Question

Find the taylor polynomial t3(x) for the function f centered at the number

a. f(x) = e^(â3x) sin(2x), a = 0

Answer 1

`e^(-3x)=\displaystyle\sum_(n=0)^\infty((-3x)^n)/(n!)=1-3x+9x^2+\cdots`

`\sin2x=\displaystyle\sum_(n=0)^\infty((-1)^n(2x)^(2n+1))/((2n+1)!)=2x-\frac{4x^3}3+\cdots`


`e^(-3x)\sin2x=\left(1-3x+9x^2+\cdots\right)\left(2x-\frac{4x^3}3+\cdots\right)`

`\approx T_3(x)=(1-3x+9x^2)\left(2x-\frac{4x^3}3\right)`

`T_3(x)=2x-6x^2+\left(18-\frac43\right)x^3`

`T_3(x)=2x-6x^2+\frac{50}3x^3`