Home›Calculus 3›Tangent planes›Problem 10.87

Tangent planes

Problem 10.87 · medium

Find the tangent plane to \( \displaystyle z = e^{x - y} \) at the point \( \displaystyle (0, 0, 1) \).

z = f(a,b) + f_x(a,b)(x − a) + f_y(a,b)(y − b).
Reviewed
\[ \left. \frac{\partial}{\partial x} e^{x - y} \right|_{\substack{ x=0\\ y=0 }} = 1 \]
f_x at the point.✓ Proved
\[ \left. \frac{\partial}{\partial y} e^{x - y} \right|_{\substack{ x=0\\ y=0 }} = -1 \]
f_y at the point.✓ Proved
\[ x - y + 1 \]
The plane.Reviewed

Answer \( z = x - y + 1 \)

✓ Nihil obstat Lines: 2 proved, 2 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the formula for the tangent plane, computes the partial derivatives and their values at the point accurately, and substitutes them to obtain the correct equation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the formula for the tangent plane, computes the partial derivatives and their values at the point accurately, and substitutes them to obtain the correct equation.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the tangent plane formula and computes the partial derivatives accurately. The final equation matches the stated answer.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the tangent plane formula, computes the partial derivatives accurately, and substitutes the point values to arrive at the correct equation.
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.