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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.

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

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 point coordinates into the tangent plane formula. It outputs the expression 4*x + 4*y - 7, which is not equal to z (the correct equation is z = 4x + 4y - 7). The constant term -7 is derived from 5 - 4(2) - 4(1), but the equation itself is missing the 'z =' part, making it mathematically incorrect as a plane equation.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to substitute the point coordinates into the tangent plane formula. It outputs the expression 4*x + 4*y - 7, which is not equal to z (the correct equation is z = 4x + 4y - 7). The constant term -7 is derived from 5 - 4(2) - 4(1), but the equation itself is missing the 'z =' part, making it mathematically incorrect as a plane equation.
gpt-oss:20b: pass 2026-09-28
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 2026-10-13 with SymPy 1.14.0.