metnum-tugas-teori

TUGAS METODE NUMERIK

1. Menurunkan deret taylor dari fungsi :a. f(x)=cos(x)

%Deret Taylor hingga suku ke-5 :1) x0=0

syms x;f=cos(x);taylor(f,0,5)

ans =

1-1/2*x^2+1/24*x^42) x0=pi/3

syms x;f=cos(x);taylor(f,pi/3,5)

ans =

1/2-1/2*3^(1/2)*(x-1/3*pi)-1/4*(x-1/3*pi)^2+1/12*3^(1/2)*(x-1/3*pi)^3+1/48*(x-1/3*pi)^4

3) x0=pi/4syms x;f=cos(x);taylor(f,pi/4,5)

ans = 1/2*2^(1/2)-1/2*2^(1/2)*(x-1/4*pi)-1/4*2^(1/2)*(x-1/4*pi)^2+1/12*2^(1/2)*(x-1/4*pi)^3+1/48*2^(1/2)*(x-1/4*pi)^4

%Membuat Grafik f(x)=cos(x)x1=-10:0.01:10;x2=-5:0.01:25;x=-2*pi:0.01:2*pi;x3=-2*pi:0.01:2*pi;x4=-2*pi:0.01:2*pi;x5=-2*pi:0.01:2*pi;y=cos(x);y3=1-(1/2)*x3.^2+(1/24)*x3.^4;y4=1/2-1/2*3.^(1/2)*(x4-1/3*pi)-1/4*(x4-1/3*pi).^2+1/12*3^(1/2)*(x4-1/3*pi).^3+1/48*(x4-1/3*pi).^4;y5=1/2*2.^(1/2)-1/2*2.^(1/2)*(x5-1/4*pi)-1/4*2.^(1/2)*(x5-1/4*pi).^2+1/12*2.^(1/2)*(x5-1/4*pi).^3+1/48*2.^(1/2)*(x5-1/4*pi).^4;y1=0*x1;y2=0*x2;plot(x,y,'b',x3,y3,'g',x4,y4,'y',x5,y5,'r',x1,y1,'r',y2,x2,'r');gtext('x');gtext('y');title('grafik f(x)=cos(x)')grid on

1

b. f(x)=tan(x)%Deret Taylor hingga suku

ke- 5 :1) x0=

0

syms x;f=tan(x);taylor(f,0,5)

ans = x+1/3*x^3

2) x0=pi/3syms x;f=tan(x);taylor(f,pi/3,5)

ans = 3^(1/2)+4*x-4/3*pi+4*3^(1/2)*(x-1/3*pi)^2+40/3*(x-1/3*pi)^3+44/3*3^(1/2)*(x-1/3*pi)^4

3) x0=pi/4syms x;f=tan(x);taylor(f,pi/3,5)

ans = 1+2*x-1/2*pi+2*(x-1/4*pi)^2+8/3*(x-1/4*pi)^3+10/3*(x-1/4*pi)^4

%Membuat Grafik f(x)=tan(x)x1=-10:0.01:10;x2=-200:0.01:1000;x=-2*pi:0.01:2*pi;x3=-2*pi:0.01:2*pi;x4=-pi:0.01:pi;x5=-pi:0.01:pi;y=tan(x);y3=x3+1/3*x3.^3;y4=3.^(1/2)+4*x4-4/3*pi+4*3.^(1/2)*(x4-1/3*pi).^2+40/3*(x4-1/3*pi).^3+44/3*3.^(1/2)*(x4-1/3*pi).^4;y5=1+2*x5-1/2*pi+2*(x5-1/4*pi).^2+8/3*(x5-1/4*pi).^3+10/3*(x5-1/4*pi).^4;y1=0*x1;y2=0*x2;plot(x,y,'b',x3,y3,'g',x4,y4,'y',x5,y5,'r',x1,y1,'r',y2,x2,'r');gtext('x');gtext('y');title('grafik f(x)=tan(x)')grid on

2

Keterangan :

Kurva biru : f(x)Kurva hijau : x0=0Kurva kuning : x0=0.25 atau pi/3Kurva merah : x0=0.5 atau pi/4

c. f(x)=ex

%Deret Taylor hingga suku ke-5 :1) x0=0

syms x;f=exp(x);taylor(f,0,5)

ans = 1+x+1/2*x^2+1/6*x^3+1/24*x^4

2) x0=0.25syms x;f=exp(x);taylor(f,0.25,5)

ans = exp(1/4)+exp(1/4)*(x-1/4)+1/2*exp(1/4)*(x-1/4)^2+1/6*exp(1/4)*(x-1/4)^3+1/24*exp(1/4)*(x-1/4)^4

3) x0=0.5syms x;f=exp(x);taylor(f,0.5,5)

ans = exp(1/2)+exp(1/2)*(x-1/2)+1/2*exp(1/2)*(x-1/2)^2+1/6*exp(1/2)*(x-1/2)^3+1/24*exp(1/2)*(x-1/2)^4

