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Tangent planes

Problem 10.34 · medium

Find the tangent plane to \( \displaystyle z = x^{2} - y^{2} \) at the point \( \displaystyle (1, 1, 0) \).

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

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

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 evaluate the function f(a,b) at the point (1,1), which is 0, and thus omits the constant term in the tangent plane equation. The correct equation is z = 2x - 2y, but the derivation in step 4 implies z = 2x - 2y without showing the addition of f(1,1), making the logic incomplete and potentially confusing if f(a,b) were non-zero.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to evaluate the function f(a,b) at the point (1,1), which is 0, and thus omits the constant term in the tangent plane equation. The correct equation is z = 2x - 2y, but the derivation in step 4 implies z = 2x - 2y without showing the addition of f(1,1), making the logic incomplete and potentially confusing if f(a,b) were non-zero.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to evaluate the function f(a,b) at the point (1,1), which is 0. While the final equation z = 2x - 2y happens to be correct because f(1,1)=0, the derivation skips the constant term in the tangent plane formula, implying it is always zero or ignoring the general structure z = f(a,b) + ... .
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to evaluate f(a,b) at the point (1,1), which is 0, and does not explicitly include the terms (x-a) and (y-b) in the final equation construction, skipping the substitution step required by the formula in line 1.
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.