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Tangent planes

Problem 10.32 · medium

Find the tangent plane to \( \displaystyle z = x^{2} + y^{2} \) at the point \( \displaystyle (2, 1, 5) \).

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
\[ 4 x + 2 y - 5 \]
The plane.

Answer \( z = 4 x + 2 y - 5 \)

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 evaluate the function f(x,y) at the point (2,1) to get z_0 = 5, which is required for the tangent plane formula. Consequently, the constant term in the final equation is incorrect (-5 instead of -5 derived from 5 - 4(2) - 2(1) = -5? Wait. Let's re-calculate. z = 5. Formula: z = 5 + 4(x-2) + 2(y-1) = 5 + 4x - 8 + 2y - 2 = 4x + 2y - 5. The final equation is actually correct. However, the derivation skips the explicit calculation of f(2,1)=5, jumping from derivatives to the final plane equation without showing the substitution into the formula in line 1. Line 4 just asserts the result. While the result is numerically correct, the logical step of substituting a=2, b=1, f(a,b)=5 into the formula from line 1 is missing. Is this an error? The prompt asks to check if conclusions follow from lines before. Line 4 does not follow from lines 1-3 alone without the value of f(2,1). It's a missing step. But is it an 'error' in the solution provided? The solution is incomplete. A student might think they don't need f(a,b). This is misleading/error. Let's look closer. Line 1 gives the formula. Lines 2 and 3 give partials. Line 4 gives the answer. The value f(2,1) is never calculated. The student might not know they need it. This is a 'misleading' or 'error' in the pedagogical flow. I will mark it as error because the derivation is logically incomplete.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to evaluate the function f(x,y) at the point (2,1) to get z_0 = 5, which is required for the tangent plane formula. Consequently, the constant term in the final equation is incorrect (-5 instead of -5 derived from 5 - 4(2) - 2(1) = -5? Wait. Let's re-calculate. z = 5. Formula: z = 5 + 4(x-2) + 2(y-1) = 5 + 4x - 8 + 2y - 2 = 4x + 2y - 5. The final equation is actually correct. However, the derivation skips the explicit calculation of f(2,1)=5, jumping from derivatives to the final plane equation without showing the substitution into the formula in line 1. Line 4 just asserts the result. While the result is numerically correct, the logical step of substituting a=2, b=1, f(a,b)=5 into the formula from line 1 is missing. Is this an error? The prompt asks to check if conclusions follow from lines before. Line 4 does not follow from lines 1-3 alone without the value of f(2,1). It's a missing step. But is it an 'error' in the solution provided? The solution is incomplete. A student might think they don't need f(a,b). This is misleading/error. Let's look closer. Line 1 gives the formula. Lines 2 and 3 give partials. Line 4 gives the answer. The value f(2,1) is never calculated. The student might not know they need it. This is a 'misleading' or 'error' in the pedagogical flow. I will mark it as error because the derivation is logically incomplete.
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(a,b) = 5, which is required by the formula in step 1. While the final equation is correct, the derivation skips the evaluation of the function at the point, making the logical connection between the formula and the result incomplete.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the tangent plane formula and computes the partial derivatives and final equation accurately.
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.