Answer: [tex]\bold{1-\dfrac{3}{2}x^2+\dfrac{7}{8}x^4-\dfrac{61}{240}x^6+\dfrac{547}{13440}x^8+...}[/tex]
Step-by-step explanation:
First, find the derivatives at x = 0:
[tex]f(x)=cos^3(x)\\f(0)=cos^3(0)\\.\qquad =1\\\\\\f'(x)=-3\ cos^2(x)\cdot sin(x)\\f'(0)=-3\ cos^2(0)\cdot sin(0)\\.\qquad =0\\\\\\f''(x)=-3[sin(2x)\cdot sin(x)+cos^3(x)]\\f''(0)=-3[sin(2\cdot 0)\cdot sin(0)+cos^3(0)]\\.\qquad =-3\\\\\\f'''(x)=-3[-2\ cos(2x)\cdot sin(x)-cos(x)\cdot sin(x)\cdot sin(2x)-3\ cos^2(x)\cdot sin(x)]\\f'''(0)=-3[-2\ cos(2\cdot 0)\cdot sin(0)-cos(0)\cdot sin(0)\cdot sin(2\cdot 0)-3\ cos^2(0)\cdot sin(0)]\\.\qquad =0[/tex]
[tex]f^{IV}(x)=-3[8\ sin(x)\cdot sin(2x)-4\ cos(2x)\cdot cos(x)-3\ cos^3(x)]\\f^{IV}(0)=-3[8\ sin(0)\cdot sin(2\cdot 0)-4\ cos(2\cdot 0)\cdot cos(0)-3\ cos^3(0)]\\.\qquad =21\\\\\\f^V(x)=-3[16\ sin(2x)\cdot cos(2x)+20\ sin(x)\cdot cos(2x)+9\ cos^2(x)\cdot sin(x)]\\f^V(0)=-3[16\ sin(2\cdot 0)\cdot cos(2\cdot 0)+20\ sin(0)\cdot cos(2\cdot 0)+9\ cos^2(0)\cdot sin(0)]\\.\qquad =0[/tex]
[tex]f^{VI}(x)=-3[52\ cos(x)\cdot cos(2x)-65\ sin(2x)\cdot sin(2x)+9\ cos^3(x)\cdot sin(x)]\\f^{VI}(0)=-3[52\ cos(0)\cdot cos(2\cdot 0)-65\ sin(2\cdot 0)\cdot sin(2\cdot 0)+9\ cos^3(0)\cdot sin(0)]\\.\qquad =-183\\\\\\f^{VII}(x)=-3[-182\ cos(2x)\cdot sin(x)-169\ cos(x)\cdot sin(2x)-27\ cos^2(x)\cdot sin(x)]\\f^{VII}(0)=-3[-182\ cos(2\cdot 0)\cdot sin(0)-169\ cos(0)\cdot sin(2\cdot 0)-27\ cos^2(0)\cdot sin(0)]\\.\qquad =0[/tex]
[tex]f^{VIII}(x)=-3[560\ sin(x)\cdot sin(2x)-520\ cos(2x)\cdot cos(x)-27\ cos^3(x)]\\f^{VIII}(0)=-3[560\ sin(0)\cdot sin(2\cdot 0)-520\ cos(2\cdot 0)\cdot cos(0)-27\ cos^3(0)]\\.\qquad =1641[/tex]
The Maclaurin Series of f(x) at 0 is:
[tex]f(x)=f(0)+\dfrac{f'(0)}{1!}(x)+\dfrac{f''(0)}{2!}(x^2)+\dfrac{f'''(0)}{3!}(x^3)+\dfrac{f^{IV}(0)}{4!}(x^4)\\\\.\qquad \qquad +\dfrac{f'(0)}{5!}(x^5)+\dfrac{f^{VI}(0)}{6!}(x^6)+\dfrac{f^{VII}(0)}{7!}(x^7)+\dfrac{f{VIII}(0)}{8!}(x^8)+...\\\\\text{Insert the derivatives into the equation above and simplify:}\\\boxed{f(x) =1-\dfrac{3}{2}x^2+\dfrac{7}{8}x^4-\dfrac{61}{240}x^6+\dfrac{547}{13440}x^8+...}[/tex]
