Differentials
Problem 3.536 · easy
The side of a cube changes from \( \displaystyle 3 \) to \( \displaystyle \frac{301}{100} \). 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
- \[ \frac{\left. 3 x^{2} \right|_{\substack{ x=3 }}}{100} = \frac{27}{100} \]Substitute x and dx.✓ Proved
- The exact change is 270901/1000000 ≈ 0.270901; the differential is the tangent-line estimate of it.Reviewed
Answer \( \frac{27}{100} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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=3 and dx=0.01. The final sentence correctly identifies the differential as a tangent-line estimate and provides the correct exact change for context.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-09qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the differential formula dV = 3x^2 dx with x=3 and dx=0.01. The final sentence correctly identifies the differential as a tangent-line estimate and provides the correct exact change for context.qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the differential formula dV = 3x^2 dx with x=3 and dx=0.01. The final sentence accurately contextualizes the result as a tangent-line estimate.gpt-oss:20b: pass 2026-10-09
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 2026-10-09 with SymPy 1.14.0.