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Tangent planes

Problem 10.24 · medium

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

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

Answer \( z = 2 x + y - 1 \)

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. Line 4 presents the expression `2*x + y - 1` as the plane, but the correct equation is `z = 2(x-1) + 1(y-1) + 2`, which simplifies to `z = 2x + y - 1`. The provided answer `Eq(z, 2*x + y - 1)` is technically the correct equation, but the derivation in line 4 is missing the crucial step of substituting into the formula from line 1, making the logic jump unjustified and potentially confusing for a student expecting to see the substitution `z = 2 + 2(x-1) + 1(y-1)`. However, looking closer, line 4 just states the final equation. The real error is that the solution does not explicitly show the substitution step `z = 2 + 2(x-1) + 1(y-1)` which leads to `z = 2x + y - 1`. While the final answer is correct, the derivation skips the application of the formula stated in line 1. A student might not see how lines 2 and 3 lead to line 4 without the intermediate substitution. This is a 'misleading' or 'style' issue at best, but since the prompt asks for adversarial review, the lack of explicit substitution is a gap in the logical flow. Let's look at the options. Is it an error? The final answer is correct. Is it misleading? It might imply that `2*x + y - 1` is derived directly without the `z = ...` structure or the expansion. Actually, the biggest issue is that line 4 is labeled as an equation that is 'proved', but it's just the final answer. The sentence in line 1 is the formula. Lines 2 and 3 are values. Line 4 is the result. The connection is missing. It's a 'style' or 'misleading' fail because it doesn't show the work. But wait, if I have to choose between error, misleading, style. The math is correct. The logic is incomplete. I will call it 'misleading' because it hides the substitution step.

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. Line 4 presents the expression `2*x + y - 1` as the plane, but the correct equation is `z = 2(x-1) + 1(y-1) + 2`, which simplifies to `z = 2x + y - 1`. The provided answer `Eq(z, 2*x + y - 1)` is technically the correct equation, but the derivation in line 4 is missing the crucial step of substituting into the formula from line 1, making the logic jump unjustified and potentially confusing for a student expecting to see the substitution `z = 2 + 2(x-1) + 1(y-1)`. However, looking closer, line 4 just states the final equation. The real error is that the solution does not explicitly show the substitution step `z = 2 + 2(x-1) + 1(y-1)` which leads to `z = 2x + y - 1`. While the final answer is correct, the derivation skips the application of the formula stated in line 1. A student might not see how lines 2 and 3 lead to line 4 without the intermediate substitution. This is a 'misleading' or 'style' issue at best, but since the prompt asks for adversarial review, the lack of explicit substitution is a gap in the logical flow. Let's look at the options. Is it an error? The final answer is correct. Is it misleading? It might imply that `2*x + y - 1` is derived directly without the `z = ...` structure or the expansion. Actually, the biggest issue is that line 4 is labeled as an equation that is 'proved', but it's just the final answer. The sentence in line 1 is the formula. Lines 2 and 3 are values. Line 4 is the result. The connection is missing. It's a 'style' or 'misleading' fail because it doesn't show the work. But wait, if I have to choose between error, misleading, style. The math is correct. The logic is incomplete. I will call it 'misleading' because it hides the substitution step.
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. Line 4 presents the expression 2*x + y - 1, which is not equal to z (the correct equation is z = 2x + y - 1). The setup in line 1 is correct, but the execution in line 4 is algebraically incomplete and does not follow from the previous lines.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to substitute the partial derivatives and the point coordinates into the tangent plane formula. It presents the final expression `2*x + y - 1` as the plane equation without showing the derivation `z = 2 + 2(x-1) + 1(y-1)`, which simplifies to `z = 2x + y - 1`. The stated answer format `Eq(z, ...)` is also non-standard for a plane equation, but the primary error is the missing logical step connecting the derivatives to the final equation.
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.