UNIVERSIDAD DE LAS FUERZAS ARMADAS - ESPEMetodos numericos
Series de Taylor
Nombre: Solange PilcoNRC: 1427Aula: G-301Fecha: 18-10-2015
Realizar la series de Taylor de la funcion seno, coseno y logaritmo natural, alrededor de ceroy un punto distinto de cero, considerando al menos cinco terminos no nulos.
f(x)= sen(x)Cuando xo = 0
f(x) = sen(x) = 0
f (x) = cos(x) = 1
f (x) = sen(x) = 0f (x) = cos(x) = 1f iv(x) = sen(x) = 0
f v(x) = cos(x) = 1
f vi(x) = sen(x) = 0f vii(x) = cos(x) = 1f viii(x) = sen(x) = 0
f ix(x) = cos(x) = 1
f(x) = Pn(x) +Rn(x)
Pn(x) =n
k=0
fk(xo)(x xo)kk!
P9(x) = 0 + x+ 0 x3
3!+ 0 +
x5
5!+ 0 x
7
7!+ 0 +
x9
9!+ ...+
(1)nx2n+1(2n+ 1)!
P9(x) = x x3
3!+x5
5! x
7
7!+x9
9!
Codigo en MATLAB:
x = linspace(pi, pi, 100)f(x) = sin(x)p1 = xp3 = x x.3/6p5 = x x.3/6 + x.5/120p7 = x x.3/6 + x.5/120 x.7/5040p9 = x x.3/6 + x.5/120 x.7/5040 + x.9/362880
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Graficas:plot(x,f,r)grid
plot(x,f,r,x,p1,b)grid
plot(x,f,r,x,p1,b,x,p3,g)grid
2
plot(x,f,r,x,p1,b,x,p3,g,x,p5,y)grid
plot(x,f,r,x,p1,b,x,p3,g,x,p5,y,x,p7,m)grid
plot(x,f,r,x,p1,b,x,p3,g,x,p5,y,x,p7,m,x,p9,c)grid
3
f(x)= sen(x)
Cuando xo =pi
2
f(x) = sen(x) = 1
f (x) = cos(x) = 0
f (x) = sen(x) = 1f (x) = cos(x) = 0f iv(x) = sen(x) = 1
f v(x) = cos(x) = 0
f vi(x) = sen(x) = 1f vii(x) = cos(x) = 0f viii(x) = sen(x) = 1
f ix(x) = cos(x) = 0
Pn(x) =n
k=0
fk(xo)(x xo)kk!
P8(x) = 1 (x pi/2)2
2!+
(x pi/2)44!
(x pi/2)6
6!+
(x pi/2)88!
+ ...+(1)n(x pi/2)2n
2n!
Codigo en MATLAB:x = linspace(pi, pi, 100)f(x) = sin(x)p1 = 1p2 = 1 (x pi/2).2/2p4 = 1 (x pi/2).2/2 + (x pi/2).4/24p6 = 1 (x pi/2).2/2 + (x pi/2).4/24 (x pi/2).6/720p8 = 1 (x pi/2).2/2 + (x pi/2).4/24 (x pi/2).6/720 + (x pi/2).8/40320
Graficas:plot(x,f,r,x,p1,b)grid
4
Graficas:plot(x,f,r,x,p1,b,x,p2,y)grid
plot(x,f,r,x,p1,b,x,p2,y,x,p4,g)grid
plot(x,f,r,x,p1,b,x,p2,y,x,p4,g,x,p6,m)grid
5
plot(x,f,r,x,p1,b,x,p2,y,x,p4,g,x,p6,m,x,p8,c)grid
f(x)=cos(x)Cuando xo = 0
f(x) = cos(x) = 1
f (x) = sen(x) = 0f (x) = cos(x) = 1f (x) = sen(x) = 0
f iv(x) = cos(x) = 1
f v(x) = sen(x) = 0f vi(x) = cos(x) = 1f vii(x) = sen(x) = 0
f viii(x) = cos(x) = 1
f ix(x) = sen(x) = 0
Pn(x) =n
k=0
fk(xo)(x xo)kk!
P8(x) = 1 x2
2+x4
24 x
6
720+
x8
40320+ ...+
(1)nx2n2n!
