ejercicios cap5

7/15/2019 ejercicios cap5

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f(x) = -2+7x-5x2

+6x3

x f(x) Xa= 0

-3 -230 Xb= 1

-2.5 -144.5

-2 -84

-1.5 -44

-1 -20

0 -2

1 6

1.5 17.5

2 40

2.5 78

3 136

iteraciones Xa f(Xa) Xb Xc f(Xc) f(Xa)*f(Xc)

1 0 -2 1 0.5 1 -2

2 0 -2 0.5 0.25 -0.46875 0.9375

3 0.25 -0.46875 0.5 0.375 0.23828125 -0.11169434

4 0.25 -0.46875 0.375 0.3125 -0.11767578 0.05516052

5 0.3125 -0.11767578 0.375 0.34375 0.05914307 -0.00695971

6 0.3125 -0.11767578 0.34375 0.328125 -0.02948761 0.00346998

7 0.328125 -0.02948761 0.34375 0.3359375 0.01476383 -0.00043535

8 0.328125 -0.02948761 0.3359375 0.33203125 -0.00737679 0.00021752

9 0.33203125 -0.00737679 0.3359375 0.33398438 0.00368966 -2.7218E-05

10 0.33203125 -0.00737679 0.33398438 0.33300781 -0.00184451 1.3607E-05

11 0.33300781 -0.00184451 0.33398438 0.33349609 0.00092234 -1.7013E-06

12 0.33300781 -0.00184451 0.33349609 0.33325195 -0.00046115 8.5059E-0713 0.33325195 -0.00046115 0.33349609 0.33337402 0.00023058 -1.0633E-07

14 0.33325195 -0.00046115 0.33337402 0.33331299 -0.00011529 5.3165E-08

15 0.33331299 -0.00011529 0.33337402 0.33334351 5.7644E-05 -6.6457E-09

16 0.33331299 -0.00011529 0.33334351 0.33332825 -2.8822E-05 3.3229E-09

17 0.33332825 -2.8822E-05 0.33334351 0.33333588 1.4411E-05 -4.1536E-10

18 0.33332825 -2.8822E-05 0.33333588 0.33333206 -7.2055E-06 2.0768E-10

19 0.33333206 -7.2055E-06 0.33333588 0.33333397 3.6028E-06 -2.596E-11

20 0.33333206 -7.2055E-06 0.33333397 0.33333302 -1.8014E-06 1.298E-11

21 0.33333302 -1.8014E-06 0.33333397 0.33333349 9.0069E-07 -1.6225E-12

22 0.33333302 -1.8014E-06 0.33333349 0.33333325 -4.5035E-07 8.1125E-13

23 0.33333325 -4.5035E-07 0.33333349 0.33333337 2.2517E-07 -1.0141E-13

24 0.33333325 -4.5035E-07 0.33333337 0.33333331 -1.1259E-07 5.0703E-14

25 0.33333331 -1.1259E-07 0.33333337 0.33333334 5.6293E-08 -6.3379E-1526 0.33333331 -1.1259E-07 0.33333334 0.33333333 -2.8147E-08 3.1689E-15

27 0.33333333 -2.8147E-08 0.33333334 0.33333334 1.4073E-08 -3.9612E-16

28 0.33333333 -2.8147E-08 0.33333334 0.33333333 -7.0367E-09 1.9806E-16

29 0.33333333 -7.0367E-09 0.33333334 0.33333333 3.5183E-09 -2.4757E-17

30 0.33333333 -7.0367E-09 0.33333333 0.33333333 -1.7592E-09 1.2379E-17

5.2 determine las raíces reales de f(x) = -2+7x-5x2+6x3:a) gráficamenteb) utilizando el método de la bisección para localizar la raíz más pequeña. Use los valores inicialesiterando hasta que el error estimado εa se encuentre debajo de εs = 10%.



