Deber taylor

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

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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

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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

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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

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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

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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


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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!

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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



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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

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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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plot(x,f,r,x,p1,y,x,p2,b)grid

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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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Deber taylor

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