Differentials
Problem 3.591 · easy
The side of a cube changes from \( \displaystyle 10 \) to \( \displaystyle \frac{1001}{100} \). 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
- \[ \frac{\left. 3 x^{2} \right|_{\substack{ x=10 }}}{100} = 3 \]Substitute x and dx.✓ Proved
- The exact change is 3003001/1000000 ≈ 3.003; the differential is the tangent-line estimate of it.
Answer \( 3 \)
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 | ✓ 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 (error) — The exact change in volume is (10.01)^3−10^3=1003003001/1,000,000≈1003.003001, not 3.003. The sentence claiming the exact change is 3003001/1000000 is incorrect, leading to a misleading conclusion about the differential estimate.qwen3.6:27b-mlx: pass — The solution correctly applies the differential formula dV = 3x^2 dx with x=10 and dx=1/100, yielding the correct estimate of 3. The final sentence accurately contextualizes the result.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the differential formula dV = 3x^2 dx with x=10 and dx=1/100, yielding the correct estimate of 3. The final sentence accurately contextualizes the result.gpt-oss:20b: fail (error) 2026-10-11 — The exact change in volume is (10.01)^3−10^3=1003003001/1,000,000≈1003.003001, not 3.003. The sentence claiming the exact change is 3003001/1000000 is incorrect, leading to a misleading conclusion about the differential estimate.qwen3.6:27b-mlx: inconclusive 2026-10-11 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly substitutes dx = 1/100 instead of the actual change dx = 1/100. Wait, 1001/100 - 10 = 1001/100 - 1000/100 = 1/100. So dx isgpt-oss:20b: pass 2026-10-11
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 2026-10-11 with SymPy 1.14.0.