gabarito lista de exercicos 1_ zeros de funcoes_2013

Prévia do material em texto

Ÿ 1)
aL 381928 = 3
10
+
8
102
+
1
103
106 = 0.381 ´ 106
RND ® 0.382 ´ 106
bL 78.457 = 7
10
+
8
102
+
4
103
102 = 0.784 ´ 102
RND ® 0.785 ´ 102
cL - 9142.683 = 9
10
+
1
102
+
4
103
104 = 0.914 ´ 104
Ÿ 2)Escreva os seguintes números que estão no sistema binário no sistema de base 10
aL
11 = 1 × 21 + 1 × 20 = 3 d
0.11 = 1 × 2-1 + 1 × 2-2 =
1
2
+
1
4
= 0.75 d
11.11 = 3.75 d
bL 0.1011 = 1 × 2-1 + 0 × 2-2 + 1 × 2-3 + 1 × 2-4
0.1011 = 0.6875 d
cL 1.0011 = 1 + 0 × 2-1 + 0 × 2-2 + 1 × 2-3 + 1 × 2-4
1.0011 = 1.1875 d
dL 110101 = 53
eL 0.111101101 = 1
2
+
1
4
+
1
8
+
1
16
+
1
64
+
1
128
+
1
512
= 0.962890625
Ÿ 3)Escreva os seguintes números que estão no sistema decimal no sistema de binário
aL 13.25 d = D + 1
4
h = 1101.01 b
bL 0.10125
0.10125 * 2 = 0.2025 ® 0
0.2025 * 2 = 0.4050 ® 0
0.4050 * 2 = 0.8100 ® 0
0.8100 * 2 = 1.6200 ® 1
0.6200 * 2 = 1.2400 ® 1
0.2400 * 2 = 0.4800 ® 0
0.4800 * 2 = 0.96 ® 0
0.96 * 2 = 1.92 ® 1
0.92 * 2 = 1.84 ® 1
0.84 * 2 = 1.68 ® 1
0.68 * 2 = 1.36 ® 1
0.36 * 2 = 0.72 ® 0
0.72 * 2 = 1.44 ® 1
…
= 0.00011001111010111000011b
dL 13 d = D h = 1101 b
eL 12.03135 = 1100.0000100000000111b
Ÿ 4)
Ÿ a) x = cosHxL
xn+1 = xn -
f HxnL
f ¢HxnL
f HxL = cosHxL - x
f ¢HxL = -sinHxL - 1
x0 = 0.5
0.5 1.0 1.5 2.0
-2.0
-1.5
-1.0
-0.5
0.5
1.0
n x xn+1 Èxn+1-xnÈ
1 0.5 0.7552224171 0.2552224171
2 0.7552224171 0.7391416661 0.01608075096
3 0.7391416661 0.7390851339 0.00005653222907
4 0.7390851339 0.7390851332 7.056460971´ 10-10
x = 0.73908
Ÿ b) 5 LogHxL - 2 + 0.4 x = 0
xn = 0.5;
2 gabarito 01g.nb
fnHxL = x -
0.4 x + 5 log10HxL - 2
5
x logH10L + 0.4
;
n x xn+1 ÈFunçãoÈ Èxn+1-xnÈ
1 0.5 1.196856089 1.131047896 0.6968560893
2 1.196856089 1.707645456 0.1549532842 0.5107893663
3 1.707645456 1.800342052 0.00308804962 0.09269659692
4 1.800342052 1.8022647 1.237384279´ 10-6 0.001922647615
5 1.8022647 1.802265471 1.983968545´ 10-13 7.710243213´ 10-7
6 1.802265471 1.802265471 1.110223025´ 10-16 1.236788449´ 10-13
x = 1.80226547
Ÿ c) e-x2 - CosHxL=0
2 4 6 8 10
-1.0
-0.5
0.5
1.0
f'@xD = -2 x ã-x2 + Sin@xD
fn@xD = xn -
ã-x
2
- Cos@xD
2 x ã-x2 + Sin@xD
fn2@x_D := x - ã-x
2
- Cos@xD
2 x ã-x2 + Sin@xD;
Um dos zeros está entre:
fB1
2
F = -0.098
f@2D = +0.67
gabarito 01g.nb 3
xn = 1.0;
n x xn+1 ÈFHxLÈ Èxn+1-xnÈ
1 1. 1.109320061 0.15315055 0.1093200606
2 1.109320061 1.20854264 0.1222798506 0.09922257901
3 1.20854264 1.290274358 0.08763025042 0.08173171837
4 1.290274358 1.350741518 0.05698526485 0.06046715953
5 1.350741518 1.391109964 0.03432372084 0.0403684464
6 1.391109964 1.415880902 0.0195983603 0.02477093815
