Taylor series about a point
Problem 7.406 · medium
Find the Taylor polynomial of degree 3 for \( \displaystyle f(x) = x^{3} - 2 x \) centered at \( \displaystyle a = 2 \).
- \[ \left[\begin{matrix}\left. x^{3} - 2 x \right|_{\substack{ x=2 }}\\\left. \frac{d}{d x} \left(x^{3} - 2 x\right) \right|_{\substack{ x=2 }}\\\left. \frac{d^{2}}{d x^{2}} \left(x^{3} - 2 x\right) \right|_{\substack{ x=2 }}\\\left. \frac{d^{3}}{d x^{3}} \left(x^{3} - 2 x\right) \right|_{\substack{ x=2 }}\end{matrix}\right] = \left[\begin{matrix}4\\10\\12\\6\end{matrix}\right] \]f and its first three derivatives at a.✓ Proved
- T(x) = Σ f⁽ʲ⁾(a)/j! · (x − a)ʲ.
- \[ \lim_{x \to 2^+}\left(\frac{x^{3} - 12 x - \left(x - 2\right)^{3} - 6 \left(x - 2\right)^{2} + 16}{\left(x - 2\right)^{3}}\right) = 0 \]T agrees with f to third order at a.✓ Proved
Answer \( T_3(x) = 10 x + \left(x - 2\right)^{3} + 6 \left(x - 2\right)^{2} - 16 \)
Lines: 2 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | Not checked | — | a sentence; read, not computed |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | SymPy's series() expansion about a |
Reviewers
gpt-oss:20b: fail (misleading) — The limit test in step 3 uses an incorrect expression for the difference between f and the proposed polynomial (it contains –12x and +16 instead of –2x and 0). Consequently the polynomial claimed to be the third‑degree Taylor polynomial is not correct; a student would learn a false result.qwen3.6:27b-mlx: fail (error) — The stated answer contains a linear term '10x' instead of the constant term '-10'. The Taylor polynomial should be -10 + 10(x-2) + 6(x-2)^2 + (x-2)^3, which simplifies to x^3 - 2x. The provided answer 10x + (x-2)^3 + 6(x-2)^2 - 16 simplifies to x^3 - 2x + 20x - 20, which is incorrect. The verification in step 3 checks a different polynomial (with +16 instead of -16 and no 10x term) that actually matches f(x), but this polynomial is not the one stated in the answer.
Every verdict on record (4)
gpt-oss:20b: fail (misleading) 2026-10-07 — The limit test in step 3 uses an incorrect expression for the difference between f and the proposed polynomial (it contains –12x and +16 instead of –2x and 0). Consequently the polynomial claimed to be the third‑degree Taylor polynomial is not correct; a student would learn a false result.qwen3.6:27b-mlx: fail (error) 2026-10-07 — The stated answer contains a linear term '10x' instead of the constant term '-10'. The Taylor polynomial should be -10 + 10(x-2) + 6(x-2)^2 + (x-2)^3, which simplifies to x^3 - 2x. The provided answer 10x + (x-2)^3 + 6(x-2)^2 - 16 simplifies to x^3 - 2x + 20x - 20, which is incorrect. The verification in step 3 checks a different polynomial (with +16 instead of -16 and no 10x term) that actually matches f(x), but this polynomial is not the one stated in the answer.gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — The stated answer contains a linear term '10x' instead of the constant term '-10'. The correct Taylor polynomial is T_3(x) = -10 + 10(x-2) + 6(x-2)^2 + (x-2)^3. The provided solution fails to construct the polynomial from the computed coefficients and presents an incorrect final expression.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/taylor_series_center, checked 2026-10-07 with SymPy 1.14.0.