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Tangent planes

Problem 10.25 · medium

Find the tangent plane to \( \displaystyle z = x^{2} y + 1 \) at the point \( \displaystyle (1, 2, 3) \).

z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
\[ \left. \frac{\partial}{\partial x} \left(x^{2} y + 1\right) \right|_{\substack{ x=1\\ y=2 }} = 4 \]
f_x at the point.✓ Proved
\[ \left. \frac{\partial}{\partial y} \left(x^{2} y + 1\right) \right|_{\substack{ x=1\\ y=2 }} = 1 \]
f_y at the point.✓ Proved
\[ 4 x + y - 3 \]
The plane.

Answer \( z = 4 x + y - 3 \)

Lines: 2 proved, 2 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. How we verify

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	Not checked	—	both sides are the same expression: the line states a value, and the working that produced it is not on the line
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 partial derivatives and point coordinates into the tangent plane formula from line 1. Line 4 presents the final equation without showing the necessary algebraic steps connecting it to the formula and the computed derivatives.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to substitute the partial derivatives and point coordinates into the tangent plane formula from line 1. Line 4 presents the final equation without showing the necessary algebraic steps connecting it to the formula and the computed derivatives.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to substitute the partial derivatives and point coordinates into the tangent plane formula from line 1. Line 4 presents the final equation but skips the necessary algebraic step of substituting f_x=4, f_y=1, a=1, b=2, and z_0=3 into the formula, making the derivation incomplete and logically disconnected.
gpt-oss:20b: fail (misleading) 2026-09-27 — The final sentence states the plane as "4*x + y - 3" without the z‑variable, so a student would incorrectly think that is the equation of the plane rather than z = 4x + y - 3.
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to substitute the partial derivatives into the tangent plane formula from step 1. It lists the derivatives and the final equation but omits the crucial algebraic step of combining them (i.e., z = 3 + 4(x-1) + 1(y-2)), making the derivation logically incomplete.
gpt-oss:20b: pass 2026-09-26

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 2026-10-13 with SymPy 1.14.0.