7 1.415880902 1.430191939 0.01081944996 0.01431103697
8 1.430191939 1.438147169 0.00585587258 0.007955230094
9 1.438147169 1.44246951 0.003134388443 0.004322341085
10 1.44246951 1.444787957 0.001667549188 0.002318446691
11 1.444787957 1.446022812 0.0008842741638 0.001234854929
12 1.446022812 1.446678032 0.0004681004545 0.0006552199831
13 1.446678032 1.447024991 0.0002475652294 0.0003469595591
14 1.447024991 1.44720852 0.0001308662066 0.0001835284469
15 1.44720852 1.447305544 0.00006915965757 0.00009702426867
x = 1.44730554
Ÿ dL x3 - x - 5=0
0.5 1.0 1.5 2.0 2.5 3.0
-5
5
10
15
fn@xD = x - x3 - x - 5
3 x2 - 1
;
xn = 2.0;
n x xn+1 ÈFHxLÈ Èxn+1-xnÈ
1 2. 1.909090909 0.04883546206 0.09090909091
2 1.909090909 1.90417486 0.0001382952717 0.004916049009
3 1.90417486 1.904160859 1.11978693´ 10-9 0.00001400083339
x = 1.90416085
Ÿ 5) ãx - x2 + 4
x = -2.032531738 ± 0.00006104
Ÿ 6) 5
x ? 2.236099243
4 gabarito 01g.nb
x = 2.2360 ± 0.000076
Ÿ 8L 265
fn@xD = x - x5 - 26
5 x4
;
xn = 3.0;
n x xn+1 ÈFHxLÈ Èxn+1-xnÈ
1 3. 2.464197531 64.86101634 0.5358024691
2 2.464197531 2.112384701 16.05959316 0.3518128296
3 2.112384701 1.951070545 2.272542303 0.1613141562
4 1.951070545 1.919705199 0.07190142514 0.03136534594
5 1.919705199 1.918646362 0.00007927253425 0.001058837539
6 1.918646362 1.918645192 9.667999734´ 10-11 1.169965351´ 10-6
7 1.918645192 1.918645192 0. 1.426858631´ 10-12
8 1.918645192 1.918645192 0. 0.
x = 1.91864 ± 1.2 ´ 10-6
Obs: Na máquina, fazendo com precisao de 10 temos:
NB 265 , 10F
1.918645192
Ÿ 10) J x
2
M2 - SinHxL=0
In[16]:= fn@x_D := x - I
x
2
M2 - Sin@xD
x
2
- Cos@xD ;
In[23]:= xn =
2 - 1.5
2
+ 1.5
Out[23]= 1.75
In[24]:= Calculos = TableB:n, xnn = xn, xn = fn@xnD, AbsB xn
2
2
- Sin@xnDF, Abs@xnn - xnD>, 8n, 5<F
Out[24]= 981, 1.75, 1.957321875, 0.03155280944, 0.2073218748<,
82, 1.957321875, 1.934046551, 0.0003871027911, 0.02327532417<,
93, 1.934046551, 1.933753809, 6.147857079´ 10-8, 0.0002927412905=,
94, 1.933753809, 1.933753763, 1.33226763´ 10-15, 4.65071166´ 10-8=,
95, 1.933753763, 1.933753763, 1.110223025´ 10-16, 1.110223025´ 10-15==
In[25]:= Insert@Calculos, 8"n", "x", "xn+1", "ÈFHxLÈ", "ÈErroÈ"<, 1D  TableForm
Out[25]//TableForm=
n x xn+1 ÈFHxLÈ ÈErroÈ
1 1.75 1.957321875 0.03155280944 0.2073218748
2 1.957321875 1.934046551 0.0003871027911 0.02327532417
3 1.934046551 1.933753809 6.147857079´ 10-8 0.0002927412905
4 1.933753809 1.933753763 1.33226763´ 10-15 4.65071166´ 10-8
5 1.933753763 1.933753763 1.110223025´ 10-16 1.110223025´ 10-15
gabarito 01g.nb 5
x = 1.933754
Método Bissecção:
Out[33]=
1 1. 2. 3. 0.09070257317 1.
