calc_num_exercicios_04_posicao_falsa_respostas

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INSTITUTO FEDERAL MINAS GERAIS (IFMG) - CAMPUS BAMBUÍ
Cálculo Numérico - Equações Não Lineares
Prof. Marcos Roberto Ribeiro
Lista de Exercícios 04
Exercício 1:
Utilize o método da posição falsa para calcular a raiz aproximada das seguintes equações. Considere um
número máximo de 20 iterações e � < 10−5.
(a) f(x) = ln(x), intervalo [0.1, 3]
Solução:
k a b � x(k) f(x(k)) f(a)× f(x(k))
0 0.10000 3.00000 nan 2.06328 0.72430 -1.66775
1 0.10000 2.06328 0.29482 1.59349 0.46593 -1.07283
2 0.10000 1.59349 0.18727 1.34214 0.29427 -0.67758
3 0.10000 1.34214 0.11716 1.20139 0.18348 -0.42247
4 0.10000 1.20139 0.07257 1.12010 0.11342 -0.26116
5 0.10000 1.12010 0.04466 1.07221 0.06973 -0.16055
6 0.10000 1.07221 0.02738 1.04364 0.04271 -0.09835
7 0.10000 1.04364 0.01674 1.02645 0.02611 -0.06012
8 0.10000 1.02645 0.01022 1.01607 0.01594 -0.03670
9 0.10000 1.01607 0.00624 1.00977 0.00972 -0.02238
10 0.10000 1.00977 0.00380 1.00594 0.00593 -0.01365
11 0.10000 1.00594 0.00232 1.00362 0.00361 -0.00832
12 0.10000 1.00362 0.00141 1.00220 0.00220 -0.00507
13 0.10000 1.00220 0.00086 1.00134 0.00134 -0.00309
14 0.10000 1.00134 0.00052 1.00082 0.00082 -0.00188
15 0.10000 1.00082 0.00032 1.00050 0.00050 -0.00115
16 0.10000 1.00050 0.00019 1.00030 0.00030 -0.00070
17 0.10000 1.00030 0.00012 1.00018 0.00018 -0.00043
18 0.10000 1.00018 0.00007 1.00011 0.00011 -0.00026
19 0.10000 1.00011 0.00004 1.00007 0.00007 -0.00016
(b) f(x) = ex, intervalo [0, 5]
Solução: Analisando o gráfico da função, podemos constatar que a mesma não possui raizes.
(c) f(x) = cos(x), intervalo [0.5, 4]
Solução:
k a b � x(k) f(x(k)) f(a)× f(x(k))
0 0.50000 4.00000 nan 2.50593 -0.80468 -0.70617
1 0.50000 2.50593 0.62046 1.54643 0.02436 +0.02138
2 1.54643 2.50593 0.01791 1.57463 -0.00383 -0.00009
3 1.54643 1.57463 0.00244 1.57080 0.00000 +0.00000
4 1.57080 1.57463 0.00000 1.57080 -0.00000 -0.00000
1/4
(d) f(x) = x2 + 2x+ 1, intervalo [−3, 0.5]
Solução:
k a b � x(k) f(x(k)) f(a)× f(x(k))
0 -3.00000 0.50000 nan 5.00000 36.00000 +144.00000
1 5.00000 0.50000 4.80000 0.20000 1.44000 +51.84000
2 0.20000 0.50000 0.53333 -0.33333 0.44444 +0.64000
3 -0.33333 0.50000 0.20513 -0.53846 0.21302 +0.09467
4 -0.53846 0.50000 0.10860 -0.64706 0.12457 +0.02654
5 -0.64706 0.50000 0.06723 -0.71429 0.08163 +0.01017
6 -0.71429 0.50000 0.04571 -0.76000 0.05760 +0.00470
7 -0.76000 0.50000 0.03310 -0.79310 0.04281 +0.00247
8 -0.79310 0.50000 0.02508 -0.81818 0.03306 +0.00142
9 -0.81818 0.50000 0.01966 -0.83784 0.02630 +0.00087
10 -0.83784 0.50000 0.01582 -0.85366 0.02142 +0.00056
11 -0.85366 0.50000 0.01301 -0.86667 0.01778 +0.00038
12 -0.86667 0.50000 0.01088 -0.87755 0.01499 +0.00027
13 -0.87755 0.50000 0.00924 -0.88679 0.01282 +0.00019
14 -0.88679 0.50000 0.00794 -0.89474 0.01108 +0.00014
15 -0.89474 0.50000 0.00690 -0.90164 0.00967 +0.00011
