Tangent planes
Problem 10.33 · medium
Find the tangent plane to \( \displaystyle z = x^{2} + y^{2} \) at the point \( \displaystyle (1, 1, 2) \).
- 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
- \[ 2 x + 2 y - 2 \]The plane.
Answer \( z = 2 x + 2 y - 2 \)
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 states the plane as "2*x + 2*y - 2" without the z‑variable, which could mislead a student into thinking the plane equation is incomplete or incorrect.qwen3.6:27b-mlx: fail (error) — The solution fails to explicitly calculate or state the value of f(a,b) = 2, which is required by the formula in step 1 to derive the final equation. While the final algebraic result is correct, the logical step from the formula to the specific plane equation is missing the constant term's derivation.
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) = 2, which is required by the formula in step 1 to derive the final equation. While the final algebraic result is correct, the logical step from the formula to the specific plane equation is missing the constant term's derivation.gpt-oss:20b: fail (misleading) 2026-09-28 — The final sentence states the plane as "2*x + 2*y - 2" without the z‑variable, which could mislead a student into thinking the plane equation is incomplete or incorrect.qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly calculate or state the value of the function f(1,1) = 2, which is required by the formula in step 1. While the final equation is correct, the derivation skips the evaluation of the constant term, making the logical connection between the formula and the result incomplete.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly calculate or state the value of f(1,1) = 2, which is required by the formula in step 1. Consequently, step 4 presents the expression for the plane's height but does not show the assembly of the full equation z = 2*x + 2*y - 2 from the components f(a,b), f_x, and f_y.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.