Maximum rate of change
Problem 10.338 · medium
Find the maximum rate of change of \( \displaystyle f(x, y) = e^{x} \sin{\left(y \right)} \) at \( \displaystyle (-1, 2) \), and the direction in which it occurs.
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} e^{x} \sin{\left(y \right)}\\\frac{\partial}{\partial y} e^{x} \sin{\left(y \right)}\end{matrix}\right] = \left[\begin{matrix}e^{x} \sin{\left(y \right)}\\e^{x} \cos{\left(y \right)}\end{matrix}\right] \]∇f.✓ Proved
- \[ \left[\begin{matrix}\frac{\sin{\left(2 \right)}}{e}\\\frac{\cos{\left(2 \right)}}{e}\end{matrix}\right] \]∇f(-1, 2).✓ Proved
- \[ \sqrt{\frac{\cos^{2}{\left(2 \right)}}{e^{2}} + \frac{\sin^{2}{\left(2 \right)}}{e^{2}}} = e^{-1} \]The maximum rate of change is ‖∇f‖, in the direction of ∇f.✓ Proved
Answer \( \|\nabla f\| = e^{-1}\ \text{in the direction of}\ \left\langle \frac{\sin{\left(2 \right)}}{e}, \frac{\cos{\left(2 \right)}}{e} \right\rangle \)
Every line of this solution was proved by the computer algebra system SymPy. 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.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ 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 largest directional difference quotient over 3600 directions |
Reviewers
gpt-oss:20b: fail (misleading) — The direction given is not a unit vector; it should be <sin 2, cos 2>, not <sin 2/e, cos 2/e>.qwen3.6:27b-mlx: pass
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: fail (misleading) 2026-10-04 — The direction given is not a unit vector; it should be <sin 2, cos 2>, not <sin 2/e, cos 2/e>.qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the gradient, evaluates it at the given point, and identifies the maximum rate of change as the magnitude of the gradient and the direction as the gradient vector itself. The final answer matches the stated result.gpt-oss:20b: fail (misleading) 2026-10-04 — The solution claims the direction of maximum increase is the vector ⟨sin 2/e, cos 2/e⟩, which is not a unit vector. The correct unit direction is ⟨sin 2, cos 2⟩, obtained by normalizing ∇f at (–1,2).
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/max_rate_of_change, checked 2026-10-04 with SymPy 1.14.0.