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Tangent planes

Problem 10.85 · medium

Find the tangent plane to \( \displaystyle z = x^{2} - y^{2} \) at the point \( \displaystyle (2, 1, 3) \).

z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
\[ \left. \frac{\partial}{\partial x} \left(x^{2} - y^{2}\right) \right|_{\substack{ x=2\\ y=1 }} = 4 \]
f_x at the point.✓ Proved
\[ \left. \frac{\partial}{\partial y} \left(x^{2} - y^{2}\right) \right|_{\substack{ x=2\\ y=1 }} = -2 \]
f_y at the point.✓ Proved
\[ 4 x - 2 y - 3 \]
The plane.

Answer \( z = 4 x - 2 y - 3 \)

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 explicitly calculate or state the value of f(2,1), which is required by the formula in line 1. While the final equation is correct, the derivation skips the crucial step of evaluating the function at the point, making the transition from the general formula to the specific plane equation logically incomplete.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly calculate or state the value of f(2,1), which is required by the formula in line 1. While the final equation is correct, the derivation skips the crucial step of evaluating the function at the point, making the transition from the general formula to the specific plane equation logically incomplete.
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 (2,1) to find the z-coordinate of the tangent plane's base. It incorrectly assumes the constant term is -3 without showing f(2,1)=3, and the final equation format is ambiguous regarding the variable z.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to evaluate the function f(x,y) at the point (2,1) to obtain the constant term in the tangent plane equation. The formula requires z = f(a,b) + ..., but the final equation 4x - 2y - 3 lacks the f(2,1) = 3 term, resulting in an incorrect plane equation (should be z = 4x - 2y - 3, wait, let me recheck. f(2,1) = 2^2 - 1^2 = 3. Formula: z = 3 + 4(x-2) - 2(y-1) = 3 + 4x - 8 - 2y + 2 = 4x - 2y - 3. The final equation is actually correct. However, the derivation in step 4 is presented as '4*x - 2*y - 3, 4*x - 2*y - 3' without showing the substitution of f(a,b) or the expansion. The sentence in step 1 states the formula, but the jump to step 4 skips the crucial step of calculating f(2,1) and substituting into the formula. While the final answer is numerically correct, the solution text implies the plane is just the linear part or skips the constant calculation entirely in the narrative flow. Let's look closer. Step 4 says 'The plane' is 4x - 2y - 3. It does not show z = ... It just gives the expression. The stated answer is Eq(z, 4*x - 2*y - 3). The solution line 4 is an equation '4*x - 2*y - 3, 4*x - 2*y - 3'. This is likely a tautology check by the CAS. The missing piece is the explicit calculation of f(2,1) and the assembly of the full equation. A student might think the constant term comes from somewhere else or is zero. The setup in step 1 is correct. The derivatives are correct. The final result is correct. But the logical step from 1, 2, 3 to 4 is missing the evaluation of f(2,1). Is this an 'error' or 'misleading'? It's a gap in the solution. The prompt asks to find what is wrong. The solution does not explicitly calculate f(2,1). If a student follows this, they might not know where the '-3' came from if they didn't do it mentally. However, usually 'error' is reserved for wrong math. This is incomplete reasoning. Let's look at the 'Stated answer'. It is Eq(z, 4*x - 2*y - 3). The solution line 4 is just the RHS. It doesn't state z = RHS. It's a bit ambiguous. But the biggest issue is the missing f(a,b) calculation. I will mark it as error because the derivation is incomplete/incorrectly presented as if the constant term wasn't needed or was derived magically.
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.