ejercicio 1 del método de bisección. la ecuación … · 7, 1.109375000, k5.079502106 8,...
TRANSCRIPT
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Ejercicio 1 del método de bisección. La ecuación estudiada por Leonardo de Pisa.
Es importante la terminología.
f d x/x3C2$ x2C10$xK20;f := x/x3C2 x2C10 xK20
f 1 ; K7
f 2 ;16
Métodos de Maple.
Hallamos las raíces.
fsolve f ;1.368808108
solve f x ;13
352C6 39301/3
K26
3 352C6 39301/3 K
23
, K16
352C6 39301/3
C13
3 352C6 39301/3
K23C
12
I 3 13
352C6 39301/3
C26
3 352C6 39301/3 , K
16
352C6 39301/3
C13
3 352C6 39301/3
K23K
12
I 3 13
352C6 39301/3
C26
3 352C6 39301/3
plot f x , x = 1 ..2 ;
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(6)(6)
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x1.2 1.4 1.6 1.8 2
K4
K3
K2
K1
0
1
2
3
a d 1; b d 2; n d 7; h dbKa
2nK1 ;
a := 1b := 2n := 7
h :=164
for i from 1 to 2nK1 do print i, evalf aCi$h , evalf f aC i$h ; end do;1, 1.015625000, K6.7331504822, 1.031250000, K6.4638366703, 1.046875000, K6.1920356754, 1.062500000, K5.9177246095, 1.078125000, K5.6408805856, 1.093750000, K5.361480713
7, 1.109375000, K5.0795021068, 1.125000000, K4.7949218759, 1.140625000, K4.50771713310, 1.156250000, K4.21786499011, 1.171875000, K3.92534256012, 1.187500000, K3.63012695313, 1.203125000, K3.33219528214, 1.218750000, K3.03152465815, 1.234375000, K2.72809219416, 1.250000000, K2.42187500017, 1.265625000, K2.11285018918, 1.281250000, K1.80099487319, 1.296875000, K1.48628616320, 1.312500000, K1.16870117221, 1.328125000, K0.848217010522, 1.343750000, K0.524810791023, 1.359375000, K0.198459625224, 1.375000000, 0.130859375025, 1.390625000, 0.463169097926, 1.406250000, 0.798492431627, 1.421875000, 1.13685226428, 1.437500000, 1.47827148429, 1.453125000, 1.82277298030, 1.468750000, 2.17037963931, 1.484375000, 2.52111434932, 1.500000000, 2.87500000033, 1.515625000, 3.23205947934, 1.531250000, 3.59231567435, 1.546875000, 3.95579147336, 1.562500000, 4.32250976637, 1.578125000, 4.69249343938, 1.593750000, 5.06576538139, 1.609375000, 5.44234848040, 1.625000000, 5.82226562541, 1.640625000, 6.20553970342, 1.656250000, 6.59219360443, 1.671875000, 6.98225021444, 1.687500000, 7.37573242245, 1.703125000, 7.77266311646, 1.718750000, 8.17306518647, 1.734375000, 8.57696151748, 1.750000000, 8.984375000
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(10)(10)
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(8)(8)
49, 1.765625000, 9.39532852250, 1.781250000, 9.80984497151, 1.796875000, 10.2279472452, 1.812500000, 10.6496582053, 1.828125000, 11.0750007654, 1.843750000, 11.5039978055, 1.859375000, 11.9366722156, 1.875000000, 12.3730468857, 1.890625000, 12.8131446858, 1.906250000, 13.2569885359, 1.921875000, 13.7046012960, 1.937500000, 14.1560058661, 1.953125000, 14.6112251362, 1.968750000, 15.0702819863, 1.984375000, 15.53319931
64, 2., 16.
for i from 1 to 2nK1 do m i d evalf aC i$h ; end do:m 23 ; m 24 ;
1.3593750001.375000000
k d 3; en d k$ ln 10
ln 2; evalf en ;
k := 3
en :=3 ln 10
ln 29.965784285
c7 = m 23 Cm 24
2;
c7 = 1.367187500
(1)(1)
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(3)(3)
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Ejercicio 2 del método de bisección. El intervalo es (0,1).
f d x/x5K3$ x2C1;f := x/x5K3 x2C1
f 0 ; 1
f 1 ;K1
Métodos de Maple.
