Tangent planes
Problem 10.37 · hard
Find the tangent plane to \( \displaystyle z = e^{x - y} \) at the point \( \displaystyle (1, -1, e^{2}) \).
- z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
- \[ \left. \frac{\partial}{\partial x} e^{x - y} \right|_{\substack{ x=1\\ y=-1 }} = e^{2} \]f_x at the point.✓ Proved
- \[ \left. \frac{\partial}{\partial y} e^{x - y} \right|_{\substack{ x=1\\ y=-1 }} = - e^{2} \]f_y at the point.✓ Proved
- \[ \left(x - 1\right) e^{2} - \left(y + 1\right) e^{2} + e^{2} = x e^{2} - y e^{2} - e^{2} \]The plane.✓ Proved
Answer \( z = x e^{2} - y e^{2} - e^{2} \)
Lines: 3 proved, 1 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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: passqwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final equation is incorrect. The constant term should be 0, not -exp(2). The expansion of (x-1)e^2 - (y+1)e^2 + e^2 simplifies to x*e^2 - y*e^2
Every verdict on record (6)
qwen3.6:27b-mlx: inconclusive 2026-09-28 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The final equation is incorrect. The constant term should be 0, not -exp(2). The expansion of (x-1)e^2 - (y+1)e^2 + e^2 simplifies to x*e^2 - y*e^2gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the tangent plane formula, computes the partial derivatives and their values at the given point, and simplifies the resulting equation accurately.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the tangent plane formula, computes the partial derivatives and their values at the given point, and simplifies the resulting 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 2026-09-28 with SymPy 1.14.0.