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# Examen final

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### Examen final

1. 1. 1.- Dados los siguientes datos,encuentre el trabajo isotérmico (𝑤 = ∫ 𝑃𝑑𝑉) Realizado sobreel gas cuando se comprime de 23L a3l. clear clc syms x Y c1 c2 c3 c4 c5 D X=[3 8 13 18 23] % V,l f=[12.5 3.5 1.8 1.4 1.2] % P,atm plot(X, f, 'p') C=[c1, c2, c3, c4, c5] y=Y*ones(5,5) for i=1:5 for j=1:5 s(i, j)=sum(X.^(i-1).*X.^(j-1)) end end p=inv(s) d=D*ones(1,5) for i=1:5 D(i)=sum(f.*X.^(i-1)) end c=p*D' f1=c(1,1)+c(2,1)*x+c(3,1)*x^2+c(4,1)*x^3+c(5,1)*x^4 hold on fplot(char(f1), [23 3], 'r') F=f1 syms x Funcion=vpa(int(F, x, [23, 3])) POLINOMIO f1 = F=(132511742741563028118449706779*x^4)/405648192073033408478945025720320 - (137666855188831736408209924901*x^3)/6338253001141147007483516026880 + (21140978790370187092321969137*x^2)/39614081257132168796771975168 - (576967209289482091394341575591*x)/99035203142830421991929937920 + 2548627377380721607314386298391/99035203142830421991929937920 EVALUANDO EN EL RANGO DE [23 A 3] F, x, [23 3]= - 60.955555383300052355524086801562 EVALUANDO EN EL RANGO DE [3 A 23] F, x, [3 23]= 60.955555383300052355524086801562
2. 2. 2., LA VISCOSIDAD DINÁMICA DEL AGUA u(10^-3N.S/m^2) SE REALCIONA CON LA TEMPERATUA T(C), DE LA SIGUIENTE MANERA: a) Grafique los datos B)CALCULE LA VISCOSIDAD DINÁMICA POR INTERPOLACIÓN A LA TEMPERATURA T=7.5 C u= 1.4068632289341517857142857142857 clear; clc; syms x X=[0 5 10 20 30 40]%volumen f=[1.787 1.519 1.307 1.002 0.7975 0.6529] plot(X, f, 'p') n=numel (X)-1; P=0 for k=1:n+1 I=1 for i=1:n+1
3. 3. if i~=k I=I*(x-X(i))/(X(k)-X(i)); end end P=P+f(k)*I; end vpa(subs (P,x,7.5)) pretty (P) X = 0 5 10 20 30 40 f = 1.7870 1.5190 1.3070 1.0020 0.7975 0.6529 P = 0 I = 1 I = 1 I = 1 I = 1 I = 1 I = 1 ans =
4. 4. 1.4068632289341517857142857142857 6529 x (x - 5) (x - 10) (x - 20) (x - 30) ----------------------------------------- 84000000000 319 x (x - 5) (x - 10) (x - 20) (x - 40) - ---------------------------------------- 600000000 167 x (x - 5) (x - 10) (x - 30) (x - 40) + ---------------------------------------- 100000000 1307 x (x - 5) (x - 20) (x - 30) (x - 40) - ----------------------------------------- 300000000 217 x (x - 10) (x - 20) (x - 30) (x - 40) + ----------------------------------------- 46875000 / x | - - 1 | (x - 10) (x - 20) (x - 30) (x - 40) 1787 5 / - -------------------------------------------------- 240000000 Published withMATLAB® R2015b C)USE EL AJUSTE DE CURVAS PARA HACER EL CÁLCULO ANTERIOR clear clc syms x Y c1 c2 c3 c4 c5 D X=[0 5 10 20 30 40]%volumen f=[1.787 1.519 1.307 1.002 0.7975 0.6529] plot(X, f, 'p') C=[c1, c2, c3, c4, c5] y=Y*ones(5,5) for i=1:5 for j=1:5 s(i, j)=sum(X.^(i-1).*X.^(j-1)) end end p=inv(s) d=D*ones(1,5) for i=1:5 D(i)=sum(f.*X.^(i-1)) end c=p*D' f1=c(1,1)+c(2,1)*x+c(3,1)*x^2+c(4,1)*x^3+c(5,1)*x^4 hold on F=f1 syms x vpa(subs (F,x,7.5))
5. 5. ans = 1.4068564099042179949929734811461 3. RESUELVA EL SIGUIENTE SISTEMA DE ECUACIONES 4X-Y+Z-2=0 -X+6Y-3Z-11=0 X-4Y+5Z+5=0 clear;clc; A=[4 -1 1;-1 6 -3;1 -4 5];%INGRESO LAS TRES ECUACIONES B(:,1)=[2 11 -5]; [n m]=size(A) for i=1:n for j=1:n I(i,j)=0 if i==j I(i,j)=1 end end end M=[A,I] for i=1:n M(i,:)=M(i,:)/M(i,i) for j=1:n if j~=i M(j,:)=M(j,:)-M(j,i)*M(i,:) end end