Hallamos las raíces.
fsolve f ;K0.5610700072
solve f x ;RootOf _Z5K3 _Z2C1, index = 1 , RootOf _Z5K3 _Z2C1, index = 2 , RootOf _Z5K3 _Z2
C1, index = 3 , RootOf _Z5K3 _Z2C1, index = 4 , RootOf _Z5K3 _Z2C1, index = 5
plot f x , x = 0 ..1 ;
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(6)(6)
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x0.2 0.4 0.6 0.8 1
K1
K0.5
0
0.5
1
a d 0; b d 1; n d 7; h dbKa
2nK1 ;
a := 0b := 1n := 7
h :=164
for i from 1 to 2nK1 do print i, evalf aCi$h , evalf f aC i$h ; end do;1, 0.01562500000, 0.99926757912, 0.03125000000, 0.99707034233, 0.04687500000, 0.99340842944, 0.06250000000, 0.98828220375, 0.07812500000, 0.98169236356, 0.09375000000, 0.9736400545
7, 0.1093750000, 0.96412698098, 0.1250000000, 0.95315551769, 0.1406250000, 0.940728821810, 0.1562500000, 0.926850944811, 0.1718750000, 0.911526943612, 0.1875000000, 0.894762992913, 0.2031250000, 0.876566496714, 0.2187500000, 0.856946200115, 0.2343750000, 0.835912301216, 0.2500000000, 0.813476562517, 0.2656250000, 0.789652423018, 0.2812500000, 0.764455109819, 0.2968750000, 0.737901750020, 0.3125000000, 0.710011482221, 0.3281250000, 0.680805568622, 0.3437500000, 0.650307506323, 0.3593750000, 0.618543139724, 0.3750000000, 0.585540771525, 0.3906250000, 0.551331275126, 0.4062500000, 0.515948206227, 0.4218750000, 0.479427914128, 0.4375000000, 0.441809654229, 0.4531250000, 0.403135699230, 0.4687500000, 0.363451451131, 0.4843750000, 0.322805552732, 0.5000000000, 0.281250000033, 0.5156250000, 0.238840253134, 0.5312500000, 0.195635348635, 0.5468750000, 0.151698011036, 0.5625000000, 0.107094764737, 0.5781250000, 0.0618960456938, 0.5937500000, 0.01617631316
39, 0.6093750000, K0.0299858385740, 0.6250000000, K0.0765075683641, 0.6406250000, K0.123301676542, 0.6562500000, K0.170276492843, 0.6718750000, K0.217335765244, 0.6875000000, K0.264378547745, 0.7031250000, K0.311299088446, 0.7187500000, K0.357986718447, 0.7343750000, K0.404325739548, 0.7500000000, K0.4501953125
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49, 0.7656250000, K0.495469345750, 0.7812500000, K0.540016382951, 0.7968750000, K0.583699491852, 0.8125000000, K0.626376152053, 0.8281250000, K0.667898143654, 0.8437500000, K0.708111435255, 0.8593750000, K0.746856072056, 0.8750000000, K0.783966064557, 0.8906250000, K0.819269276258, 0.9062500000, K0.852587312559, 0.9218750000, K0.883735408160, 0.9375000000, K0.912522316061, 0.9531250000, K0.938750195362, 0.9687500000, K0.962214499763, 0.9843750000, K0.9827038655
64, 1., K1.
