1)

aL 381928 = 310

+8

102+

1

103106 = 0.381 106

RND 0.382 106

bL 78.457 = 710

+8

102+

4

103102 = 0.784 102

RND 0.785 102

cL - 9142.683 = 910

+1

102+

4

103104 = 0.914 104

2)Escreva os seguintes nmeros que esto no sistema binrio no sistema de base 10aL11 = 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 2-40.1011 = 0.6875 d

cL 1.0011 = 1 + 0 2-1 + 0 2-2 + 1 2-3 + 1 2-41.0011 = 1.1875 d

dL 110101 = 53

eL 0.111101101 = 12+1

4+1

8+

1

16+

1

64+

1

128+

1

512= 0.962890625

3)Escreva os seguintes nmeros que esto no sistema decimal no sistema de binrio

aL 13.25 d = D + 14

h = 1101.01 b

bL 0.101250.10125 * 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 beL 12.03135 = 1100.0000100000000111b

4)

a) x = cosHxL

xn+1 = xn -f HxnLf 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 xn+1-xn1 0.5 0.7552224171 0.25522241712 0.7552224171 0.7391416661 0.016080750963 0.7391416661 0.7390851339 0.000056532229074 0.7390851339 0.7390851332 7.056460971 10-10

x = 0.73908

b) 5 LogHxL - 2 + 0.4 x = 0

xn = 0.5;

2 gabarito 01g.nb

fnHxL = x -0.4 x + 5 log10HxL - 2

5x logH10L + 0.4

;

n x xn+1 Funo xn+1-xn1 0.5 1.196856089 1.131047896 0.69685608932 1.196856089 1.707645456 0.1549532842 0.51078936633 1.707645456 1.800342052 0.00308804962 0.092696596924 1.800342052 1.8022647 1.237384279 10-6 0.0019226476155 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

c) e-x2 - CosHxL=0

2 4 6 8 10

-1.0

-0.5

0.5

1.0

f'@xD = -2 x -x2 + Sin@xDfn@xD = xn -

-x2- Cos@xD

2 x -x2 + Sin@xDfn2@x_D := x - -x

2- Cos@xD

2 x -x2 + Sin@xD;

Um dos zeros est entre:

fB12F = -0.098

f@2D = +0.67

gabarito 01g.nb 3

xn = 1.0;

n x xn+1 FHxL xn+1-xn1 1. 1.109320061 0.15315055 0.10932006062 1.109320061 1.20854264 0.1222798506 0.099222579013 1.20854264 1.290274358 0.08763025042 0.081731718374 1.290274358 1.350741518 0.05698526485 0.060467159535 1.350741518 1.391109964 0.03432372084 0.04036844646 1.391109964 1.415880902 0.0195983603 0.024770938157 1.415880902 1.430191939 0.01081944996 0.014311036978 1.430191939 1.438147169 0.00585587258 0.0079552300949 1.438147169 1.44246951 0.003134388443 0.00432234108510 1.44246951 1.444787957 0.001667549188 0.00231844669111 1.444787957 1.446022812 0.0008842741638 0.00123485492912 1.446022812 1.446678032 0.0004681004545 0.000655219983113 1.446678032 1.447024991 0.0002475652294 0.000346959559114 1.447024991 1.44720852 0.0001308662066 0.000183528446915 1.44720852 1.447305544 0.00006915965757 0.00009702426867

x = 1.44730554

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 - 53 x2 - 1

;

xn = 2.0;

n x xn+1 FHxL xn+1-xn1 2. 1.909090909 0.04883546206 0.090909090912 1.909090909 1.90417486 0.0001382952717 0.0049160490093 1.90417486 1.904160859 1.11978693 10-9 0.00001400083339

x = 1.90416085

5) x - x2 + 4x = -2.032531738 0.00006104

6) 5x ? 2.236099243

4 gabarito 01g.nb

x = 2.2360 0.000076

8L 265

fn@xD = x - x5 - 265 x4

;

xn = 3.0;

n x xn+1 FHxL xn+1-xn1 3. 2.464197531 64.86101634 0.53580246912 2.464197531 2.112384701 16.05959316 0.35181282963 2.112384701 1.951070545 2.272542303 0.16131415624 1.951070545 1.919705199 0.07190142514 0.031365345945 1.919705199 1.918646362 0.00007927253425 0.0010588375396 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 mquina, fazendo com precisao de 10 temos:

