Differentials
Problem 3.594 · easy
The radius of a sphere changes from \( \displaystyle 2 \) to \( \displaystyle \frac{41}{20} \). Use differentials to estimate the change in its volume, \( \displaystyle dV \).
- \[ \frac{d}{d x} \frac{4 \pi x^{3}}{3} = 4 \pi x^{2} \]dy = f′(x) dx.✓ Proved
- \[ \frac{\left. 4 \pi x^{2} \right|_{\substack{ x=2 }}}{20} = \frac{4 \pi}{5} \]Substitute x and dx.✓ Proved
- The exact change is 4921*pi/6000 ≈ 2.57663; the differential is the tangent-line estimate of it.Reviewed
Answer \( \frac{4 \pi}{5} \)
✓ 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 identifies the derivative of the volume formula, substitutes the correct values for radius and change in radius, and provides a valid comparison with the exact change.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the derivative of the volume formula, substitutes the correct values for radius and change in radius, and provides a valid comparison with the exact change.gpt-oss:20b: pass 2026-10-11qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the derivative of the volume formula, substitutes the correct values for radius and change in radius, and accurately calculates the differential. The final sentence correctly contextualizes the differential as an estimate of the exact change.gpt-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.