for i from 1 to 2nK1 do m i d evalf aC i$h ; end do:m 38 ; m 39 ;
0.59375000000.6093750000
k d 3; en d k$ ln 10
ln 2; evalf en ;
k := 3
en :=3 ln 10
ln 29.965784285
c7 d m 38 Cm 39
2;
c7 := 0.6015625000
f c7 ;K0.006854537
evalf % ;K0.006854537
evalf38
26;
0.5937500000
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(3)(3)
(1)(1)
(4)(4)
Ejercicio 3 del método de bisección. Se buscan raíces de 3x-e^x.
f d x/3$xKexp x ;f := x/3 xKex
f 0 ; K1
evalf f 1 ;0.281718172
evalf f 2 ;K1.389056099
plot f x , x = 0 ..2 ;
x0.5 1 1.5 2
K1.2
K1.0
K0.8
K0.6
K0.4
K0.2
0
0.2
a d 0; b d 1; n d 7; h dbKa
2nK1 ;
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a := 0b := 1n := 7
h :=164
for i from 1 to 2nK1 do print i, evalf aCi$h , evalf f aC i$h ; end do;1, 0.01562500000, K0.96887270902, 0.03125000000, K0.93799340703, 0.04687500000, K0.90736600204, 0.06250000000, K0.87699445905, 0.07812500000, K0.84688280706, 0.09375000000, K0.81703514007, 0.1093750000, K0.78745561508, 0.1250000000, K0.75814845309, 0.1406250000, K0.729117945010, 0.1562500000, K0.700368446011, 0.1718750000, K0.671904383012, 0.1875000000, K0.643730249013, 0.2031250000, K0.615850612014, 0.2187500000, K0.588270108015, 0.2343750000, K0.560993448016, 0.2500000000, K0.534025417017, 0.2656250000, K0.507370875018, 0.2812500000, K0.481034759019, 0.2968750000, K0.455022083020, 0.3125000000, K0.429337941021, 0.3281250000, K0.403987507022, 0.3437500000, K0.37897603523, 0.3593750000, K0.35430886424, 0.3750000000, K0.32999141525, 0.3906250000, K0.30602919526, 0.4062500000, K0.28242780027, 0.4218750000, K0.25919291128, 0.4375000000, K0.23633029929, 0.4531250000, K0.21384582730, 0.4687500000, K0.19174545031, 0.4843750000, K0.17003521732, 0.5000000000, K0.14872127133, 0.5156250000, K0.12780985334, 0.5312500000, K0.10730730235, 0.5468750000, K0.087220057
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36, 0.5625000000, K0.06755465737, 0.5781250000, K0.04831774638, 0.5937500000, K0.02951607239, 0.6093750000, K0.01115648940, 0.6250000000, 0.00675404341, 0.6406250000, 0.02420845042, 0.6562500000, 0.04119955043, 0.6718750000, 0.05772004744, 0.6875000000, 0.07376253045, 0.7031250000, 0.08931947246, 0.7187500000, 0.10438322747, 0.7343750000, 0.11894602748, 0.7500000000, 0.13299998349, 0.7656250000, 0.14653708450, 0.7812500000, 0.15954918951, 0.7968750000, 0.17202803152, 0.8125000000, 0.18396521353, 0.8281250000, 0.19535220454, 0.8437500000, 0.20618034055, 0.8593750000, 0.21644082056, 0.8750000000, 0.22612470657, 0.8906250000, 0.23522291758, 0.9062500000, 0.24372623059, 0.9218750000, 0.25162527760, 0.9375000000, 0.25891054261, 0.9531250000, 0.26557235962, 0.9687500000, 0.27160091163, 0.9843750000, 0.276986225
64, 1., 0.281718172
for i from 1 to 2nK1 do m i d evalf aC i$h ; end do:m 39 ; m 40 ;
0.60937500000.6250000000
k d 3; en d k$ ln 10
ln 2; evalf en ;
k := 3
en :=3 ln 10
ln 29.965784285
c7 d m 39 Cm 40
2;
c7 := 0.6171875000
f c7 ;
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(11)(11)
(10)(10)
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K0.002144652
evalf % ;K0.006854537
evalf38
26 ;
0.5937500000
Raíz en el intervalo [1,2].
