Home›Calculus 3›Tangent planes›Problem 10.90

Tangent planes

Problem 10.90 · medium

Find the tangent plane to \( \displaystyle z = x y \) at the point \( \displaystyle (2, 1, 2) \).

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

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

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: pass
qwen3.6:27b-mlx: fail (error) — The solution fails to substitute the function value f(2,1) = 2 into the tangent plane formula. It incorrectly sets the constant term to 0 (implied by the result x + 2y - 2) instead of 2, resulting in the plane z = x + 2y - 2 instead of the correct z = x + 2y - 2 + 2 = x + 2y.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to substitute the function value f(2,1) = 2 into the tangent plane formula. It incorrectly sets the constant term to 0 (implied by the result x + 2y - 2) instead of 2, resulting in the plane z = x + 2y - 2 instead of the correct z = x + 2y - 2 + 2 = x + 2y.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to substitute the point coordinates (a, b) = (2, 1) and the function value f(2,1) = 2 into the tangent plane formula. It incorrectly equates the expression x + 2y - 2 with the plane equation, omitting the constant term derived from f(a,b) - f_x(a,b)a - f_y(a,b)b, which results in the wrong plane z = x + 2y - 2 instead of z = x + 2y - 2 (wait, let me re-calculate: z = 2 + 1(x-2) + 2(y-1) = 2 + x - 2 + 2y - 2 = x + 2y - 2. Actually, the algebra in line 4 is correct for the final expression, but the sentence in line 1 is just a formula, and line 4 jumps to the result without showing the substitution step. However, looking closely at line 4: 'x + 2*y - 2, x + 2*y - 2'. This is an identity check, not a derivation. The solution skips the crucial step of substituting a=2, b=1, f(a,b)=2 into the formula from line 1. A student would not learn how to apply the formula. But is it 'wrong'? The final answer is correct. The prompt asks to find what is wrong. The derivation is missing. Is that an error? Yes, it's a logical gap. But let's look closer. Line 1 states the formula. Lines 2 and 3 compute partials. Line 4 states the plane. It never uses the formula from line 1 with the values from 2 and 3. It's a non-sequitur. However, usually 'error' implies the math is wrong. The math is right, the logic is skipped. Let's look at the 'Stated answer': Eq(z, x + 2*y - 2). The solution line 4 is 'x + 2*y - 2, x + 2*y - 2'. This looks like a CAS check that the expression is equal to itself. It does not derive the plane. This is a 'misleading' or 'error' in reasoning. Let's call it 'error' because the solution does not actually solve the problem, it just asserts the answer.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to substitute the partial derivatives and the point coordinates into the tangent plane formula from step 1. It jumps directly to the final equation without showing the substitution z = 2 + 1(x-2) + 2(y-1), making the derivation incomplete and logically disconnected.
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.