2 1. 1.5 2. -0.4349949866 0.5
3 1.5 1.75 2. -0.2183609469 0.25
4 1.75 1.875 2. -0.07517953161 0.125
5 1.875 1.9375 2. 0.004962281638 0.0625
6 1.875 1.90625 1.9375 -0.03581379306 0.03125
7 1.90625 1.921875 1.9375 -0.01560141284 0.015625
8 1.921875 1.9296875 1.9375 -0.005363397452 0.0078125
9 1.9296875 1.93359375 1.9375 -0.000211505375 0.00390625
10 1.93359375 1.935546875 1.9375 0.002372652588 0.001953125
11 1.93359375 1.934570313 1.935546875 0.001079889555 0.0009765625
12 1.93359375 1.934082031 1.934570313 0.0004340210562 0.00048828125
13 1.93359375 1.933837891 1.934082031 0.0001112150796 0.000244140625
14 1.93359375 1.93371582 1.933837891 -0.00005015583826 0.0001220703125
15 1.93371582 1.933776855 1.933837891 0.00003052694807 0.00006103515625
16 1.93371582 1.933746338 1.933776855 -9.815113251´ 10-6 0.00003051757813
17 1.933746338 1.933761597 1.933776855 0.00001035575037 0.00001525878906
18 1.933746338 1.933753967 1.933761597 2.702768014´ 10-7 7.629394531´ 10-6
x = 1.9337
Ÿ 7) 53
x3 - 5 = 0
xn+1 = xn -
xn - xn-1
f HxnL - f Hxn - 1L
f HxnL
um maximo: xn = 2 ×2 ×2 = 8
um minimo: xn-1 = 1.5 ×1.5 ×1.5 =3.375
981.675675676, -0.2948887529<, 81.70470233, -0.04611792773<,
81.710083383, 0.0009424974791<, 91.709975615, -2.912370056´ 10-6=,
91.709975947, -1.829745244´ 10-10=, 91.709975947, 8.881784197´ 10-16==
x > 1.70996
obs :
NB 53 F
1.709975947
6 gabarito 01g.nb
Ÿ 9) x3 - 2 x2 + 2 x - 5= 0
x0 = 2; x-1 = -2
Out[59]=
xn xn+1 erro
-2. 0.3421052632 6.846153846
0.3421052632 0.4578018321 0.252721944
0.4578018321 1.224238104 0.6260516393
1.224238104 0.6488285049 0.886843896
0.6488285049 0.7408028831 0.1241549948
0.7408028831 0.8187751087 0.09523033225
0.8187751087 0.8033058267 0.01925702694
0.8033058267 0.8044336475 0.001402005956
0.8044336475 0.8044534159 0.00002457364387
0.8044534159 0.8044533884 3.419458367´ 10-8
0.8044533884 0.8044533884 8.188110462´ 10-13
x > 0.8044533
Ÿ 12) Ln(x) - x + 2=0 Ε [3,4]
In[60]:= xn@nD = 3; xn@n - 1D = 4;
Out[68]=
xn xn+1 erro
3 3.138438589 0.04411065722
3.138438589 3.146281039 0.002492609421
3.146281039 3.14619317 0.00002792850659
3.14619317 3.146193221 1.606294995´ 10-8
3.146193221 3.146193221 1.044519495´ 10-13
x > 3.14619
Ÿ 13)
x2 - 3 x + ãx = 2
-2 -1 1 2
2
4
6
8
In[160]:= xn@nD = 2; xn@n - 1D = 1;
Método dassecantes :
gabarito 01g.nb 7
Out[167]=
xn xn+1 FHxL erro
2 1.274412356 3.389056099 0.5693507605
1.274412356 1.387008355 -0.6225111896 0.08117903461
1.387008355 1.454999563 -0.2343758921 0.04672936665
