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Tangent planes

Problem 10.27 · 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).
Reviewed
\[ \left. \frac{\partial}{\partial x} \left(x^{2} + y^{2}\right) \right|_{\substack{ x=1\\ y=-1 }} = 2 \]
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.Reviewed

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

Lines: 2 proved, 2 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly applies the tangent plane formula and computes the partial derivatives and final equation accurately.

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 and final equation accurately.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to evaluate the function f(x,y) at the point (1, -1) to find the z-coordinate of the tangent plane's offset. The formula in line 1 requires f(a,b), but line 4 jumps directly to the linear terms, resulting in an incorrect constant term (-2 instead of 0).
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly calculate or state the value of f(a,b) = 2, which is required by the formula in step 1. While the final equation is correct, the derivation skips the evaluation of the function at the point, making the transition from the formula to the final equation logically incomplete.
gpt-oss:20b: fail (misleading) 2026-09-26 — The first sentence incorrectly states the linearization formula by omitting the constant term f(a,b). While the subsequent algebra yields the correct plane, the sentence could mislead a student into thinking the constant term is unnecessary.

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.