16 -0.90164 0.50000 0.00605 -0.90769 0.00852 +0.00008
17 -0.90769 0.50000 0.00535 -0.91304 0.00756 +0.00006
18 -0.91304 0.50000 0.00476 -0.91781 0.00676 +0.00005
19 -0.91781 0.50000 0.00427 -0.92208 0.00607 +0.00004
(e) f(x) = e(2−x2) × (x+ 1)2, intervalo [−3,−0.5]
Solução:
k a b � x(k) f(x(k)) f(a)× f(x(k))
0 -3.00000 -0.50000 nan -3.00635 0.00353 +0.00001
1 -3.00635 -0.50000 0.00617 -3.01253 0.00343 +0.00001
2 -3.01253 -0.50000 0.00600 -3.01852 0.00332 +0.00001
3 -3.01852 -0.50000 0.00583 -3.02435 0.00323 +0.00001
4 -3.02435 -0.50000 0.00567 -3.03003 0.00314 +0.00001
5 -3.03003 -0.50000 0.00553 -3.03555 0.00305 +0.00001
6 -3.03555 -0.50000 0.00538 -3.04094 0.00297 +0.00001
7 -3.04094 -0.50000 0.00525 -3.04619 0.00289 +0.00001
8 -3.04619 -0.50000 0.00512 -3.05131 0.00281 +0.00001
9 -3.05131 -0.50000 0.00500 -3.05631 0.00274 +0.00001
10 -3.05631 -0.50000 0.00488 -3.06119 0.00267 +0.00001
11 -3.06119 -0.50000 0.00477 -3.06596 0.00261 +0.00001
12 -3.06596 -0.50000 0.00466 -3.07062 0.00255 +0.00001
13 -3.07062 -0.50000 0.00456 -3.07518 0.00249 +0.00001
14 -3.07518 -0.50000 0.00446 -3.07964 0.00243 +0.00001
15 -3.07964 -0.50000 0.00436 -3.08400 0.00238 +0.00001
16 -3.08400 -0.50000 0.00427 -3.08828 0.00232 +0.00001
17 -3.08828 -0.50000 0.00419 -3.09246 0.00227 +0.00001
18 -3.09246 -0.50000 0.00410 -3.09656 0.00222 +0.00001
19 -3.09656 -0.50000 0.00402 -3.10059 0.00218 +0.00000
2/4
(f) f(x) = x3 + 3x− 1, intervalo [−8, 7]
Solução:
k a b � x(k) f(x(k)) f(a)× f(x(k))
0 -8.00000 7.00000 nan 0.95000 2.70737 -1453.86037
1 -8.00000 0.95000 0.04490 0.90510 2.45678 -1319.29200
2 -8.00000 0.90510 0.04056 0.86455 2.23984 -1202.79656
3 -8.00000 0.86455 0.03682 0.82773 2.05028 -1101.00284
4 -8.00000 0.82773 0.03358 0.79415 1.88330 -1011.33443
5 -8.00000 0.79415 0.03073 0.76342 1.73517 -931.78841
6 -8.00000 0.76342 0.02823 0.73519 1.60295 -860.78404
7 -8.00000 0.73519 0.02600 0.70919 1.48428 -797.05681
8 -8.00000 0.70919 0.02401 0.68519 1.37725 -739.58293
9 -8.00000 0.68519 0.02222 0.66297 1.28031 -687.52442
10 -8.00000 0.66297 0.02060 0.64237 1.19216 -640.18854
11 -8.00000 0.64237 0.01914 0.62322 1.11173 -596.99741
12 -8.00000 0.62322 0.01782 0.60541 1.03811 -557.46499
13 -8.00000 0.60541 0.01660 0.58880 0.97054 -521.17933
14 -8.00000 0.58880 0.01549 0.57331 0.90836 -487.78879
15 -8.00000 0.57331 0.01448 0.55883 0.85101 -456.99127
16 -8.00000 0.55883 0.01354 0.54529 0.79800 -428.52567
17 -8.00000 0.54529 0.01268 0.53261 0.74891 -402.16501
18 -8.00000 0.53261 0.01188 0.52073 0.70337 -377.71093
19 -8.00000 0.52073 0.01115 0.50958 0.66106 -354.98929
(g) f(x) = x2 + sin(x), intervalo [−4, 3]
Solução:
k a b � x(k) f(x(k)) f(a)× f(x(k))
0 -4.00000 3.00000 nan 11.40211 129.08973 +2163.13115
1 11.40211 3.00000 3.83205 2.35969 6.27275 +809.74797
2 2.35969 3.00000 1.40028 0.95940 1.73930 +10.91020
3 0.95940 3.00000 0.47951 0.47990 0.69199 +1.20358
4 0.47990 3.00000 0.20640 0.27350 0.34490 +0.23867