1.454999563 1.445836439 0.03650663372 0.00633759365
1.445836439 1.446236039 -0.00166462923 0.0002763028493
1.446236039 1.446238687 -0.00001095912569 1.831098053´ 10-6
x = 1.446238
Método de Newton:
In[176]:= xn = 1.00;
Out[178]=
n xn xn+1 FHxL erro
0 1. 1.745930121 1.541711356 0.7459301206
1 1.745930121 1.498189631 0.2235861771 0.2477404895
2 1.498189631 1.448169929 0.008006202531 0.0500197024
3 1.448169929 1.446241495 0.00001162624072 0.001928434193
4 1.446241495 1.446238686 2.463851345´ 10-11 2.808538916´ 10-6
5 1.446238686 1.446238686 8.881784197´ 10-16 5.951905635´ 10-12
x = 1.446238
Metodo bissecção
In[195]:= a = 1.0; b = 2.0;
Out[197]=
n a x b FHxL Ε
0 1. 1.5 2. 0.2316890703 0.25
1 1. 1.25 1.5 -0.6971570425 0.125
2 1.25 1.375 1.5 -0.2792982771 0.0625
3 1.375 1.4375 1.5 -0.03593649386 0.03125
4 1.4375 1.46875 1.5 0.09477855606 0.015625
5 1.4375 1.453125 1.46875 0.02865485134 0.0078125
6 1.4375 1.4453125 1.453125 -0.003831348585 0.00390625
7 1.4453125 1.44921875 1.453125 0.01236399297 0.001953125
8 1.4453125 1.447265625 1.44921875 0.004254398448 0.0009765625
9 1.4453125 1.446289063 1.447265625 0.0002085459752 0.00048828125
10 1.4453125 1.445800781 1.446289063 -0.001812145797 0.000244140625
x = 1.446238
Método ponto fixo:
8 gabarito 01g.nb
Θ@xD = -2 + ãx + x2
3
-2 -1 1 2 3
-2
2
4
6
8
xn = 1.5;
TableA9n, xnn = xn, xn = fn@xnD, NA xn2 - 3 xn + ãxn - 2E, Abs@xn - xnnD=, 8n, 5<E  TableForm
1 1.5 1.57722969 0.597489117 0.07722969011
2 1.57722969 1.776392729 1.734897295 0.199163039
3 1.776392729 2.354691828 7.015379921 0.5782990984
4 2.354691828 4.693151801 115.143023 2.338459974
5 4.693151801 43.07415948 5.091781695´ 1018 38.38100768
Diverge, pois:
¶ J 1
3
Ix2 + ãx - 2MN
¶ x
> 1
Para x Ε (1,2);
Ÿ 14)
 
In[76]:= R = 140; L = 260 ´ 10-3; c = 25; Vm = 24; im = 0.15;
O método da sacante precisa de dois valores iniciais, uma forma de obter-los é observar a resposta em frequência do circuito:
Primeiro escrevemos o circuito no dominio da frequência:
Vm =
1
s c
+ R + s L im \ im =
Vm
J 1
s c
+ R + s LN
onde s = ü Ω
gabarito 01g.nb 9
In[157]:= LogLinearPlotBAbsB VmJ 1
ü Ω c
+ R + ü Ω LNF, 8Ω, 1, 1000<, GridLines ® AutomaticF
Out[157]=
5 10 50 100 500 1000
0.12
0.13
0.14
0.15
0.16
0.17
Podemos fazer x-1 = 10 e x0 = 100
im Š
Vm
R2 + JH2 Π fL L - 1H2 Π fL cN
2
im R2 + H2 Π fL L - 1H2 Π fL c
2
2
Š Vm2
A função fica: (lembre que frequencia negativa não é válido fisicamente, apenas pode-se considerá-la quando levar em
consideracao um atraso de fase de -Π)
im2 R2 + H2 Π fL L - 1H2 Π fL c - Vm