6. 6. end MIA=M(:,4:6) X=MIA*B n = 3 m = 3 I = 0 I = 1 I = 1 0 I = 1 0 0 I = 1 0 0 0 0 0 I = 1 0 0 0 0 0 I = 1 0 0 0 1 0
7. 7. I = 1 0 0 0 1 0 I = 1 0 0 0 1 0 0 0 0 I = 1 0 0 0 1 0 0 0 0 I = 1 0 0 0 1 0 0 0 0 I = 1 0 0 0 1 0 0 0 1 M = 4 -1 1 1 0 0 -1 6 -3 0 1 0 1 -4 5 0 0 1 M = 1.0000 -0.2500 0.2500 0.2500 0 0 -1.0000 6.0000 -3.0000 0 1.0000 0 1.0000 -4.0000 5.0000 0 0 1.0000 M =
8. 8. 1.0000 -0.2500 0.2500 0.2500 0 0 0 5.7500 -2.7500 0.2500 1.0000 0 1.0000 -4.0000 5.0000 0 0 1.0000 M = 1.0000 -0.2500 0.2500 0.2500 0 0 0 5.7500 -2.7500 0.2500 1.0000 0 0 -3.7500 4.7500 -0.2500 0 1.0000 M = 1.0000 -0.2500 0.2500 0.2500 0 0 0 1.0000 -0.4783 0.0435 0.1739 0 0 -3.7500 4.7500 -0.2500 0 1.0000 M = 1.0000 0 0.1304 0.2609 0.0435 0 0 1.0000 -0.4783 0.0435 0.1739 0 0 -3.7500 4.7500 -0.2500 0 1.0000 M = 1.0000 0 0.1304 0.2609 0.0435 0 0 1.0000 -0.4783 0.0435 0.1739 0 0 0 2.9565 -0.0870 0.6522 1.0000 M = 1.0000 0 0.1304 0.2609 0.0435 0 0 1.0000 -0.4783 0.0435 0.1739 0 0 0 1.0000 -0.0294 0.2206 0.3382 M = 1.0000 0 0 0.2647 0.0147 -0.0441 0 1.0000 -0.4783 0.0435 0.1739 0 0 0 1.0000 -0.0294 0.2206 0.3382 M = 1.0000 0 0 0.2647 0.0147 -0.0441 0 1.0000 0 0.0294 0.2794 0.1618 0 0 1.0000 -0.0294 0.2206 0.3382
9. 9. MIA = 0.2647 0.0147 -0.0441 0.0294 0.2794 0.1618 -0.0294 0.2206 0.3382 X = 0.9118 2.3235 0.6765 X=0.9118 Y= 2.3235 Z= 0.6765 4. ENCUENTRE LA SOLUCIÓNAPROXIMADA DE LA SIGUIENTEECUACIÓNDIFERENCIAL clear;clc; h=0.1 x=0:h:1 syms X n=numel(x) for i=1:n y(i)=sym(['y',num2str(i)])%[genera ya lamtriz y1 y2 y3....yn] end y(1)=0 y(end)=0% lo que cambia for i=1:n-2 f(i,1)=((y(i+2)-2*y(i+1)+y(i))/h^2)+pi^2/4*y(i)-pi^2/16*cos(pi*x(i)/4) end R=solve(f)%resuelve el sistema de ecuaciones V(:,1)=sort(symvar(f))%ordenq los vectores R=struct2cell(R) R=cell2num(R) y=vpa(subs(y,V,R)) plot(x,y) n=numel (x); P=0 for k=1:n I=1 for i=1:n if i~=k I=I*(X-x(i))/(x(k)-x(i)); end end P=P+y(k)*I; end P=expand(P)
10. 10. hold on fplot(char(P),[0 1],'r') h = 0.1000 x = Columns 1 through 7 0 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 Columns 8 through 11 0.7000 0.8000 0.9000 1.0000 n = 11 y = y1 y = [ y1, y2] y = [ y1, y2, y3] y = [ y1, y2, y3, y4] y = [ y1, y2, y3, y4, y5] y = [ y1, y2, y3, y4, y5, y6] y =
11. 11. [ y1, y2, y3, y4, y5, y6, y7] y = [ y1, y2, y3, y4, y5, y6, y7, y8] y = [ y1, y2, y3, y4, y5, y6, y7, y8, y9] y = [ y1, y2, y3, y4, y5, y6, y7, y8, y9, y10] y = [ y1, y2, y3, y4, y5, y6, y7, y8, y9, y10, y11] y = [ 0, y2, y3, y4, y5, y6, y7, y8, y9, y10, y11] y = [ 0, y2, y3, y4, y5, y6, y7, y8, y9, y10, 0] f = 100*y3 - 200*y2 - 2778046668940015/4503599627370496 f = 100*y3 - 200*y2 - 2778046668940015/4503599627370496 (115368037353202415*y2)/1125899906842624 - 200*y3 + 100*y4 - 5538965756371755/9007199254740992 f = 100*y3 - 200*y2 - 2778046668940015/4503599627370496 (115368037353202415*y2)/1125899906842624 - 200*y3 + 100*y4 - 5538965756371755/9007199254740992 (115368037353202415*y3)/1125899906842624 - 200*y4 + 100*y5 - 5487688609082427/9007199254740992 f =
