Taylor series of function around specific value

Question

I have no idea of how to tackle this question. It would be much appreciated if someone could help me: "Find the Taylor series of $$f(x)=5+2x-4x^2+x^3$$ (a) around 0;

(b) around 1."

Answer 1

Notice that the Taylor series of a polynomial is the polynomial itself!

In this case, if you want to find the coefficients of the Taylor expansion centered in $x_0$ you can just replace $x$ with $t+x_0$ (so that $t=x-x_0$). For $x_0=0$ nothing happens. For $x_0=1$: $$5+2x-4x^2+x^3=5+2(t+1)-4(t+1)^2+(t+1)^3=4-3t-t^2+t^3\\=4-3(x-1)-(x-1)^2+(x-1)^3.$$ Check if you obtain the same result by using the derivatives.

Answer 2

$$ f(0) = 5 \\ f'(0) = 2 \\ f''(0) = -8 \\ f'''(0) = 6 \\ f^{(n)}(0) = 0 \text{ for } n > 3 $$ Applying the formula around $0$: $$ f(x) = f(0) + f'(0)x + f''(0)x^2/2 + f'''(0)x^3/6 + f''''(0)x^4/24 ... $$ Astonishingly you get $5 + 2x -4x^2 + x^3$.

Can you do the rest ?