5 0.27350 3.00000 0.10691 0.16659 0.19357 +0.06676
6 0.16659 3.00000 0.06130 0.10529 0.11618 +0.02249
7 0.10529 3.00000 0.03727 0.06803 0.07260 +0.00844
8 0.06803 3.00000 0.02347 0.04455 0.04652 +0.00338
9 0.04455 3.00000 0.01512 0.02943 0.03030 +0.00141
10 0.02943 3.00000 0.00988 0.01956 0.01994 +0.00060
11 0.01956 3.00000 0.00651 0.01304 0.01321 +0.00026
12 0.01304 3.00000 0.00432 0.00872 0.00879 +0.00012
13 0.00872 3.00000 0.00288 0.00584 0.00587 +0.00005
14 0.00584 3.00000 0.00192 0.00391 0.00393 +0.00002
15 0.00391 3.00000 0.00129 0.00263 0.00263 +0.00001
16 0.00263 3.00000 0.00086 0.00176 0.00176 +0.00000
17 0.00176 3.00000 0.00058 0.00118 0.00118 +0.00000
18 0.00118 3.00000 0.00039 0.00079 0.00079 +0.00000
19 0.00079 3.00000 0.00026 0.00053 0.00053 +0.00000
3/4
(h) f(x) = ex − x2 − 10, intervalo [−8, 6]
Solução:
k a b � x(k) f(x(k)) f(a)× f(x(k))
0 -8.00000 6.00000 nan -5.59869 -41.34158 +3059.26301
1 -5.59869 6.00000 1.20247 -4.39622 -29.31442 +1211.90444
2 -4.39622 6.00000 0.78801 -3.60821 -22.99204 +673.99844
3 -3.60821 6.00000 0.58070 -3.02750 -19.11732 +439.54627
4 -3.02750 6.00000 0.45833 -2.56917 -16.52405 +315.89550
5 -2.56917 6.00000 0.37865 -2.19052 -14.68653 +242.68086
6 -2.19052 6.00000 0.32326 -1.86726 -13.33212 +195.80250
7 -1.86726 6.00000 0.28290 -1.58436 -12.30513 +164.05343
8 -1.58436 6.00000 0.25242 -1.33195 -11.51012 +141.63356
9 -1.33195 6.00000 0.22874 -1.10321 -10.88526 +125.29068
10 -1.10321 6.00000 0.20993 -0.89328 -10.38863 +113.08294
11 -0.89328 6.00000 0.19469 -0.69858 -9.99073 +103.78998
12 -0.69858 6.00000 0.18215 -0.51644 -9.67007 +96.61100
13 -0.51644 6.00000 0.17166 -0.34478 -9.41050 +91.00016
14 -0.34478 6.00000 0.16276 -0.18202 -9.19955 +86.57234
15 -0.18202 6.00000 0.15512 -0.02690 -9.02726 +83.04674
16 -0.02690 6.00000 0.14847 0.12157 -8.88551 +80.21188
17 0.12157 6.00000 0.14259 0.26416 -8.76745 +77.90326
18 0.26416 6.00000 0.13733 0.40148 -8.66715 +75.98876
19 0.40148 6.00000 0.13254 0.53403 -8.57940 +74.35891
(i) f(x) =
√
x− cos(x), intervalo [0.1, 9]
Solução:
k a b � x(k) f(x(k)) f(a)× f(x(k))
0 0.10000 9.00000 nan 1.41617 1.03602 -0.70323
1 0.10000 1.41617 0.79519 0.62099 -0.02528 +0.01716
2 0.62099 1.41617 0.01894 0.63993 -0.00219 +0.00006
3 0.63993 1.41617 0.00163 0.64156 -0.00019 +0.00000
40.64156 1.41617 0.00014 0.64170 -0.00002 +0.00000
5 0.64170 1.41617 0.00001 0.64171 -0.00000 +0.00000
6 0.64171 1.41617 0.00000 0.64171 -0.00000 +0.00000
(j) f(x) = x− 3− x−x, intervalo [0, 9]
Solução:
k a b � x(k) f(x(k)) f(a)× f(x(k))
0 0.00000 9.00000 nan 3.60000 0.59006 -2.36025
1 0.00000 3.60000 0.14752 3.13721 0.10953 -0.43811
2 0.00000 3.13721 0.02738 3.05360 0.02052 -0.08207
3 0.00000 3.05360 0.00513 3.03802 0.00383 -0.01531
4 0.00000 3.03802 0.00096 3.03511 0.00071 -0.00285
5 0.00000 3.03511 0.00018 3.03457 0.00013 -0.00053
6 0.00000 3.03457 0.00003 3.03447 0.00002 -0.00010
7 0.00000 3.03447 0.00001 3.03445 0.00000 -0.00002
4/4