Differentials
Problem 3.439 · easy
The side of a cube changes from \( \displaystyle 12 \) to \( \displaystyle \frac{241}{20} \). Use differentials to estimate the change in its volume, \( \displaystyle dV \).
- \[ \frac{d}{d x} x^{3} = 3 x^{2} \]dy = f′(x) dx.✓ Proved
- \[ \frac{\left. 3 x^{2} \right|_{\substack{ x=12 }}}{20} = \frac{108}{5} \]Substitute x and dx.✓ Proved
- The exact change is 173521/8000 ≈ 21.6901; the differential is the tangent-line estimate of it.Reviewed
Answer \( \frac{108}{5} \)
Lines: 2 proved, 1 reviewed. 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | 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: passqwen3.6:27b-mlx: pass — The solution correctly applies the differential formula dV = 3x^2 dx with x=12 and dx=0.05. The final sentence correctly contextualizes the result as an estimate.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the differential formula dV = 3x^2 dx with x=12 and dx=0.05. The final sentence correctly contextualizes the result as an estimate.gpt-oss:20b: fail (misleading) 2026-10-06 — The sentence claiming the exact change is 173521/8000 ≈ 21.6901 is incorrect; the true exact change is (241/20)^3−12^3=13,997,521/8000≈1749.69. The differential estimate 108/5≈21.6 is correct, but the exact value stated is false.qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly applies the differential formula dV = 3x^2 dx with x=12 and dx=0.05. The final sentence correctly contextualizes the result as an estimate.
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 2026-10-06 with SymPy 1.14.0.