Codigo en MATLAB:
x = linspace(pi, pi, 100)f = cos(x)p0 = 1p2 = 1 x.2/2p4 = 1 x.2/2 + x.4/24p6 = 1 x.2/2 + x.4/24 x.6/720p8 = 1 x.2/2 + x.4/24 x.6/720 + x.8/40320
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Graficas:
plot(x,f,r)grid
plot(x,f,r,x,p0,y)grid
plot(x,f,r,x,p0,y,x,p2,b)grid
7
Graficas:plot(x,f,r,x,p0,y,x,p2,b,x,p4,m)grid
plot(x,f,r,x,p0,y,x,p2,b,x,p4,m,x,p6,g)grid
plot(x,f,r,x,p0,y,x,p2,b,x,p4,m,x,p6,g,x,p8,c)grid
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f(x)=cos(x)Cuando xo = pi/2
f(x) = cos(x) = 0
f (x) = sen(x) = 1f (x) = cos(x) = 0f (x) = sen(x) = 1
f iv(x) = cos(x) = 0
f v(x) = sen(x) = 1f vi(x) = cos(x) = 0f vii(x) = sen(x) = 1
f viii(x) = cos(x) = 0
f ix(x) = sen(x) = 1
Pn(x) =n
k=0
fk(xo)(x xo)kk!
P8(x) = (x pi/2) + (x pi/2)3
6 (x pi/2)
5
120+
(x pi/2)75040
(x pi/2)9
362880
Codigo en MATLAB:
x = linspace(pi, pi, 100)f = cos(x)p1 = (x pi/2)p3 = (x pi/2) + (x pi/2).3/6p5 = (x pi/2) + (x pi/2).3/6 (x pi/2).5/120p7 = (x pi/2) + (x pi/2).3/6 (x pi/2).5/120 + (x pi/2).7/5040p9 = (x pi/2) + (x pi/2).3/6 (x pi/2).5/120 + (x pi/2).7/5040 (x pi/2).9/362880
Graficas:plot(x,f,r)grid
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Graficas:plot(x,f,r,x,p1,y)grid
plot(x,f,r,x,p1,y,x,p3,g)grid
plot(x,f,r,x,p1,y,x,p3,g,x,p5,b)grid
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Graficas:plot(x,f,r,x,p1,y,x,p3,g,x,p5,b,x,p7,m)grid
plot(x,f,r,x,p1,y,x,p3,g,x,p5,b,x,p7,m,x,p9,c)grid
f(x)= ln(x)Cuando xo = 1/2
f(x) = ln(x) = 0,69f (x) =
1
x=
1
2
f (x) = 1x2
= 14
f (x) =2
x3=
1
4
f iv(x) = 6x4
= 38
f v(x) =24
x5=
3
4
Pn(x) =n
k=0
fk(xo)(x xo)kk!
11
P5(x) = 0,69 + x 1/22
(x+ 1/4)2
8+
(x 1/4)324
3(x+ 3/8)4
192+
3(x 3/4)5480
Codigo en MATLAB:
x = linspace(0, 2, 100)f = log(x)p1 = 0,69 + (x 1/2).1/2p2 = 0,69 + (x 1/2).1/2 (x+ 1/4).2/8p3 = 0,69 + (x 1/2).1/2 (x+ 1/4).2/8 + (x 1/4).3/24p4 = 0,69 + (x 1/2).1/2 (x+ 1/4).2/8 + (x 1/4).3/24 3 (x+ 3/8).4/192p5 = 0,69+(x1/2).1/2(x+1/4).2/8+(x1/4).3/243(x+3/8).4/192+3(x3/4).5/480
Graficas:plot(x,f,r)grid
plot(x,f,r,x,p1,y)grid
12
Graficas:plot(x,f,r,x,p1,y,x,p2,c)grid
plot(x,f,r,x,p1,y,x,p2,c,x,p3,b)grid
plot(x,f,r,x,p1,y,x,p2,c,x,p3,b,x,p4,m)grid
13
plot(x,f,r,x,p1,y,x,p2,c,x,p3,b,x,p4,m,x,p5,g)grid
f(x)= ln(x)
Cuando xo = 1
f(x) = ln(x) = 0
f (x) =1
x= 1
f (x) = 1x2
= 1
f (x) =2
x3= 2
f iv(x) = 6x4
= 6
f v(x) =24
x5= 24
Pn(x) =n
k=0
fk(xo)(x xo)kk!
P5(x) = (x 1) (x+ 1)2
2+
(x 2)33
(x+ 6)4
4+
(x 24)55
Codigo en MATLAB:
x = linspace(0, 2, 50)f = log(x)p1 = (x 1)p2 = (x 1) (x+ 1).2/2p3 = (x 1) (x+ 1).2/2 + (x 2).3/3p4 = (x 1) (x+ 1).2/2 + (x 2).3/3 (x+ 6).4/4p5 = (x 1) (x+ 1).2/2 + (x 2).3/3 (x+ 6).4/4 + (x 24).5/5
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Graficas:plot(x,f,r)grid
plot(x,f,r,x,p1,y)grid
plot(x,f,r,x,p1,y,x,p2,b)grid
15
Graficas:plot(x,f,r,x,p1,y,x,p2,b,x,p3,m)grid
plot(x,f,r,x,p1,y,x,p2,b,x,p3,m,x,p4,c)grid
plot(x,f,r,x,p1,y,x,p2,b,x,p3,m,x,p4,c,x,p5,g)grid
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