a d 1; b d 2; n d 6; h dbKa
2n;
a := 1b := 2n := 6
h :=164
for i from 1 to 2n do print i, evalf aC i$h , evalf f aCi$h ; end do;1, 1.015625000, 0.2857864612, 1.031250000, 0.2891806443, 1.046875000, 0.2918901034, 1.062500000, 0.2939040565, 1.078125000, 0.2952115506, 1.093750000, 0.2958014617, 1.109375000, 0.2956624878, 1.125000000, 0.2947831519, 1.140625000, 0.29315179410, 1.156250000, 0.29075657211, 1.171875000, 0.28758545812, 1.187500000, 0.28362623213, 1.203125000, 0.27886648314, 1.218750000, 0.27329360615, 1.234375000, 0.26689479416, 1.250000000, 0.25965704317, 1.265625000, 0.25156713918, 1.281250000, 0.24261166419, 1.296875000, 0.23277698720, 1.312500000, 0.22204926221, 1.328125000, 0.21041442722, 1.343750000, 0.19785819523, 1.359375000, 0.18436605824, 1.375000000, 0.16992327725, 1.390625000, 0.154514881
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(15)(15)
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26, 1.406250000, 0.13812566527, 1.421875000, 0.12074018228, 1.437500000, 0.10234274429, 1.453125000, 0.08291741430, 1.468750000, 0.06244800631, 1.484375000, 0.04091807832, 1.500000000, 0.018310930
33, 1.515625000, K0.00539040434, 1.531250000, K0.03020315335, 1.546875000, K0.05614482036, 1.562500000, K0.08323318237, 1.578125000, K0.11148629838, 1.593750000, K0.14092250939, 1.609375000, K0.17156044840, 1.625000000, K0.20341903741, 1.640625000, K0.23651750042, 1.656250000, K0.27087536243, 1.671875000, K0.30651245544, 1.687500000, K0.34344892545, 1.703125000, K0.38170523346, 1.718750000, K0.42130216547, 1.734375000, K0.46226083148, 1.750000000, K0.50460267649, 1.765625000, K0.54834948250, 1.781250000, K0.59352337451, 1.796875000, K0.64014682552, 1.812500000, K0.68824266253, 1.828125000, K0.73783407254, 1.843750000, K0.78894460755, 1.859375000, K0.84159819056, 1.875000000, K0.89581912057, 1.890625000, K0.95163208058, 1.906250000, K1.00906213959, 1.921875000, K1.06813476460, 1.937500000, K1.12887582161, 1.953125000, K1.19131158562, 1.968750000, K1.25546874263, 1.984375000, K1.321374402
64, 2., K1.389056099
for i from 1 to 2n do m i d evalf aC i$h ; end do:
c7 dm 32 Cm 33
2;
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(15)(15)c7 := 1.507812500
Iteraciones para error menor o igual que 10^{-4}.
k d 4; en d k$ ln 10
ln 2; evalf en ;
k := 4
en :=4 ln 10
ln 213.28771238
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Ejercicio 4 del método de bisección. Ecuación de Kepler.