12. 12. 100*y3 - 200*y2 - 2778046668940015/4503599627370496 (115368037353202415*y2)/1125899906842624 - 200*y3 + 100*y4 - 5538965756371755/9007199254740992 (115368037353202415*y3)/1125899906842624 - 200*y4 + 100*y5 - 5487688609082427/9007199254740992 (115368037353202415*y4)/1125899906842624 - 200*y5 + 100*y6 - 2701289018338233/4503599627370496 f = 100*y3 - 200*y2 - 2778046668940015/4503599627370496 (115368037353202415*y2)/1125899906842624 - 200*y3 + 100*y4 - 5538965756371755/9007199254740992 (115368037353202415*y3)/1125899906842624 - 200*y4 + 100*y5 - 5487688609082427/9007199254740992 (115368037353202415*y4)/1125899906842624 - 200*y5 + 100*y6 - 2701289018338233/4503599627370496 (115368037353202415*y5)/1125899906842624 - 200*y6 + 100*y7 - 5284158774134893/9007199254740992 f = 100*y3 - 200*y2 - 2778046668940015/4503599627370496 (115368037353202415*y2)/1125899906842624 - 200*y3 + 100*y4 - 5538965756371755/9007199254740992 (115368037353202415*y3)/1125899906842624 - 200*y4 + 100*y5 - 5487688609082427/9007199254740992 (115368037353202415*y4)/1125899906842624 - 200*y5 + 100*y6 - 2701289018338233/4503599627370496 (115368037353202415*y5)/1125899906842624 - 200*y6 + 100*y7 - 5284158774134893/9007199254740992 (115368037353202415*y6)/1125899906842624 - 200*y7 + 100*y8 - 5133160915589677/9007199254740992 f = 100*y3 - 200*y2 - 2778046668940015/4503599627370496 (115368037353202415*y2)/1125899906842624 - 200*y3 + 100*y4 - 5538965756371755/9007199254740992 (115368037353202415*y3)/1125899906842624 - 200*y4 + 100*y5 - 5487688609082427/9007199254740992 (115368037353202415*y4)/1125899906842624 - 200*y5 + 100*y6 - 2701289018338233/4503599627370496 (115368037353202415*y5)/1125899906842624 - 200*y6 + 100*y7 - 5284158774134893/9007199254740992 (115368037353202415*y6)/1125899906842624 - 200*y7 + 100*y8 - 5133160915589677/9007199254740992 (115368037353202415*y7)/1125899906842624 - 200*y8 + 100*y9 - 4950515413050633/9007199254740992
13. 13. f = 100*y3 - 200*y2 - 2778046668940015/4503599627370496 (115368037353202415*y2)/1125899906842624 - 200*y3 + 100*y4 - 5538965756371755/9007199254740992 (115368037353202415*y3)/1125899906842624 - 200*y4 + 100*y5 - 5487688609082427/9007199254740992 (115368037353202415*y4)/1125899906842624 - 200*y5 + 100*y6 - 2701289018338233/4503599627370496 (115368037353202415*y5)/1125899906842624 - 200*y6 + 100*y7 - 5284158774134893/9007199254740992 (115368037353202415*y6)/1125899906842624 - 200*y7 + 100*y8 - 5133160915589677/9007199254740992 (115368037353202415*y7)/1125899906842624 - 200*y8 + 100*y9 - 4950515413050633/9007199254740992 (115368037353202415*y8)/1125899906842624 - 200*y9 + 100*y10 - 2368674168388353/4503599627370496 f = 100*y3 - 200*y2 - 2778046668940015/4503599627370496 (115368037353202415*y2)/1125899906842624 - 200*y3 + 100*y4 - 5538965756371755/9007199254740992 (115368037353202415*y3)/1125899906842624 - 200*y4 + 100*y5 - 5487688609082427/9007199254740992 (115368037353202415*y4)/1125899906842624 - 200*y5 + 100*y6 - 2701289018338233/4503599627370496 (115368037353202415*y5)/1125899906842624 - 200*y6 + 100*y7 - 5284158774134893/9007199254740992 (115368037353202415*y6)/1125899906842624 - 200*y7 + 100*y8 - 5133160915589677/9007199254740992 (115368037353202415*y7)/1125899906842624 - 200*y8 + 100*y9 - 4950515413050633/9007199254740992 (115368037353202415*y8)/1125899906842624 - 200*y9 + 100*y10 - 2368674168388353/4503599627370496 (115368037353202415*y9)/1125899906842624 - 200*y10 - 4494973932678371/9007199254740992 R = y2: [1x1 sym] y3: [1x1 sym] y4: [1x1 sym] y5: [1x1 sym] y6: [1x1 sym] y7: [1x1 sym] y8: [1x1 sym] y9: [1x1 sym] y10: [1x1 sym] V = y2
