Differentials
Problem 3.744 · easy
Find the differential \( \displaystyle dy \) of \( \displaystyle y = - x^{3} - x + 2 \), and evaluate it at \( \displaystyle x = 3 \), \( \displaystyle dx = \frac{1}{100} \).
- \[ \frac{d}{d x} \left(- x^{3} - x + 2\right) = - 3 x^{2} - 1 \]dy = f′(x) dx.✓ Proved
- \[ \frac{\left. - 3 x^{2} - 1 \right|_{\substack{ x=3 }}}{100} = - \frac{7}{25} \]Substitute x and dx.✓ Proved
- The exact change is -280901/1000000 ≈ -0.280901; the differential is the tangent-line estimate of it.
Answer \( dy = - 3 x^{2} - 1\,dx = - \frac{7}{25} \)
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. How we verify
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | a sentence; read, not computed |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | central difference quotient of f at x times dx |
Reviewers
gpt-oss:20b: fail (misleading) — The sentence claims the exact change is -280901/1000000 ≈ -0.280901, but the true difference f(3.01)-f(3) is -0.280301. This misleads the reader about the exact value.qwen3.6:27b-mlx: pass — The solution correctly computes the differential and evaluates it. The final sentence provides correct context regarding the relationship between the differential and the exact change, which is pedagogically sound.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-26 — The solution correctly computes the differential and evaluates it. The final sentence provides correct context regarding the relationship between the differential and the exact change, which is pedagogically sound.gpt-oss:20b: fail (misleading) 2026-10-26 — The sentence claims the exact change is -280901/1000000 ≈ -0.280901, but the true difference f(3.01)-f(3) is -0.280301. This misleads the reader about the exact value.qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly computes the differential and evaluates it. The final sentence provides correct context regarding the relationship between the differential and the exact change, without introducing errors.gpt-oss:20b: fail (misleading) 2026-10-25 — The sentence claims the exact change is -280901/1000000, which is incorrect (the true change is about -0.280301). It also incorrectly states that the differential is the tangent‑line estimate of it, which is true, but the preceding numeric claim misleads the reader.qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly computes the differential and evaluates it. The final sentence provides correct context regarding the relationship between the differential and the exact change, without introducing errors.gpt-oss:20b: fail (misleading) 2026-10-25 — The sentence claims the exact change is -280901/1000000 ≈ -0.280901, but the true change from x=3 to x=3.01 is -0.280301. The statement is therefore misleading.
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/differentials, checked 2026-10-26 with SymPy 1.14.0.