-3, -230

-144.5

-84

-44-20

-2 6 17.540

78

136

-250

-200

-150

-100

-50

0

50

100

150

200

-4 -3 -2 -1 0 1 2 3 4

f(x)=-2+7x-5x^2+6x^3

xl = 0 y xu = 1



f(x) = -11-22x+17x2-2.5x

3

x f(x) Xa= -0.5

-3 275.5 Xb= 0

-2.5 189.3125

-2 121

-1.5 68.6875

-1 30.5

-0.5 4.5625

0 -11

0.5 -18.0625

1 -18.5

1.5 -14.1875

2 -7

2.5 1.1875

3 8.5

3.5 13.0625

4 13

4.5 6.4375

5 -8.5

iteraciones Xa f(Xa) Xb f(Xb) Xc

1 4.5 6.4375 5 -8.5 4.71548117

2 4.71548117 1.13657915 5 -8.5 4.74903853

3 4.74903853 0.16136889 5 -8.5 4.753714164 4.75371416 0.02217511 5 -8.5 4.75435501

5 4.75435501 0.00303353 5 -8.5 4.75444265

6 4.75444265 0.00041473 5 -8.5 4.75445463

7 4.75445463 5.6694E-05 5 -8.5 4.75445627

8 4.75445627 7.7502E-06 5 -8.5 4.75445649

5.4 calcule las raíces reales de f(x) = -11-22x+17x2-2.5x3:a) gráficamenteb) utilizando el método de la falsa posición con un valor de εs correspondiente a tresmás pequeña.



f(Xc) f(Xa)*f(Xc)

1.13657915 30.35591

0.16136889 5.39765818

0.02217511 0.767101560.00303353 0.10542833

0.00041473 0.01442274

5.6694E-05 0.0019718

7.7502E-06 0.00026955

1.0595E-06 3.6848E-05

-50

0

50

100

150

200

250

300

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

f(x) = -11-22x+17x^2-2.5x^3

cifras significativas para determinar la raíz



3 3.5



f(x) = 0,7-lnx2

x f(x) Xa= 0.5

-3 1.49722458 Xb= 2

-2.5 1.13258146

-2 0.68629436

-1.5 0.11093022

-1 -0.7

-0.5 -2.08629436

0 #NUM!

0.5 -2.08629436

1 -0.7

1.5 0.11093022

2 0.68629436

2.5 1.13258146

3 1.49722458

iteraciones Xa f(Xa) Xb Xc f(Xc) f(Xa)*f(Xc)

1 0.5 2.08629436 2 1.25 0.2537129 0.52931979

2 1.25 0.2537129 2 1.625 -0.27101563 -0.06876016

3 1.25 0.2537129 1.625 1.4375 -0.02581099 -0.00654858

4 1.25 0.2537129 1.4375 1.34375 0.10907157 0.02767287

5 1.34375 0.10907157 1.4375 1.390625 0.04049343 0.00441668

6 1.390625 0.04049343 1.4375 1.4140625 0.00706647 0.00028615

7 1.4140625 0.00706647 1.4375 1.42578125 -0.00943982 -6.6706E-05

8 1.4140625 0.00706647 1.42578125 1.41992188 -0.0012037 -8.5059E-06

9 1.4140625 0.00706647 1.41992188 1.41699219 0.00292711 2.0684E-0510 1.41699219 0.00292711 1.41992188 1.41845703 0.00086063 2.5192E-06

