Tangent planes
Problem 10.35 · medium
Find the tangent plane to \( \displaystyle z = x y \) at the point \( \displaystyle (0, 0, 0) \).
- z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
- \[ \left. \frac{\partial}{\partial x} x y \right|_{\substack{ x=0\\ y=0 }} = 0 \]f_x at the point.✓ Proved
- \[ \left. \frac{\partial}{\partial y} x y \right|_{\substack{ x=0\\ y=0 }} = 0 \]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 (misleading) — The final sentence claims "0, 0 -- The plane" which does not state the plane equation; it should read "z = 0". This omission could mislead a student into thinking the plane is represented by two zeros rather than the correct linear equation.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 (misleading) 2026-09-28 — The final sentence claims "0, 0 -- The plane" which does not state the plane equation; it should read "z = 0". This omission could mislead a student into thinking the plane is represented by two zeros rather than the correct linear equation.qwen3.6:27b-mlx: pass 2026-09-27 — 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 (misleading) 2026-09-27 — The final step incorrectly states "0, 0" as the plane; it should provide the full equation z = 0, derived from f(a,b)=0 and the zero partial derivatives. The conclusion is not properly justified.qwen3.6:27b-mlx: pass 2026-09-26 — 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) 2026-09-26 — The solution never writes the plane equation; it only lists "0, 0" which does not represent the tangent plane. The correct tangent plane at (0,0,0) is z = 0, but the solution omits it and mislabels the final line.
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.