Please help me with this problem?

Question

If f(0)=0, f'(0)= 1, f"(0)=0, and f'''(0)=2, then which of the following is the third order Taylor polynomial generated by f(x) at x=0? 

 

a) 2x^3+x

b) (1/3)x^3+(1/2)x

c) (2/3)x^3+x

d) 2x^3-x

e) (1/3)x^3+x

 

Please show all your work. 

Answer 1

The third order Taylor polynomial generated by f(x) at x=0 is of the form

P₃(x) = f(0) + f'(0) x + (1/2!) f''(0) x² + (1/3!) f'''(0) x³.

 

With f(0)=0, f'(0)= 1, f"(0)=0, and f'''(0)=2 this becomes

P₃(x) =  x + (1/3!) 2 x³ = x + (1/3) x³ .