what would i do for such a problem dealing with taylor series??

Question

find the taylor polynomial Tsubn (x) for the function f at the number a.
f(x)= xe^-2x, a=0, n=3

my teacher said to compute the taylor polynomial. i'm not sure what that means

Answer 1

The Taylor series expansion is an easier-to-calculate approximation of the function of interest, so we use it instead of the much more complex function.

What your teacher is asking for is a 3-term (n=3) polynomial that approximates f(x), near the point x=0 (that's from a=0)

Here's a start:
1. Look up the Taylor series expansion of e^(k x), where k=-2
2. With the first 3 terms calculated, multiply them by x, since your function was f(x) = x e^(k x)

That's what your teacher is looking for.

Answer 2

f´(x) = f(a) +(x-a) f´(a) +(x-a)^2 f¨¨(a)/2 +(x-a)^3/3!f´´´(a)
x*e^-2x= x-2x^2+2x^3-4/3 x^4