Home›Calculus 3›Tangent planes›Problem 10.92

Tangent planes

Problem 10.92 · medium

Find the tangent plane to \( \displaystyle z = x^{2} y + 1 \) at the point \( \displaystyle (0, 0, 1) \).

z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
\[ \left. \frac{\partial}{\partial x} \left(x^{2} y + 1\right) \right|_{\substack{ x=0\\ y=0 }} = 0 \]
f_x at the point.✓ Proved
\[ \left. \frac{\partial}{\partial y} \left(x^{2} y + 1\right) \right|_{\substack{ x=0\\ y=0 }} = 0 \]
f_y at the point.✓ Proved
\[ 1 \]
The plane.

Answer \( z = 1 \)

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 computed partial derivatives into the tangent plane formula. Line 4 asserts the result is z=1 without showing the substitution of f_x=0 and f_y=0 into the equation from Line 1, skipping the logical step that connects the derivatives to the final answer.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to substitute the computed partial derivatives into the tangent plane formula. Line 4 asserts the result is z=1 without showing the substitution of f_x=0 and f_y=0 into the equation from Line 1, skipping the logical step that connects the derivatives to the final answer.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Line 4 claims the plane is z=1, but fails to substitute the computed partial derivatives (both 0) and the point coordinates into the formula from Line 1. While the final answer happens to be correct, the derivation is missing the crucial substitution step, making the logic incomplete.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to substitute the computed partial derivatives into the tangent plane formula. It lists the derivatives as 0 but then asserts the plane is z=1 without showing the substitution step z = 1 + 0(x-0) + 0(y-0).
gpt-oss:20b: fail (style) 2026-09-26 — Sentence 4 is unclear – it should state the plane equation z=1, not “1, 1”. The conclusion is correct, but the wording is ambiguous.

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.