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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. 

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Andre W. | Friendly tutor for ALL math and physics coursesFriendly tutor for ALL math and physics ...
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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 .

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