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Tangent planes

Problem 10.186 · medium

Find the tangent plane to \( \displaystyle z = x^{2} y + 1 \) at the point \( \displaystyle (2, 1, 5) \).

z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
\[ \left. \frac{\partial}{\partial x} \left(x^{2} y + 1\right) \right|_{\substack{ x=2\\ y=1 }} = 4 \]
f_x at the point.✓ Proved
\[ \left. \frac{\partial}{\partial y} \left(x^{2} y + 1\right) \right|_{\substack{ x=2\\ y=1 }} = 4 \]
f_y at the point.✓ Proved
\[ 4 x + 4 y - 7 \]
The plane.✓ Proved

Answer \( z = 4 x + 4 y - 7 \)

Lines: 3 proved, 1 not checked. 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	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
3	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
4	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the gap between surface and plane shrinks quadratically near the point

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to substitute the point coordinates into the linearization formula. It provides the partial derivatives but does not show the substitution of (a,b) = (2,1) and z_0 = 5 into the equation z = 5 + 4(x-2) + 4(y-1), which is necessary to derive the final result.

Every verdict on record (4)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to substitute the point coordinates into the linearization formula. It provides the partial derivatives but does not show the substitution of (a,b) = (2,1) and z_0 = 5 into the equation z = 5 + 4(x-2) + 4(y-1), which is necessary to derive the final result.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the formula for the tangent plane, computes the partial derivatives and their values at the given point, and substitutes them into the formula to derive the correct equation.
gpt-oss:20b: pass 2026-09-27

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/tangent_plane, checked 2026-09-27 with SymPy 1.14.0.