Differentials
Problem 3.394 · easy
The side of a cube changes from \( \displaystyle 7 \) to \( \displaystyle \frac{349}{50} \). 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
- \[ - \frac{\left. 3 x^{2} \right|_{\substack{ x=7 }}}{50} = - \frac{147}{50} \]Substitute x and dx.✓ Proved
- The exact change is -366451/125000 ≈ -2.93161; the differential is the tangent-line estimate of it.Reviewed
Answer \( - \frac{147}{50} \)
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 identifies the derivative, substitutes the initial value and the change in x (dx = 349/50 - 7 = -1/50), and calculates the differential. The final sentence correctly contextualizes the result as an estimate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the derivative, substitutes the initial value and the change in x (dx = 349/50 - 7 = -1/50), and calculates the differential. The final sentence correctly contextualizes the result as an estimate.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the formula for the differential of volume, substitutes the initial side length and the change in side length accurately, and provides a valid comparison with the exact change.gpt-oss:20b: fail (misleading) 2026-10-04 — The exact change is computed incorrectly; the correct exact change is (349/50)^3−7^3≈−3.7316, not −2.9316. The differential estimate of −147/50 is correct, but the sentence claiming the exact change is 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 2026-10-04 with SymPy 1.14.0.