Diberikan fungsi kuadrat dengan rumus h(x) = ax^2+bx+c. Tentukanlah konstanta a,b,danc jika diketahui bahwa h(1) = 7 , h^1 (1)=2 ,dan h^2 (1)=-5.

Diberikan fungsi kuadrat dengan rumus h(x) = ax^2+bx+c. Tentukanlah konstanta a,b,danc jika diketahui bahwa h(1) = 7 , h^1 (1)=2 ,dan h^2 (1)=-5.

Jawab:

h(x) = ax^2+bx+c
h(1) = 7
a+b+c=7 ……..Pers 1)

h^1 (x)=2ax+b
Maka:
h^1 (1)=2
2a+b=2 …….Pers 2)

h^2 (1)=2a
h^2 (1)=-5
2a= -5
a= – 5/2
Maka:
2a+b=2
2( – 5/2)+b=2
-5+b=2
b = 7

sehingga
a+b+c=7
-5/2+7+c=7
c = 5/2

Tentukan suku Pertama hingga suku ketiga dari deret MacLaurin fungsi h(x) = (cos x) ln (1+x)

Tentukan suku Pertama hingga suku ketiga dari deret MacLaurin fungsi
h(x) = (cos x) ln (1+x)

Jawab:

Deret MacLaurin;
f(x) = f(0) + f^1 (0)(x)+(f^2 (0))/2! x^2+(f^3 (0))/3! x^3+(f^4 (0))/4! x^4+⋯

maka:
cos x = 1 – x^2/2!+x^4/4!-x^6/6!+⋯
ln(1+x)= x-x^2/2+x^3/3-x^4/4+⋯

Sehingga deret MacLaurin dari h(x) = (cos x) ln (1+x) adalah

h(x) = (cos x) ln (1+x)

h(x) = (1 – x^2/2!+x^4/4!-x^6/6!+⋯) (x-x^2/2+x^3/3-x^4/4+⋯)

h(x) = x – x^2/2!+x^3/3-x^4/4– x^3/2!+x^4/4-x^5/6+x^6/8+x^5/4!-x^6/48+⋯

h(x) = x – x^2/2-x^3/6-x^4/4+x^4/4-3x^5/24+(5x^6)/48+⋯

h(x) = x – x^2/2 – x^3/6 – 3x^5/24 + (5x^6)/48 +⋯

Deret MacLaurin

Deret MacLaurin:

1/(1-x)=1+x+x^2+x^3+⋯

1/(1+x)=1-x+x^2-x^3+⋯

1/(1-x^2 )=1+x^2+x^4+x^6+⋯

1/(1-x)^2 =1+2x+3x^2+4x^3+⋯

ln(1+x) = x – x^2/2+x^3/3-x^4/4+⋯

e^x=1+x+x^2/2!+x^3/3!+⋯

1/tan⁡x =x-x^3/3+x^5/5-x^7/7+⋯

ln(1-x) = x + x^2/2+x^3/3+x^4/4+⋯

sin x = x – x^3/3!+x^5/5!-x^7/7!+⋯

cos x = 1 – x^2/2!+x^4/4!-x^6/6!+⋯

ln(1-x)^2=x+x^2+x^3+x^4+⋯

sinh x = x + x^3/3!+x^5/5!+x^7/7!+⋯

ln(1+x)=x-x^2/2+x^3/3-x^4/4+⋯

cosh x = 1 + x^2/2!+x^4/4!+x^6/6!+⋯

Tentukan deret MacLaurin dari f(x) = e^(-x) cos⁡(x) 

Tentukan deret MacLaurin dari f(x) = e^(-x) cos⁡(x)

Jawab:

Deret MacLaurin;
f(x) = f(0) + f^1 (0)(x)+(f^2 (0))/2! x^2+(f^3 (0))/3! x^3+(f^4 (0))/4! x^4+⋯

Maka:
f(x) = e^(-x) cos⁡x
f^1 (x)= – e^(-x) cos⁡x-e^(-x) sin⁡x
f^2 (x)= e^(-x) cos⁡x+e^(-x) sin⁡x+e^(-x) sin⁡x-e^(-x) cos⁡x =2e^(-x) sin⁡x
f^3 (x)=-2e^(-x) sin⁡x+2e^(-x) cos x
f^4 (x)= 2e^(-x) sin⁡x-2e^(-x) cos ⁡x-2e^(-x) cos⁡x – 2e^(-x) sin⁡x= -4e^(-x) cos⁡x

Maka:
f(0) = e^(-0) cos⁡0=1
f^1 (0)= -e^(-0) cos⁡0-e^(-0) sin⁡0=-1-0=-1
f^2 (0)=2e^(-0) sin⁡0=0
f^3 (0)= -2e^(-0) sin⁡0+2e^(-0) cos 0=0+2=2
f^4 (0)= -4e^(-0) cos⁡0=-4

Maka: deret macLaurinnya
f(x) = f(0) + f^1 (0)(x)+(f^2 (0))/2! x^2+(f^3 (0))/3! x^3+(f^4 (0))/4! x^4+⋯
f(x) = 1 + (-1)(x)+0/2! x^2+2/3! x^3+(-4)/4! x^4+⋯
f(x) = 1 -x+0+1/3 x^3-1/6 x^4+⋯
f(x) = 1 – x+1/3 x^3-1/6 x^4+⋯

Tentukan deret MacLaurin dari f(x) = e^x cos⁡(x)

Tentukan deret MacLaurin dari f(x) = e^x cos⁡(x)

Jawab:

Deret MacLaurin;
f(x) = f(0) + f^1 (0)(x)+(f^2 (0))/2! x^2+(f^3 (0))/3! x^3+(f^4 (0))/4! x^4+⋯

Maka:
f(x) = e^x cos⁡(x)
f^1 (x)= e^x cos(⁡x)  -e^x sin⁡(x)
f^2 (x)= e^x cos⁡x-e^x sin⁡x-e^x sin⁡(x) – e^x cos⁡(x) =-2e^x sin⁡(x)
f^3 (x)=-2e^x sin⁡x-2e^x cos (⁡x)
f^4 (x)= -2e^x sin⁡x-2e^x cos (⁡x) – 2e^x cos⁡(x) + 2e^x sin⁡x= -4e^x cos(⁡x)

Maka:
f(0) = e^0 cos⁡0=1
f^1 (0)= e^0 cos⁡0-e^0 sin⁡0 =1- 0 =1
f^2 (0)=-2e^0 sin⁡0=0
f^3 (0)= -2e^0 sin⁡0-2e^0 cos 0 = 0 – 2 =-2
f^4 (0)= -4e^0 cos⁡0=-4

Maka: deret macLaurinnya
f(x) = f(0) + f^1 (0)(x)+(f^2 (0))/2! x^2+(f^3 (0))/3! x^3+(f^4 (0))/4! x^4+⋯
f(x) = 1 + (1)(x)+0/2! x^2+(-2)/3! x^3+(-4)/4! x^4+⋯
f(x) = 1 + x+0- 1/3 x^3-1/6 x^4+⋯
f(x) = 1 + x-1/3 x^3-1/6 x^4+⋯