Differentials
Problem 3.440 · medium
The radius of a circle changes from \( \displaystyle 2 \) to \( \displaystyle \frac{99}{50} \). 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
- \[ - \frac{\left. 2 \pi x \right|_{\substack{ x=2 }}}{50} = - \frac{2 \pi}{25} \]Substitute x and dx.✓ Proved
- The exact change is -199*pi/2500 ≈ -0.250071; the differential is the tangent-line estimate of it.Reviewed
Answer \( - \frac{2 \pi}{25} \)
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
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: pass 2026-10-06gpt-oss:20b: pass 2026-10-06qwen3.6:27b-mlx: fail (error) 2026-10-06 — The problem states the radius changes from 2 to 99/50 (1.98), so the change in radius dx is -0.02. The solution correctly calculates the differential as -2*pi/25, but Sentence 3 incorrectly claims the 'exact change' is -199*pi/2500. The exact change is A(1.98) - A(2) = pi(1.98^2 - 2^2) = pi(3.9204 - 4) = -0.0796pi = -199pi/2500. Wait, let me re-calculate. 1.98^2 = 3.9204. 3.9204 - 4 = -0.0796. -0.0796 = -796/10000 = -199/2500. So the exact change IS -199pi/2500. Why did I think it was an error? Let me re-read the prompt's stated answer. Stated answer: -2pi/25. -2pi/25 = -0.08pi. The exact change is -0.0796pi. The differential is -0.08pi. The sentence says 'The exact change is -199*pi/2500 ... the differential is the tangent-line estimate of it.' This is actually correct. Is there a subtle error? Let's check the sign. Radius changes from 2 to 1.98. dx = -0.02. dA = 2*pi*2*(-0.02) = -0.08pi = -2pi/25. The solution is correct. Why would it fail? Maybe the sentence is misleading? No, it's accurate. Let me look closer at the 'Stated answer'. The problem asks for dA. The solution provides dA. The sentence compares exact and differential. This is standard. Is there a typo in the sentence? '-199*pi/2500'. 199/2500 = 0.0796. Correct. Is the verdict 'pass'? Let me check the constraints again. 'Review worked calculus solutions adversarially'. Is there anything wrong? The problem says 'radius ... changes from 2 to 99/50'. 99/50 = 1.98. dx = 1.98 - 2 = -0.02. dA = 2*pi*r*dr = 2*pi*2*(-0.02) = -0.08pi. -0.08 = -8/100 = -2/25. So dA = -2pi/25. The solution is correct. Sentence 3 is correct. Verdict: pass.
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.