Linear approximation in two variables
Problem 10.419 · hard
Find the linear approximation of \( \displaystyle f(x, y) = \ln{\left(x - 3 y \right)} \) at \( \displaystyle (7, 2) \), and use it to estimate \( \displaystyle f\left(\frac{141}{20}, \frac{19}{10}\right) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \ln{\left(x - 3 y \right)}\\\frac{\partial}{\partial y} \ln{\left(x - 3 y \right)}\end{matrix}\right] = \left[\begin{matrix}\frac{1}{x - 3 y}\\- \frac{3}{x - 3 y}\end{matrix}\right] \]f_x and f_y.✓ Proved
- \[ \left[\begin{matrix}0\\1\\-3\end{matrix}\right] \]f, f_x and f_y at the point.✓ Proved
- \[ x - 3 y - 1 \]The linearization.✓ Proved
- \[ \frac{7}{20} \]The estimate; the true value is about 0.30010.✓ Proved
Answer \( L(x, y) = x - 3 y - 1,\ f \approx \frac{7}{20} = 0.35000 \)
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 |
| 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 | L agrees with f to within O(h²) in four directions at h = 1e−5 |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly evaluate the partial derivatives at the point (7, 2) in the text, jumping from the symbolic derivatives to the linearization formula. While the final formula is correct, the intermediate step showing the evaluation f_x(7,2)=1/2 and f_y(7,2)=-3/2 is missing, making the derivation opaque and potentially misleading about how the coefficients were obtained.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly evaluate the partial derivatives at the point (7, 2) in the text, jumping from the symbolic derivatives to the linearization formula. While the final formula is correct, the intermediate step showing the evaluation f_x(7,2)=1/2 and f_y(7,2)=-3/2 is missing, making the derivation opaque and potentially misleading about how the coefficients were obtained.qwen3.6:27b-mlx: inconclusive 2026-10-08 — reviewer returned a non-objectgpt-oss:20b: pass 2026-10-08
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/linear_approx_2var, checked 2026-10-08 with SymPy 1.14.0.