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Tangent planes

Problem 10.84 · medium

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

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

Answer \( z = 2 x - 4 y + 3 \)

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 explicitly calculate or state the value of f(a,b) = -3, which is required by the formula in step 1. While the final equation is correct, the derivation skips the crucial step of evaluating the function at the point, making the transition from steps 2/3 to step 4 logically incomplete and potentially confusing for a student.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly calculate or state the value of f(a,b) = -3, which is required by the formula in step 1. While the final equation is correct, the derivation skips the crucial step of evaluating the function at the point, making the transition from steps 2/3 to step 4 logically incomplete and potentially confusing for a student.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly calculate or state the value of f(a,b) = -3, which is required by the formula in step 1 to derive the constant term in step 4. While the final equation is correct, the logical gap makes it impossible to verify that the constant '3' was derived correctly from the formula rather than guessed.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly calculate or state the value of f(1, 2) = -3, which is required by the formula in step 1. While the final constant in step 4 is correct, the derivation skips the evaluation of the function at the point, making the transition from the formula to the result unjustified.
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.