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.

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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New Wilmington, PA

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

P_{3}(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

P_{3}(x) = x + (1/3!) 2 x³ = x + (1/3) x³ .

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