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Tangent planes

Problem 10.30 · medium

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

z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
Reviewed
\[ \left. \frac{\partial}{\partial x} \left(x^{2} y + 1\right) \right|_{\substack{ x=1\\ y=-1 }} = -2 \]
f_x at the point.✓ Proved
\[ \left. \frac{\partial}{\partial y} \left(x^{2} y + 1\right) \right|_{\substack{ x=1\\ y=-1 }} = 1 \]
f_y at the point.✓ Proved
\[ - 2 x + y + 3 \]
The plane.Reviewed

Answer \( z = - 2 x + y + 3 \)

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 uses the correct formula for the tangent plane but fails to substitute the specific point values (a=1, b=-1) into the linear terms (x-a) and (y-b). This results in the equation z = -2x + y + 3, which is incorrect; the correct equation is z = -2(x-1) + 1(y+1) = -2x + y + 1.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution incorrectly calculates the partial derivative with respect to x. For f(x,y) = x^2y + 1, f_x = 2xy, so f_x(1, -1) = -2. However, the equation in line 2 claims the result is -2, but the label says 'f_x at the point' and the value -2 is actually correct for f_x(1,-1). Wait, let me re-read. f_x = 2xy. At (1, -1), 2(1)(-1) = -2. So line 2 is correct. f_y = x^2. At (1, -1), 1^2 = 1. So line 3 is correct. The formula is z = f(a,b) + f_x(x-a) + f_y(y-b). f(1,-1) = 1^2(-1)+1 = 0. So z = 0 + (-2)(x-1) + 1(y-(-1)) = -2x + 2 + y + 1 = -2x + y + 3. The stated answer is Eq(z, -2*x + y + 3). This matches. Why did I think it was wrong? Let me check the prompt's provided solution text again. Line 2 says 'Subs(Derivative(x**2*y + 1, x), (x, y), (1, -1)), -2'. This is correct. Line 3 says 'Subs(Derivative(x**2*y + 1, y), (x, y), (1, -1)), 1'. This is correct. Line 4 says '-2*x + y + 3, -2*x + y + 3'. This is the final plane equation. The setup in line 1 is correct. The calculations are correct. The final answer is correct. Is there a subtle error? The problem asks for the tangent plane. The solution provides the equation. The steps are logical. Wait, look at line 2 again. The derivative of x^2*y with respect to x is 2xy. Evaluated at (1, -1) is -2. Correct. Look at line 3. Derivative with respect to y is x^2. Evaluated at (1, -1) is 1. Correct. The plane equation is z = 0 - 2(x-1) + 1(y+1) = -2x + 2 + y + 1 = -2x + y + 3. Correct. Why would this fail? Maybe the format? No, the instructions say to check sentences. Line 1 is a sentence. It states the formula. It is correct. The equations are marked proved. I should not re-check them. But wait, if the equations are proved, I assume they are correct. If they are correct, the solution is correct. Let me look closer at the prompt's provided solution. Is it possible the user made a typo in the 'proved' lines that I'm supposed to catch? No, the prompt says 'do not re-check the algebra of a line marked proved'. So I must trust lines 2, 3, 4 are algebraically correct. Line 1 is the only sentence. It is correct. Therefore, the verdict should be pass. Let me double check the 'Stated answer'. Eq(z, -2*x + y + 3). This is z = -2x + y + 3. My derivation matches. So it is pass.
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.