NB 265 , 10F1.918645192

10) J x2M2 - SinHxL=0

In[16]:= fn@x_D := x - Ix

2M2 - 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 xn2

2

- Sin@xnDF, Abs@xnn - xnD>, 8n, 5

x = 1.933754

Mtodo Bisseco:

Out[33]=

1 1. 2. 3. 0.09070257317 1.2 1. 1.5 2. -0.4349949866 0.53 1.5 1.75 2. -0.2183609469 0.254 1.75 1.875 2. -0.07517953161 0.1255 1.875 1.9375 2. 0.004962281638 0.06256 1.875 1.90625 1.9375 -0.03581379306 0.031257 1.90625 1.921875 1.9375 -0.01560141284 0.0156258 1.921875 1.9296875 1.9375 -0.005363397452 0.00781259 1.9296875 1.93359375 1.9375 -0.000211505375 0.0039062510 1.93359375 1.935546875 1.9375 0.002372652588 0.00195312511 1.93359375 1.934570313 1.935546875 0.001079889555 0.000976562512 1.93359375 1.934082031 1.934570313 0.0004340210562 0.0004882812513 1.93359375 1.933837891 1.934082031 0.0001112150796 0.00024414062514 1.93359375 1.93371582 1.933837891 -0.00005015583826 0.000122070312515 1.93371582 1.933776855 1.933837891 0.00003052694807 0.0000610351562516 1.93371582 1.933746338 1.933776855 -9.815113251 10-6 0.0000305175781317 1.933746338 1.933761597 1.933776855 0.00001035575037 0.0000152587890618 1.933746338 1.933753967 1.933761597 2.702768014 10-7 7.629394531 10-6

x = 1.9337

7) 53

x3 - 5 = 0

xn+1 = xn -xn - xn-1

f HxnL - f Hxn - 1Lf HxnL

um maximo: xn = 2 2 2 = 8um minimo: xn-1 = 1.5 1.5 1.5 =3.375

981.675675676, -0.2948887529

9) x3 - 2 x2 + 2 x - 5= 0x0 = 2; x-1 = -2

Out[59]=

xn xn+1 erro

-2. 0.3421052632 6.8461538460.3421052632 0.4578018321 0.2527219440.4578018321 1.224238104 0.62605163931.224238104 0.6488285049 0.8868438960.6488285049 0.7408028831 0.12415499480.7408028831 0.8187751087 0.095230332250.8187751087 0.8033058267 0.019257026940.8033058267 0.8044336475 0.0014020059560.8044336475 0.8044534159 0.000024573643870.8044534159 0.8044533884 3.419458367 10-8

0.8044533884 0.8044533884 8.188110462 10-13

x > 0.8044533

12) Ln(x) - x + 2=0 [3,4]In[60]:= xn@nD = 3; xn@n - 1D = 4;

Out[68]=

xn xn+1 erro

3 3.138438589 0.044110657223.138438589 3.146281039 0.0024926094213.146281039 3.14619317 0.000027928506593.14619317 3.146193221 1.606294995 10-8

3.146193221 3.146193221 1.044519495 10-13

x > 3.14619

13)

x2 - 3 x + x = 2

-2 -1 1 2

2

4

6

8

In[160]:= xn@nD = 2; xn@n - 1D = 1;Mtodo das secantes :

gabarito 01g.nb 7

Out[167]=

xn xn+1 FHxL erro2 1.274412356 3.389056099 0.5693507605

1.274412356 1.387008355 -0.6225111896 0.081179034611.387008355 1.454999563 -0.2343758921 0.046729366651.454999563 1.445836439 0.03650663372 0.006337593651.445836439 1.446236039 -0.00166462923 0.00027630284931.446236039 1.446238687 -0.00001095912569 1.831098053 10-6

x = 1.446238

Mtodo de Newton:

