Tangent planes
Problem 10.36 · medium
Find the tangent plane to \( \displaystyle z = x^{2} - y^{2} \) at the point \( \displaystyle (0, 0, 0) \).
- z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
- \[ \left. \frac{\partial}{\partial x} \left(x^{2} - y^{2}\right) \right|_{\substack{ x=0\\ y=0 }} = 0 \]f_x at the point.✓ Proved
- \[ \left. \frac{\partial}{\partial y} \left(x^{2} - y^{2}\right) \right|_{\substack{ x=0\\ y=0 }} = 0 \]f_y at the point.✓ Proved
- \[ 0 \]The plane.
Answer \( z = 0 \)
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: fail (error) — The solution never substitutes the point (0,0) into the general tangent‑plane formula, and the final line “0, 0” does not correctly state the plane equation (it should be z = 0).qwen3.6:27b-mlx: pass — The solution correctly applies the tangent plane formula and computes the partial derivatives at the origin, leading to the correct result z=0.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the tangent plane formula and computes the partial derivatives at the origin, leading to the correct result z=0.gpt-oss:20b: fail (error) 2026-09-28 — The solution never substitutes the point (0,0) into the general tangent‑plane formula, and the final line “0, 0” does not correctly state the plane equation (it should be z = 0).qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to substitute the computed partial derivatives into the tangent plane formula. It lists the derivatives and then abruptly states the final equation without showing the substitution step z = 0 + 0(x-0) + 0(y-0).gpt-oss:20b: inconclusive 2026-09-27 — reviewer response could not be parsed: {"verdict":"fail","severity":"error","notes":"The final step incorrectly states the tangent plane as \"0, 0\" instead of the correct equation \(z=0\)."}qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly calculate or state the value of f(0,0) required by the tangent plane formula in line 1. While the final result is correct because f(0,0) happens to be 0, the derivation skips a necessary term, making the logic incomplete.gpt-oss:20b: fail (misleading) 2026-09-26 — The final sentence claims the plane is "0, 0" which is ambiguous and does not correctly state the tangent plane equation. The correct tangent plane at (0,0,0) is z = 0, but the solution fails to present it clearly.
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.