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Differentials

Problem 3.535 · medium

The radius of a circle changes from \( \displaystyle 11 \) to \( \displaystyle \frac{109}{10} \). 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
\[ - \frac{\left. 2 \pi x \right|_{\substack{ x=11 }}}{10} = - \frac{11 \pi}{5} \]
Substitute x and dx.✓ Proved
The exact change is -219*pi/100 ≈ -6.88009; the differential is the tangent-line estimate of it.
Reviewed

Answer \( - \frac{11 \pi}{5} \)

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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the differential formula dA = 2πr dr with r=11 and dr=-1/10. The final sentence accurately contextualizes the result as a linear approximation.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-09
qwen3.6:27b-mlx: pass 2026-10-09 — The solution correctly applies the differential formula dA = 2πr dr with r=11 and dr=-1/10. The final sentence accurately contextualizes the result as a linear approximation.
qwen3.6:27b-mlx: fail (error) 2026-10-09 — The problem states the radius changes from 11 to 10.9, so dx = -0.1. The solution incorrectly uses dx = -0.1 (implied by the division by 10 and negative sign) but the final answer is negative, implying a decrease. However, the radius decreases, so the area should decrease, meaning dA should be negative. Wait, let's re-read carefully. Radius goes from 11 to 10.9. dx = 10.9 - 11 = -0.1. dA = 2*pi*r*dx = 2*pi*11*(-0.1) = -2.2*pi = -11*pi/5. The calculation is actually correct. Why did I think it was wrong? Let me re-read the prompt's stated answer. Stated answer: -11*pi/5. The solution calculates -11*pi/5. The sentence says 'The exact change is -219*pi/100'. Let's check exact change. A(11) = 121pi. A(10.9) = 10.9^2 * pi = 118.81 * pi. Change = 118.81pi - 121pi = -2.19pi = -219/100 pi. This is correct. The differential estimate is -2.2pi = -220/100 pi. The sentence is just a comment. Is there an error? The prompt asks to review the solution. The solution lines are: 1. Derivative. 2. Substitution. 3. Sentence. The substitution line says '-Subs(2*pi*x, x, 11)/10'. This implies dx = -1/10. This is correct. The result is -11pi/5. This matches the stated answer. The sentence is correct. So why would it fail? Maybe the 'unchecked' equations are not the issue. The issue might be that the solution doesn't explicitly state dx = -0.1 in the text, but the equation shows the substitution. Is that misleading? No. Is there a sign error in the problem statement interpretation? 'Changes from 11 to 109/10'. 109/10 = 10.9. Decrease. dA should be negative. The answer is negative. Everything seems correct. Wait, look at line 2: '-Subs(2*pi*x, x, 11)/10'. The negative sign is explicit. This corresponds to dx = -1/10. This is correct. The verdict should be pass. Let me double check if I missed a subtle error. 'Use differentials to estimate the change in its area, dA.' The solution provides dA. The sentence compares it to the exact change. This is good practice. I see no error.
gpt-oss:20b: pass 2026-10-09

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 2026-10-09 with SymPy 1.14.0.