In[176]:= xn = 1.00;

Out[178]=

n xn xn+1 FHxL erro0 1. 1.745930121 1.541711356 0.74593012061 1.745930121 1.498189631 0.2235861771 0.24774048952 1.498189631 1.448169929 0.008006202531 0.05001970243 1.448169929 1.446241495 0.00001162624072 0.0019284341934 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 bisseco

In[195]:= a = 1.0; b = 2.0;

Out[197]=

n a x b FHxL 0 1. 1.5 2. 0.2316890703 0.251 1. 1.25 1.5 -0.6971570425 0.1252 1.25 1.375 1.5 -0.2792982771 0.06253 1.375 1.4375 1.5 -0.03593649386 0.031254 1.4375 1.46875 1.5 0.09477855606 0.0156255 1.4375 1.453125 1.46875 0.02865485134 0.00781256 1.4375 1.4453125 1.453125 -0.003831348585 0.003906257 1.4453125 1.44921875 1.453125 0.01236399297 0.0019531258 1.4453125 1.447265625 1.44921875 0.004254398448 0.00097656259 1.4453125 1.446289063 1.447265625 0.0002085459752 0.0004882812510 1.4453125 1.445800781 1.446289063 -0.001812145797 0.000244140625

x = 1.446238

Mtodo ponto fixo:

8 gabarito 01g.nb

@xD = -2 + x + x23

-2 -1 1 2 3

-2

2

4

6

8

xn = 1.5;

TableA9n, xnn = xn, xn = fn@xnD, NA xn2 - 3 xn + xn - 2E, Abs@xn - xnnD=, 8n, 5 1

Para x (1,2);

14)

In[76]:= R = 140; L = 260 10-3; c = 25; Vm = 24; im = 0.15;

O mtodo da sacante precisa de dois valores iniciais, uma forma de obter-los observar a resposta em frequncia do circuito:

Primeiro escrevemos o circuito no dominio da frequncia:

Vm =1

s c+ R + s L im \ im =

Vm

J 1s c

+ R + s LNonde s =

gabarito 01g.nb 9

In[157]:= LogLinearPlotBAbsB VmJ 1 c

+ R + LNF, 8, 1, 1000

-g x2

2 Iv02 cos2HLM+ h + x tanHL - y 0

Tabela de valores a funo:

0.01 -6.5502492480.3241592654 -1.3446070060.6383185307 3.2201920050.9524777961 6.1037986841.266637061 -14.22800319

a = 0.3; b = 0.64;

1 0.3 0.47 0.64 0.8486644367 0.172 0.3 0.385 0.47 -0.4159551941 0.0853 0.385 0.4275 0.47 0.2211237631 0.04254 0.385 0.40625 0.4275 -0.09623513818 0.021255 0.40625 0.416875 0.4275 0.06273976922 0.0106256 0.40625 0.4115625 0.416875 -0.01667392707 0.00531257 0.4115625 0.41421875 0.416875 0.02305136942 0.002656258 0.4115625 0.412890625 0.41421875 0.003193331869 0.0013281259 0.4115625 0.4122265625 0.412890625 -0.006739145077 0.000664062510 0.4122265625 0.4125585938 0.412890625 -0.001772618456 0.0003320312511 0.4125585938 0.4127246094 0.412890625 0.0007104287462 0.00016601562512 0.4125585938 0.4126416016 0.4127246094 -0.0005310768451 0.000083007812513 0.4126416016 0.4126831055 0.4127246094 0.00008968045299 0.0000415039062514 0.4126416016 0.4126623535 0.4126831055 -0.0002206970705 0.0000207519531315 0.4126623535 0.4126727295 0.4126831055 -0.00006550802733 0.0000103759765616 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

= 0.4126771119 rad

gabarito 01g.nb 11