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Differentials

Problem 3.593 · medium

The radius of a circle changes from \( \displaystyle 9 \) to \( \displaystyle \frac{89}{10} \). 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
\[ - \frac{\left. 2 \pi x \right|_{\substack{ x=9 }}}{10} = - \frac{9 \pi}{5} \]
Substitute x and dx.✓ Proved
The exact change is -179*pi/100 ≈ -5.62345; the differential is the tangent-line estimate of it.
Reviewed

Answer \( - \frac{9 \pi}{5} \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the differential formula dA = 2πr dr with r=9 and dr=-1/10. The final sentence correctly contextualizes the result as a linear approximation.

Every verdict on record (4)

qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly applies the differential formula dA = 2πr dr with r=9 and dr=-1/10. The final sentence correctly contextualizes the result as a linear approximation.
gpt-oss:20b: pass 2026-10-11
qwen3.6:27b-mlx: pass 2026-10-11 — The solution correctly identifies the derivative, substitutes the initial radius and the change in radius (dx = -1/10), and calculates the correct differential estimate. The final sentence provides context without introducing errors.
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.