11 1.41845703 0.00086063 1.41992188 1.41918945 -0.0001718 -1.4786E-07

12 1.41845703 0.00086063 1.41918945 1.41882324 0.00034435 2.9636E-07

13 1.41882324 0.00034435 1.41918945 1.41900635 8.6257E-05 2.9703E-08

14 1.41900635 8.6257E-05 1.41918945 1.4190979 -4.2777E-05 -3.6898E-09

15 1.41900635 8.6257E-05 1.4190979 1.41905212 2.1739E-05 1.8752E-09

16 1.41905212 2.1739E-05 1.4190979 1.41907501 -1.0519E-05 -2.2867E-10

17 1.41905212 2.1739E-05 1.41907501 1.41906357 5.61E-06 1.2196E-10

18 1.41906357 5.61E-06 1.41907501 1.41906929 -2.4545E-06 -1.377E-11

19 1.41906357 5.61E-06 1.41906929 1.41906643 1.5777E-06 8.8511E-12

20 1.41906643 1.5777E-06 1.41906929 1.41906786 -4.384E-07 -6.9167E-13

21 1.41906643 1.5777E-06 1.41906786 1.41906714 5.6967E-07 8.9879E-13

22 1.41906714 5.6967E-07 1.41906786 1.4190675 6.5637E-08 3.7391E-14

23 1.4190675 6.5637E-08 1.41906786 1.41906768 -1.8638E-07 -1.2233E-14

24 1.4190675 6.5637E-08 1.41906768 1.41906759 -6.0371E-08 -3.9626E-15

25 1.4190675 6.5637E-08 1.41906759 1.41906755 2.6327E-09 1.728E-16

26 1.41906755 2.6327E-09 1.41906759 1.41906757 -2.8869E-08 -7.6004E-17

27 1.41906755 2.6327E-09 1.41906757 1.41906756 -1.3118E-08 -3.4536E-17

28 1.41906755 2.6327E-09 1.41906756 1.41906755 -5.2428E-09 -1.3803E-17

29 1.41906755 2.6327E-09 1.41906755 1.41906755 -1.3051E-09 -3.4358E-18

30 1.41906755 2.6327E-09 1.41906755 1.41906755 6.6382E-10 1.7476E-18

5.6 determine la raíz real de ln x^2 = 0.7:a) gráficamenteb) empleando tres iteraciones en el método de la bisección con los valores iniciales xl = 0.5 y xu =2.c) usando tres iteraciones del método de la falsa posición, con los mismos valores iniciales de b).



f(x) = -11-22x+17x2-2.5x

3

x f(x) Xa= 0.5

-3 -1.49722458 Xb= 2

-2.5 -1.13258146

-2 -0.68629436

-1.5 -0.11093022

-1 0.7

-0.5 2.086294360 #NUM!

0.5 2.08629436

1 0.7

1.5 -0.11093022

2 -0.68629436

2.5 -1.13258146

3 -1.49722458


1 0.5 2.08629436 2 -0.68629436 1.62870745

2 1.62870745 -0.27557345 2 -0.68629436 1.37958848

3 1.62870745 -0.27557345 1.37958848 0.05642949 1.42193045

4 1.62870745 -0.27557345 1.42193045 -0.00403084 1.41886101

5 1.62870745 -0.27557345 1.41886101 0.00029112 1.41908246

6 1.62870745 -0.27557345 1.41908246 -2.1009E-05 1.41906647

7 1.62870745 -0.27557345 1.41906647 1.5162E-06 1.41906763

8 1.62870745 -0.27557345 1.41906763 -1.0942E-07 1.41906754

9 1.62870745 -0.27557345 1.41906754 7.8971E-09 1.41906755

10 1.62870745 -0.27557345 1.41906755 -5.6993E-10 1.41906755

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

-3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3

f(x)=0,7-ln(x^2)



f(Xc) f(Xa)*f(Xc)

-0.27557345 3.39796317

0.05642949 -0.38017795

-0.00403084 -0.39184628

0.00029112 -0.39100042

-2.1009E-05 -0.39106144

1.5162E-06 -0.39105704

-1.0942E-07 -0.39105736

7.8971E-09 -0.39105733

-5.6993E-10 -0.39105734

4.1132E-11 -0.39105734

3.5



f(x) = (√15)-x

x f(x) Xa= 3

Xb= 4

2 1.87298335

3 0.87298335

4 -0.12701665

5 -1.12701665

6 -2.12701665


1 3 0.87298335 4 -0.12701665 3.87298335

2 3.87298335 0 4 -0.12701665 3.87298335

3 3.87298335 0 4 -0.12701665 3.87298335

4 3.87298335 0 4 -0.12701665 3.87298335

5 3.87298335 0 4 -0.12701665 3.87298335

6 3.87298335 0 4 -0.12701665 3.872983357 3.87298335 0 4 -0.12701665 3.87298335

8 3.87298335 0 4 -0.12701665 3.87298335

5.8 calcule la raíz cuadrada positiva de 15 usando el método de la falsa posición conUse los valores iniciales xl = 3 y xu = 4.



