Differentials
Problem 3.391 · medium
The radius of a circle changes from \( \displaystyle 9 \) to \( \displaystyle \frac{449}{50} \). Use differentials to estimate the change in its area, \( \displaystyle dA \).
- \[ \frac{d}{d x} \pi x^{2} = 2 \pi x \]dy = f′(x) dx.✓ Proved
- \[ - \frac{\left. 2 \pi x \right|_{\substack{ x=9 }}}{50} = - \frac{9 \pi}{25} \]Substitute x and dx.✓ Proved
- The exact change is -899*pi/2500 ≈ -1.12972; the differential is the tangent-line estimate of it.Reviewed
Answer \( - \frac{9 \pi}{25} \)
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 differential formula dA = 2πr dr, substitutes r=9 and dr=-1/50, and obtains the correct result. The final sentence accurately describes the relationship between the differential and the exact change.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly identifies the differential formula dA = 2πr dr, substitutes r=9 and dr=-1/50, and obtains the correct result. The final sentence accurately describes the relationship between the differential and the exact change.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the differential formula dA = 2πr dr with r=9 and dr=-1/50. The final sentence correctly identifies the differential as a linear approximation of the exact change.gpt-oss:20b: fail (misleading) 2026-10-04 — The exact change is computed incorrectly; it should be -901π/2500, not -899π/2500. The differential estimate -9π/25 is correct, but the sentence claiming the exact change is wrong misleads the reader.
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.