How would I answer this Taylor polynomials question?? Help!

Question

Let c₀+c₁(x-a)+c₂(x-a)²+….+c_n(x-a)ⁿ be the Taylor series of the function f(x)=x²sin(x). For a=0 determine the value of c_3.

c₃=?

Answer 1

You need the derivatives of x²sinx to f''':

f' = 2xsinx + x²cosx

f'' = 2sinx + x sinx + 2xcosx +2xcosx - x²sinx = (2-x²)sinx +4xcosx

f''' = (2-x²)cosx - 2xsinx +4cosx -4xsinx = (6-x²)cosx -6xsinx (checked)

The c₃ term is f'''(0)/3!

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