2
Š 0
Out[155]=
xn xn+1 F@xD erro
100 29.52962646 465.4662638 2.386429563
29.52962646 40.1546866 -82.63983873 0.2646032392
40.1546866 49.27955311 -38.18128086 0.1851653664
49.27955311 47.26449216 10.82133788 0.04263371635
47.26449216 47.41288755 -0.8602701653 0.003129853523
47.41288755 47.41581332 -0.01663320285 0.00006170459398
47.41581332 47.41580865 0.00002658455128 9.846398778´ 10-8
f = 47.4158086 Hz
Ÿ 15)
vo = 15.2; x1 = 18.2; h = 1.82; y = 2.1; g = 9.0;
10 gabarito 01g.nb
-
g x2
2 Iv02 cos2HΘLM
+ h + x tanHΘL - y‡ 0
Tabela de valores a função:
0.01 -6.550249248
0.3241592654 -1.344607006
0.6383185307 3.220192005
0.9524777961 6.103798684
1.266637061 -14.22800319
a = 0.3; b = 0.64;
1 0.3 0.47 0.64 0.8486644367 0.17
2 0.3 0.385 0.47 -0.4159551941 0.085
3 0.385 0.4275 0.47 0.2211237631 0.0425
4 0.385 0.40625 0.4275 -0.09623513818 0.02125
5 0.40625 0.416875 0.4275 0.06273976922 0.010625
6 0.40625 0.4115625 0.416875 -0.01667392707 0.0053125
7 0.4115625 0.41421875 0.416875 0.02305136942 0.00265625
8 0.4115625 0.412890625 0.41421875 0.003193331869 0.001328125
9 0.4115625 0.4122265625 0.412890625 -0.006739145077 0.0006640625
10 0.4122265625 0.4125585938 0.412890625 -0.001772618456 0.00033203125
11 0.4125585938 0.4127246094 0.412890625 0.0007104287462 0.000166015625
12 0.4125585938 0.4126416016 0.4127246094 -0.0005310768451 0.0000830078125
13 0.4126416016 0.4126831055 0.4127246094 0.00008968045299 0.00004150390625
14 0.4126416016 0.4126623535 0.4126831055 -0.0002206970705 0.00002075195313
15 0.4126623535 0.4126727295 0.4126831055 -0.00006550802733 0.00001037597656
16 0.4126727295 0.4126779175 0.4126831055 0.00001208628318 5.187988281´ 10-6
17 0.4126727295 0.4126753235 0.4126779175 -0.00002671085449 2.593994141´ 10-6
18 0.4126753235 0.4126766205 0.4126779175 -7.312281258´ 10-6 1.29699707´ 10-6
19 0.4126766205 0.412677269 0.4126779175 2.387002058´ 10-6 6.484985352´ 10-7
20 0.4126766205 0.4126769447 0.412677269 -2.462639326´ 10-6 3.242492676´ 10-7
21 0.4126769447 0.4126771069 0.412677269 -3.781856406´ 10-8 1.621246338´ 10-7
22 0.4126771069 0.4126771879 0.412677269 1.174591764´ 10-6 8.106231691´ 10-8
23 0.4126771069 0.4126771474 0.4126771879 5.683866038´ 10-7 4.053115846´ 10-8
24 0.4126771069 0.4126771271 0.4126771474 2.65284019´ 10-7 2.026557921´ 10-8
25 0.4126771069 0.412677117 0.4126771271 1.137327292´ 10-7 1.013278961´ 10-8
26 0.4126771069 0.4126771119 0.412677117 3.795708126´ 10-8 5.066394804´ 10-9
Θ = 0.4126771119 rad
gabarito 01g.nb 11