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Tangent planes

Problem 10.28 · medium

Find the tangent plane to \( \displaystyle z = x y \) at the point \( \displaystyle (1, 1, 1) \).

z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
\[ \left. \frac{\partial}{\partial x} x y \right|_{\substack{ x=1\\ y=1 }} = 1 \]
f_x at the point.✓ Proved
\[ \left. \frac{\partial}{\partial y} x y \right|_{\substack{ x=1\\ y=1 }} = 1 \]
f_y at the point.✓ Proved
\[ x + y - 1 \]
The plane.

Answer \( z = x + y - 1 \)

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to substitute the function value f(1,1)=1 into the tangent plane formula. The correct equation is z = 1 + 1(x-1) + 1(y-1), which simplifies to z = x + y - 1. However, line 1 presents the general formula, and line 4 jumps to the result without showing the substitution of f(a,b), making the derivation incomplete and potentially confusing regarding the constant term.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to substitute the function value f(1,1)=1 into the tangent plane formula. The correct equation is z = 1 + 1(x-1) + 1(y-1), which simplifies to z = x + y - 1. However, line 1 presents the general formula, and line 4 jumps to the result without showing the substitution of f(a,b), making the derivation incomplete and potentially confusing regarding the constant term.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly calculate or state the value of f(1,1) required by the tangent plane formula in step 1. While the final algebraic result is correct, the derivation skips the evaluation of the function at the point of tangency, making the logical connection between the formula and the result incomplete.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly calculate or substitute the function value f(1,1) = 1 into the tangent plane formula. While the final equation is correct, the derivation skips the constant term calculation, making the transition from the formula to the result logically incomplete.
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.