Differentials
Problem 3.387 · easy
The radius of a circle changes from \( \displaystyle 11 \) to \( \displaystyle \frac{1101}{100} \). 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
- \[ \frac{\left. 2 \pi x \right|_{\substack{ x=11 }}}{100} = \frac{11 \pi}{50} \]Substitute x and dx.✓ Proved
- The exact change is 2201*pi/10000 ≈ 0.691465; the differential is the tangent-line estimate of it.Reviewed
Answer \( \frac{11 \pi}{50} \)
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 applies the differential formula dA = 2πr dr with r=11 and dr=1/100. The final sentence correctly contextualizes the result as a tangent-line estimate.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the differential formula dA = 2πr dr with r=11 and dr=1/100. The final sentence correctly contextualizes the result as a tangent-line estimate.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to explicitly state the value of dx (0.01) derived from the change in radius, making the substitution in step 2 unjustified. Additionally, the 'exact change' calculated in step 3 is incorrect; the exact change is pi * ((1101/100)^2 - 11^2) = 2201*pi/10000 is actually correct, but the solution presents it as a separate fact rather than part of the estimation process, and the prompt only asks for dA, not a comparison. However, the primary error is the missing explicit definition of dx.gpt-oss:20b: pass 2026-10-04
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.