Tangent planes
Problem 10.26 · 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 (misleading) — The solution omits the constant term f(a,b)=0 and gives an incomplete plane equation “0, 0” instead of the correct z=0. This could mislead a student about how to assemble the tangent plane formula.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 solution omits the constant term f(a,b)=0 and gives an incomplete plane equation “0, 0” instead of the correct z=0. This could mislead a student about how to assemble the tangent plane formula.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 (error) 2026-09-27 — The solution omits the constant term f(a,b) and does not produce a proper plane equation; it incorrectly concludes with "0,0" instead of the correct tangent plane 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 step 1. Step 4 jumps to the final result '0' without showing the substitution of f(0,0)=0, f_x=0, and f_y=0 into the formula, making the derivation incomplete and logically disconnected.gpt-oss:20b: fail (error) 2026-09-26 — The solution omits computing f(a,b)=f(0,0)=0 and incorrectly concludes the plane as "0, 0" instead of the correct equation z=0. The missing step and unclear final statement make the solution incorrect.
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.