%Membuat Grafik f(x)=e^xx1=-5:0.01:10;x2=-2:0.01:20;x=-4:0.01:3;x3=-4:0.01:3;x4=-4:0.01:3;x5=-4:0.01:3;y=exp(x);y3=1 + x3 + 1/2*x3.^2 + 1/6*x3.^3 + 1/24*x3.^4;y4=exp(1/4)+exp(1/4)*(x4-1/4)+1/2*exp(1/4)*(x4-1/4).^2+1/6*exp(1/4)*(x4-1/4).^3+1/24*exp(1/4)*(x4-1/4).^4;y5=exp(1/2)+exp(1/2)*(x5-1/2)+1/2*exp(1/2)*(x5-1/2).^2+1/6*exp(1/2)*(x5-1/2).^3+1/24*exp(1/2)*(x5-1/2).^4;y1=0*x1;y2=0*x2;plot(x,y,'b',x3,y3,'g',x4,y4,'y',x5,y5,'r',x1,y1,'r',y2,x2,'r');gtext('x');

3

Keterangan :


gtext('y');title('grafik f(x)=e^x')grid on

d. f(x)=e-x

%Deret Taylor hingga suku ke-5 :

1) x0=0

syms x;f=exp(-x);taylor(f,0,5)

ans = 1-x+1/2*x^2-1/6*x^3+1/24*x^4

2) x0=0.25syms x;f=exp(-x);taylor(f,0.25,5)

ans = exp(-1/4)-exp(-1/4)*(x-1/4)+1/2*exp(-1/4)*(x-1/4)^2-1/6*exp(-1/4)*(x-1/4)^3+1/24*exp(-1/4)*(x-1/4)^4

3) x0=0.5syms x;f=exp(-x);taylor(f,0.5,5)

ans = exp(-1/2)-exp(-1/2)*(x-1/2)+1/2*exp(-1/2)*(x-1/2)^2-1/6*exp(-1/2)*(x-1/2)^3+1/24*exp(-1/2)*(x-1/2)^4

%Membuat Grafik f(x)=e^-xx1=-4:0.01:10;x2=-2:0.01:15;x=-2:0.01:5;x3=-2:0.01:5;x4=-2:0.01:5;x5=-2:0.01:5;y=exp(-x);y3=1 - x3 + 1/2*x3.^2 - 1/6*x3.^3 + 1/24*x3.^4;y4=exp(-1/4)-exp(-1/4)*(x4-1/4)+1/2*exp(-1/4)*(x4-1/4).^2-1/6*exp(-1/4)*(x4-1/4).^3+1/24*exp(-1/4)*(x4-1/4).^4;y5=exp(-1/2)-exp(-1/2)*(x5-1/2)+1/2*exp(-1/2)*(x5-1/2).^2-1/6*exp(-1/2)*(x5-1/2).^3+1/24*exp(-1/2)*(x5-1/2).^4;y1=0*x1;y2=0*x2;plot(x,y,'b',x3,y3,'g',x4,y4,'y',x5,y5,'r',x1,y1,'r',y2,x2,'r');gtext('x');gtext('y');title('grafik f(x)=e^-x')grid on

4

Keterangan :


e. f(x)=ln(x+1)%Deret Taylor hingga suku ke-

5 :1) x

0=

0syms x;f=log(x+1);taylor(f,0,5)

ans =

1+x+1/2*x^2+1/6*x^3+1/24*x^42) x0=0.25

syms x;f=log(x+1);taylor(f,0.25,5)

ans = log(5/4)+4/5*x-1/5-8/25*(x-1/4)^2+64/375*(x-1/4)^3-64/625*(x-1/4)^4

3) x0=0.5syms x;f=log(x+1);taylor(f,0.5,5)

ans = log(3/2)+2/3*x-1/3-2/9*(x-1/2)^2+8/81*(x-1/2)^3-4/81*(x-1/2)^4

%Membuat Grafik f(x)=ln(x+1)x1=-2:0.01:5;x2=-6:0.01:2;x=-1:0.01:2;x3=-1:0.01:2;x4=-1:0.01:2;x5=-1:0.01:2;y=log(x+1);y3=x3-x3.^2/2+x3.^3/3-x3.^4/4;y4=log(5/4)+4/5*x4-1/5-8/25*(x4-1/4).^2+64/375*(x4-1/4).^3-64/625*(x4-1/4).^4;y5=log(3/2)+2/3*x5-1/3-2/9*(x5-1/2).^2+8/81*(x5-1/2).^3-4/81*(x5-1/2).^4;y1=0*x1;y2=0*x2;plot(x,y,'b',x3,y3,'g',x4,y4,'y',x5,y5,'r',x1,y1,'r',y2,x2,'r');gtext('x');gtext('y');title('grafik f(x)=ln(x+1)')grid on

5

Keterangan :


6

Keterangan :


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