = 0.5q 3 − 3q2+ 25q + 20 derive functions for: (a) MC, (b) AC, (c) theslope of AC.4. ... 0.2q2slope =dTRdq= 60 −0.4q© 19 93, 20 03 Mike RosserTherefore,dpdq= 30 +0.5qWhen q = 8,dpdq= 30 +0.5(8) = 30 +4 =−26Also, when q = 8,p = 900 − 30 (8) + 0.25(8)2= 900 −240 + 16 =...

... L0.75(2)Taking (1) to the power of 3 L0.75=42186K0.7 3 =42 3 186 3 K2.1 (3) Setting (3) equal to (2)42 3 186 3 K2.1= 31 K0 .3 K1.8= 31 (186) 3 42 3 = 2, 692.481K = 80.471179 (4)Substituting ... TC2= (32 0q1+ 32 0q2− 0.1q21− 0.2q1q2− 0.1q22) − (36 + 0.003q 3 1) − (45 + 0.005q 3 2)= 32 0q1+ 32 0q2− 0.1q21− 0.2q1q2− 0.1q22− 36 − 0.003q 3 1− 45 − 0.005q 3 2First-order ... 1.8x−0.7y0.2z0 .3 (1)∂G∂y= 4.8x0 .3 y−0.8z0 .3 − λ = 0 λ = 4.8x0 .3 y−0.8z0 .3 (2)∂G∂z= 7.2x0 .3 y0.2z−0.7− 6λ = 0 λ = 1.2x0 .3 y0.2z−0.7 (3) ∂G∂λ= 30 0 − 4x − y −6z = 0(4)A simultaneous...

... 0 175 23. 13 Stability => UNSTABLE 11 1 157.5 25 .31 2.19 12 2 1 83. 75 22. 03 -3. 28 13 3 144 .37 5 26.95 4.92 14 4 2 03. 437 5 19.57 -7 .38 15 5 114.8 437 5 30 .64 11.07 16 6 247. 734 38 14. 03 -16.61 ... λ18K−0.6L0 .3 R0 .3 = 0 λ =80K0.618L0 .3 R0 .3 (1)∂G∂L= 35 − λ 13. 5K0.4L−0.7R0 .3 = 0 λ = 35 L0.7 13. 5K0.4R0 .3 (2)∂G∂R= 50 − λ 13. 5K0.4L0 .3 R−0.7= 0 λ =50R0.7 13. 5K0.4L0 .3 (3) ∂G∂λ= ... 247. 734 38 14. 03 -16.61 17 7 48 .39 8 438 38 .95 24.92 18 8 34 7.40 234 1.57 -37 .38 19 9 -101.1 035 2 57.64 56.06 20 10 571.65527 -26.46 -84.09 © 19 93, 20 03 Mike Rosser for price P1, corresponding...

... 19 9 265 .36 37 5 .36 20 10 265.22 37 5.22 21 11 265. 13 375. 13 22 12 265.08 37 5.08 23 13 265.05 37 5.05 24 14 265. 03 375. 03 25 15 265.02 37 5.02 26 16 265.01 37 5.01 27 17 265.01 37 5.01 28 ... = 37 5 9 t C Y 10 0 30 1.00 411.00 11 1 286.60 39 6.60 12 2 277.96 38 7.96 13 3 272.78 38 2.78 14 4 269.67 37 9.67 15 5 267.80 37 7.80 16 6 266.68 37 6.68 17 7 266.01 37 6.01 18 8 265.60 37 5.60 ... theny0= 34 = Ae0Therefore 34 = AThe definite solution is thusyt= 34 e1.5tWhen t = 7 then using this definite solution we can predicty7= 34 e1.5(7)= 34 e10.5= 34 (36 ,31 5.5) = 1, 234 , 727©...

... -0.1805 0. 031 94 0.08268 values 2 12 0. 033 98 -0.0162 -1E-17 -0.0512 0. 033 43 -1E-17 12 13 -0.07 23 0. 034 16 0.06457 0.05788 -0.02 53 -0.0591 1 14 0. 031 84 -0.0257 -0. 133 9 -0.0156 0. 033 1 0.11024 8 ... 5 93 5 24 41 9 3 0 1 31 76 6 5 0 -1 3 -7 35 7 9 11 39 23 15 0 6788 28 49 4 5 9 7 39 19 Inverse A^-1 A^-1*b = x 10 -0.04 53 0.087 83 0.11969 0 .32 077 -0.0 634 -0. 133 9 solution 5 11 0.02 431 -0.0509 ... remainby (−1) 3 . Therefore|C12|=(−1) 3 a21a 23 a 31 a 33 = (−1)a21a 23 a 31 a 33 Example 15. 13 Find the cofactor |C22| of the matrix A =8 23 194 436 SolutionThe...