Find the Taylor polynomial of order 2 for
𝑓
(
𝑥
)
=
−
7
𝑥
3
+
9
𝑥
2
−
2
𝑥
−
8
around
𝑥
=
0
.

Question

Find the Taylor polynomial of order 2 for 
𝑓
(
𝑥
)
=
−
7
𝑥
3
+
9
𝑥
2
−
2
𝑥
−
8
 around 
𝑥
=
0
.

BlackTom AI · Accepted Answer

Taylor polynomial of order 2 centered at 0 uses f(0), f'(0), and f''(0)/2 to form T2(x) = f(0) + f'(0)x + (f''(0)/2) x^2.

1) Relevant concepts:
- The Taylor polynomial of degree 2 about x = 0 (Maclaurin polynomial) is T2(x) = f(0) + f'(0)x + (f''(0)/2)x^2.
- For a polynomial f(x), this is equivalent to truncating f(x) after degree 2 (dropping terms of degree ≥ 3).

2) Step-by-step differentiation and evaluation:
Given f(x) = −7x^3 + 9x^2 − 2x − 8.

Compute f(0):
- f(0) = −7(0)^3 + 9(0)^2 − 2(0) − 8 = −8.

Compute f'(x):
- f'(x) = derivative of −7x^3 is −21x^2; of 9x^2 is 18x; of −2x is −2; constant vanishes.
- So f'(x) = −21x^2 + 18x − 2.
- Evaluate at 0: f'(0) = −21(0)^2 + 18(0) − 2 = −2.

Compute f''(x):
- Differentiate f'(x): f''(x) = −42x + 18.
- Evaluate at 0: f''(0) = −42(0) + 18 = 18.

3) Form the Taylor polynomial:
T2(x) = f(0) + f'(0)x + (f''(0)/2)x^2
= (−8) + (−2)x + (18/2) x^2
= −8 − 2x + 9x^2.

4) Why this is correct:
This uses the exact values of f and its first two derivatives at 0, and includes all terms up to x^2; the original cubic term −7x^3 is degree 3 and therefore omitted in the order-2 polynomial.

5) Why other answer choices are incorrect:
- Choice A (−7x^3 + 9x^2 − 2x − 8) is the full cubic f(x), not truncated to degree 2.
- Choice B (−21x^2 + 18x − 2) is f'(x), not a degree-2 Taylor polynomial.
- Choice C (−7x^3 + 9x^2) omits necessary linear and constant terms and still includes a degree-3 term.

6) Conclusion:
The Taylor polynomial of order 2 about x = 0 is 9x^2 − 2x − 8, which matches the provided correct answer.

Find the Taylor polynomial of order 2 for 𝑓 ( 𝑥 ) = − 7 𝑥 3 + 9 𝑥 2 − 2 𝑥 − 8 around 𝑥 = 0 .单项选择题

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