f d x/xK0.5$sin x K0.7;f := x/xC K1 $0.5 sin x K0.7
evalf f 1 ;K0.1207354924
evalf f 2 ;0.8453512866
plot f x , x = 1 ..2 ;
x1.2 1.4 1.6 1.8 2
K0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
a d 1; b d 2; n d 4; h dbKa
2nK1 ;
a := 1
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(6)(6)
b := 2n := 4
h :=18
for i from 1 to 2nK1 do print i, evalf aCi$h , evalf f aC i$h ; end do;1, 1.125000000, K0.02613379702, 1.250000000, 0.07550769033, 1.375000000, 0.18455347154, 1.500000000, 0.30125250675, 1.625000000, 0.42573432986, 1.750000000, 0.55800702667, 1.875000000, 0.6979571092
8, 2., 0.8453512866
for i from 1 to 2nK1 do m i d evalf aC i$h ; end do:
9.965784285
c7 d m 1 Cm 2
2;
c7 := 1.187500000
Para un error menor que 5*10^{-6} hace falta $n$ mayor que:
5 $ln 5ln 2
C6;
5 ln 5ln 2
C6
evalf % ;17.60964047
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(2)(2)
(1)(1)
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Ejercicio 5. La raíz 7^{1/4} con valor menor que una centésima.
f d x/ x4K7;f := x/x4K7
evalf f 1 ;K6.
evalf f 2 ;9.
plot f x , x = 1 ..2 ;
x1.2 1.4 1.6 1.8 2
K6
K4
K2
0
2
4
6
8
a d 1; b d 2; n d 4; h dbKa
2nK1;
a := 1
(5)(5)
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(7)(7)
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(6)(6)
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(4)(4)
b := 2n := 4
h :=18
for i from 1 to 2nK1 do print i, evalf aC i$h , evalf f aC i$h ; end do;1, 1.125000000, K5.3981933592, 1.250000000, K4.5585937503, 1.375000000, K3.4255371094, 1.500000000, K1.937500000
5, 1.625000000, K0.027099609386, 1.750000000, 2.3789062507, 1.875000000, 5.359619141
8, 2., 9.
for i from 1 to 2nK1 do m i d evalf aC i$h ; end do:m 5 ; m 6 ;
1.6250000001.750000000
c4 d m 5 Cm 6
2;
c4 := 1.687500000
(2)(2)
(3)(3)
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Ejercicio 6b.
f d x/ exp x Kx2C3$xK2;f := x/exKx2C3 xK2
evalf f 0 ;K1.
evalf f 1 ;2.718281828
plot f x , x = 0 ..1 ;
x0.2 0.4 0.6 0.8 1
K1
0
1
2
a d 0; b d 1; n d 4; h dbKa
2nK1;
a := 0
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(5)(5)
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(4)(4)
(6)(6)
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> >
b := 1n := 4
h :=18
for i from 1 to 2nK1 do print i, evalf aC i$h , evalf f aC i$h ; end do;1, 0.1250000000, K0.5074765472, 0.2500000000, K0.0284745833, 0.3750000000, 0.4393664154, 0.5000000000, 0.89872127105, 0.6250000000, 1.3526209576, 0.7500000000, 1.8045000177, 0.8750000000, 2.258250294
8, 1., 2.718281828
for i from 1 to 2nK1 do m i d evalf aC i$h ; end do:
c4 d m 2 Cm 3
2;
c4 := 0.3125000000
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(3)(3)
(2)(2)
> >
Ejercicio 6c.
f d x/ 2 x$cos 2$x K xC1 2;f := x/2 x cos 2 x K xC1 2
evalf f K3 ;K9.761021720
evalf f K2 ;1.614574484
plot f x , x =K3 ..K2 ;
xK3 K2.8 K2.6 K2.4 K2.2
K8
K6
K4
K2
0
a dK3; b dK2; n d 4; h dbKa
2nK1;
a := K3
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(5)(5)
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(4)(4)
(6)(6)
b := K2n := 4
h :=18
for i from 1 to 2nK1 do print i, evalf aC i$h , evalf f aC i$h ; end do;1, K2.875000000, K8.4674813992, K2.750000000, K6.9601837593, K2.625000000, K5.3290737554, K2.500000000, K3.6683109285, K2.375000000, K2.069235226
6, K2.250000000, K0.61391890277, K2.125000000, 0.630246832
8, K2., 1.614574484
for i from 1 to 2nK1 do m i d evalf aC i$h ; end do:
c4 d m 6 Cm 7
2;
c4 := K2.187500000