14. 14. y3 y4 y5 y6 y7 y8 y9 y10 R = [1x1 sym] [1x1 sym] [1x1 sym] [1x1 sym] [1x1 sym] [1x1 sym] [1x1 sym] [1x1 sym] [1x1 sym] R = - 5479480335076111974454307016617118247814794623629915525398060954655645196105908691/16652 6140704593586610215588894933189801576823372031615697621472639563198192890675200 - 1241467964144459044791372777739847138082774634969182699644925920740032345062669869/20815 767588074198326276948611866648725197102921503951962202684079945399774111334400 - 2977950296898445163316325462680505042561611028785621032278639896313331451171197419703592 88312689767/3749835326123272314354241130007724453017593764194257367200298172281167449757 485627800856869994496000 - 1717914066534286339595969631322530063972912951137326276150630377474180181505278807593138 54193803979/1874917663061636157177120565003862226508796882097128683600149086140583724878 742813900428434997248000 - 3358608839805541139339632072712832617955056225963513807996684510102544420976820691/35029 604586466190446559122324188521566881025673128934216139862552944888114053120000 - 1942193839611116471941432436481786875894521394144609085871122170184276894456977429136656 726456353715918054836756181/211096962217868644249156937479372940942901952247599873171552 25575615611421862296161608936531431632430698090987520000 - 1522365289391611371765580911012770391270863579825783193029347838455554598899908602047449 87992545704891079147923071205088455366106463/1901392400766873798178740679832603891465408 0351348880043211536172900014069947851641627966312302727628008464002010210714710441984000 00 - 5738449853945654575928682306930829044283860604295000928808751179860006665896084138502514
15. 15. 8213329008784313205769099536721554452947617/95069620038343689908937033991630194573270401 7567444002160576808645000703497392582081398315615136381400423200100510535735522099200000 - 1430901260164460489703022676664353189263928871836348349027461869752071363788993545179389 758378177603551089097496915468760056056995714488495148659011/428155505378939281517418673 9876936007478810260904326163166428642542960608981333008777322299933586233171221230378136 8051925062480552449940652032000000 y = [ 0, -0.03290462573558555858768644316681, -0.05964074872049026843018229875493, - 0.079415495292620427042006826597999, -0.091626107128834037134502402588291, - 0.095879153631758416843133586417488, -0.092004821822429632713408409267006, - 0.080065813283865425589841603532548, -0.060360500564020454268710267968903, - 0.03342012988710820123494764952464, 0] P = 0 I = 1 I = 1 I = 1 I = 1 I = 1 I = 1 I = 1
16. 16. I = 1 I = 1 I = 1 I = 1 P = 0.000008377446453390598345464870661955*X^10 - 0.000069602164145921450316337671428029*X^9 - 0.00011698363573581745189686669582295*X^8 + 0.001533299032018222700433745025631*X^7 + 0.0040930901028205732145231521862629*X^6 - 0.024128925374348760378954901216781*X^5 - 0.058257941125119981073034651845014*X^4 + 0.17271940874980220128830367905092*X^3 + 0.26103566669973466448955614575887*X^2 - 0.3568163897314785719369594294633*X