Tangent planes
Problem 10.88 · hard
Find the tangent plane to \( \displaystyle z = e^{x - y} \) at the point \( \displaystyle (1, 2, e^{-1}) \).
- z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).Reviewed
- \[ \left. \frac{\partial}{\partial x} e^{x - y} \right|_{\substack{ x=1\\ y=2 }} = e^{-1} \]f_x at the point.✓ Proved
- \[ \left. \frac{\partial}{\partial y} e^{x - y} \right|_{\substack{ x=1\\ y=2 }} = - \frac{1}{e} \]f_y at the point.✓ Proved
- \[ \frac{x - 1}{e} - \frac{y - 2}{e} + e^{-1} = \frac{x}{e} - \frac{y}{e} + \frac{2}{e} \]The plane.✓ Proved
Answer \( z = \frac{x}{e} - \frac{y}{e} + \frac{2}{e} \)
✓ Nihil obstat Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 | ✓ 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: pass — 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.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-09-28 — 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 2026-09-28qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the tangent plane formula, computes the partial derivatives and their values at the point accurately, and simplifies the equation to match the stated answer.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.