ejercicios cap5
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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%.
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-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
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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.
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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
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3 3.5
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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).
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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
iteraciones Xa f(Xa) Xb f(Xb) Xc
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)
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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
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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
iteraciones Xa f(Xa) Xb f(Xb) Xc
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.
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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
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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
iteraciones Xa f(Xa) Xb f(Xb) Xc
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
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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
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6
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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
iteraciones Xa f(Xa) Xb f(Xb) Xc
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
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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
metodo de la regla falsa
iteraciones Xa f(Xa) Xb f(Xb) Xc
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
metodo de la regla falsa
iteraciones Xa f(Xa) Xb f(Xb) Xc
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
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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)
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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
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161.65717
-15
-10
-5
0
5
10
15
-1 -0.5 0 0.5 1 y
x
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