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Differentials

Problem 3.443 · easy

The side of a cube changes from \( \displaystyle 8 \) to \( \displaystyle \frac{399}{50} \). 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
\[ - \frac{\left. 3 x^{2} \right|_{\substack{ x=8 }}}{50} = - \frac{96}{25} \]
Substitute x and dx.✓ Proved
The exact change is -478801/125000 ≈ -3.83041; the differential is the tangent-line estimate of it.
Reviewed

Answer \( - \frac{96}{25} \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the formula for the differential, substitutes the correct values for x and dx, and provides a valid comparison with the exact change.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the formula for the differential, substitutes the correct values for x and dx, and provides a valid comparison with the exact change.
gpt-oss:20b: pass 2026-10-06
qwen3.6:27b-mlx: pass 2026-10-06 — The solution correctly identifies the differential formula, substitutes the given values (including the negative change in side length), and provides a correct sanity check comparing the estimate to the exact change.

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.