f(Xc) f(Xa)*f(Xc)

0 3.38104996

0 0

0 0

0 0

0 0

0 00 0

0 0

εs = 0.5%.

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3 3.5 4 4.5 5



f(x) = x4-8x

3-36x

2+462x-1010

x f(x) Xa= 3

-3 -2423 Xb= 4

-2.5 -2225.9375

-2 -1998

-1.5 -1751.9375

-1 -1499

0 -1010

1 -591

1.5 -419.9375

2 -278

2.5 -165.9375

3 -83

3.66666667 -13.617284

4.23809524 15.022758

4.80952381 24.3197485

5.38095238 25.5756449


1 3 -83 4 6 3.93258427

2 3 -83 3.93258427 2.73283018 3.9028571

3 3 -83 3.9028571 1.18445917 3.89015407

4 3 -83 3.89015407 0.50224906 3.88479998

5 3 -83 3.88479998 0.21098687 3.882556516 3 -83 3.88255651 0.08828349 3.88161877

7 3 -83 3.88161877 0.0368796 3.88122721

8 3 -83 3.88122721 0.01539547 3.88106379

9 3 -83 3.88106379 0.00642502 3.88099559

10 3 -83 3.88099559 0.00268104 3.88096713

11 3 -83 3.88096713 0.00111869 3.88095526

12 3 -83 3.88095526 0.00046678 3.8809503

13 3 -83 3.8809503 0.00019476 3.88094824

14 3 -83 3.88094824 8.1264E-05 3.88094737

15 3 -83 3.88094737 3.3907E-05 3.88094701

16 3 -83 3.88094701 1.4148E-05 3.88094686

17 3 -83 3.88094686 5.903E-06 3.880946818 3 -83 3.8809468 2.463E-06 3.88094677

19 3 -83 3.88094677 1.0277E-06 3.88094676

20 3 -83 3.88094676 4.2879E-07 3.88094676

5.10 calcule la raíz real positiva de f(x) = x4-8x3-36x2+462x-1010 utilizando el métodposición. Use una gráfica para escoger el valor inicial y realice el cálculo con εs = 1.0



f(Xc) f(Xa)*f(Xc)

2.73283018 -326.404494

1.18445917 -323.937139

0.50224906 -322.882788

0.21098687 -322.438398

0.08828349 -322.252190.0368796 -322.174358

0.01539547 -322.141859

0.00642502 -322.128294

0.00268104 -322.122634

0.00111869 -322.120272

0.00046678 -322.119286

0.00019476 -322.118875

8.1264E-05 -322.118704

3.3907E-05 -322.118632

1.4148E-05 -322.118602

5.903E-06 -322.11859

2.463E-06 -322.1185841.0277E-06 -322.118582

4.2879E-07 -322.118581

1.7891E-07 -322.118581

o de la falsa%.

-3000

-2500

-2000

-1500

-1000

-500

0

500

-4 -3 -2 -1 0 1 2 3 4 5



6



v = gm/c (1-e-(c/m)t

)

g= 9.8 m/s^2 m v

c= 14 kg/s 50 4.93004473

v= 35 m/s 55 2.98077472

t= 7 s 60 1.20165764

65 -0.42537367

70 -1.91674877

75 -3.28714974

80 -4.54956878

85 -5.71544225

90 -6.79481266

95 -7.79649327

Xa= 60

Xb= 65

metodo de la regla falsa


1 60 1.20165764 65 -0.42537367 63.692792

2 63.692792 -0.01373311 65 -0.42537367 63.649181

3 63.692792 -0.01373311 63.649181 0.00016294 63.6496924

4 63.692792 -0.01373311 63.6496924 -6.1592E-08 63.6496922

Xa 0

Xb 1

metodo de la biseccion

iteraciones Xa f(Xa) Xb Xc f(Xc)

