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+⋯