Suppose you are told that the twentieth order Taylor polynomial of
𝑓
(
𝑥
)
=
𝑥
2
𝑒
−
𝑥
5
about
𝑥
=
0
is
𝑥
2
−
𝑥
7
+
𝑥
12
2
−
𝑥
17
6
.

Which of the following options is correct?

Question

Suppose you are told that the twentieth order Taylor polynomial of 
𝑓
(
𝑥
)
=
𝑥
2
𝑒
−
𝑥
5
 about 
𝑥
=
0
 is
𝑥
2
−
𝑥
7
+
𝑥
12
2
−
𝑥
17
6
.

Which of the following options is correct?

BlackTom AI · Accepted Answer

For a Taylor series about 0, the coefficient of x^n equals f^{(n)}(0)/n!. The given 20th-order Taylor polynomial is
x^2 − x^7 + (1/2)x^12 − (1/6)x^17.
We use the coefficient–derivative relation and direct differentiation of the series to find the requested derivatives.

1) Relevant concepts
- A Taylor series about 0 (Maclaurin) is f(x)=Σ [f^{(n)}(0)/n!] x^n. Thus the coefficient of x^n is f^{(n)}(0)/n!.
- Equivalently, differentiating the power series termwise gives values of derivatives at 0 by evaluating coefficients.

2) Step-by-step reasoning for f′(0)
- The polynomial has no x^1 term; lowest power is x^2. Write f(x)=x^2 − x^7 + (1/2)x^12 − (1/6)x^17 + ...
- Differentiate termwise: f′(x)=2x − 7x^6 + (12/2) x^11 − (17/6) x^16 + ...
- Evaluate at x=0: f′(0)=2·0 − 7·0 + (6)·0 − (17/6)·0 = 0.
- Or use coefficient rule: coefficient of x^1 is 0 = f′(0)/1! ⇒ f′(0)=0.

3) Step-by-step reasoning for f^{(12)}(0)
- The coefficient of x^12 in the polynomial is 1/2.
- Using f^{(12)}(0)/12! = 1/2 gives f^{(12)}(0) = (1/2)·12! = 12!/2.
- Intermediate expression: f^{(12)}(0) = (1/2) × 479001600 = 239500800 (if a numeric value is desired).

4) Why the correct answer is correct
- The relations above directly yield f′(0)=0 and f^{(12)}(0)=12!/2, matching the coefficients in the given polynomial.

5) Why other options are incorrect
- Any option claiming f′(0)=2! (or 2) contradicts the absence of an x term; the correct f′(0) is 0.
- Options giving f^{(12)}(0) = −17!/6 confuse the x^17 coefficient (which yields f^{(17)}(0)=−17!/6) with the x^12 coefficient.

6) Conclusion
Therefore the correct choice is f′(0)=0 and f^{(12)}(0)=12!/2, matching the provided answer.

Suppose you are told that the twentieth order Taylor polynomial of 𝑓 ( 𝑥 ) = 𝑥 2 𝑒 − 𝑥 5 about 𝑥 = 0 is 𝑥 2 − 𝑥 7 + 𝑥 12 2 − 𝑥 17 6 . Which of the following options is correct?Single choice

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