1 0 -2 1 0.5 1

2 0 -2 0.5 0.25 -0.46875

3 0.25 -0.46875 0.5 0.375 0.23828125

4 0.25 -0.46875 0.375 0.3125 -0.117675785 0.3125 -0.11767578 0.375 0.34375 0.05914307

6 0.3125 -0.11767578 0.34375 0.328125 -0.02948761

7 0.328125 -0.02948761 0.34375 0.3359375 0.01476383

8

9

5.12 la velocidad v de caída de un paracaidista esta dada por v = gm/c (1-e^-(c/m)t)paracaidista con un coeficiente de arrastre c= 14 kg/s, calcule la masa m de éste de tsea de 35 m/s en t 0 7s. Con el método de la falsa posición determine m a un nivel d



Xa 0.5

Xb 2

iteraciones Xa f(Xa) Xb Xc f(Xc)

1 0.5 -2.08629436 2 1.25 -0.2537129

2 1.25 -0.2537129 2 1.625 0.27101563

3 1.25 -0.2537129 1.625 1.4375 0.02581099

9

Xa= 0.5

Xb= 2



1 0.5 -2.08629436 2 0.68629436 1.62870745

2 0.5 -2.08629436 1.62870745 0.27557345 1.4970143

3 0.5 -2.08629436 1.4970143 0.10694532 1.44839854

Xa= 3.5

Xb= 4



1 3.5 -26.9375 4.5 21.0625 4.06119792

2 3.5 -26.9375 4.06119792 8.68212511 3.92440842

3 3.5 -26.9375 3.92440842 2.31365744 3.8908393

4 3.5 -26.9375 3.8908393 0.5393655 3.8831672

5 3.5 -26.9375 3.8831672 0.12172294 3.88144356

6 3.5 -26.9375 3.88144356 0.02726781 3.88105783

7 3.5 -26.9375 3.88105783 0.00609827 3.88097159

8 3.5 -26.9375 3.88097159 0.00136333 3.88095231



f(Xc) f(Xa)*f(Xc)

-0.01373311 76.5369301

0.00016294 -0.87410136

-6.1592E-08 -0.87410839

2.3277E-11 -0.87410838

x y-0.6 -9.296

-0.4 -5.984

-0.2 -3.648

0 -2

0.2 -0.752

0.4 0.384

0.6 1.696

0.8 3.472

f(Xa)*f(Xc) 1 6

-2 1.2 9.568

0.9375

-0.11169434

0.05516052-0.00695971

0.00346998

-0.00043535

, donde g = 9.8. para elal forma que la velocidad

e de εs 0 0.1%.

-10

-8

-6

-4

-2

0

2

4

6

0

v ( m / s )

m(kg)

v=f(t)



x y

-3 1.49722458

-2 0.68629436

-1.5 0.11093022

-1 -0.7

1 -0.7

1.5 0.11093022

2 0.68629436

3 1.49722458

f(Xa)*f(Xc)

0.52931979

-0.06876016

-0.00654858

f(Xc) f(Xa)*f(Xc)0.27557345 -3.39796317

0.10694532 -3.1232125

0.04091699 -3.02178571

f(Xc) f(Xa)*f(Xc)8.68212511 -109.398519

2.31365744 -105.713752

0.5393655 -104.809484

0.12172294 -104.602816

0.02726781 -104.556386

0.00609827 -104.545995

0.00136333 -104.543672

0.00030476 -104.543153

-1

-0.5

0

0.5

1

1.5

2

-4 -3 -2 -1 0 1 2 3



161.65717

-15

-10

-5

0

5

10

15

-1 -0.5 0 0.5 1 y

x



4



1.5

ejercicios cap5

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