Linear approximation in two variables
Problem 10.477 · 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 \( \displaystyle f\left(\frac{141}{20}, \frac{41}{20}\right) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \ln{\left(x - 3 y \right)}\\\frac{\partial}{\partial y} \ln{\left(x - 3 y \right)}\end{matrix}\right] = \left[\begin{matrix}\frac{1}{x - 3 y}\\- \frac{3}{x - 3 y}\end{matrix}\right] \]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
- \[ - \frac{1}{10} \]The estimate; the true value is about -0.10536.✓ Proved
Answer \( L(x, y) = x - 3 y - 1,\ f \approx - \frac{1}{10} = -0.10000 \)
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 state the linearization formula $L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$, making the derivation of the final expression opaque. Furthermore, the intermediate step showing the gradient vector as a 3x1 matrix including the function value is non-standard and confusingly presented without explanation.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to explicitly state the linearization formula $L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$, making the derivation of the final expression opaque. Furthermore, the intermediate step showing the gradient vector as a 3x1 matrix including the function value is non-standard and confusingly presented without explanation.gpt-oss:20b: pass 2026-10-10qwen3.6:27b-mlx: fail (error) 2026-10-10 — The solution fails to explicitly state the linear approximation formula $L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$, making the derivation of the linearization in step 3 appear as an unexplained identity rather than a calculated result. Additionally, step 2 presents the values of $f$, $f_x$, and $f_y$ at the point $(7,2)$ as a trivial matrix equality without showing the substitution or calculation, obscuring the critical step of evaluating the derivatives at the specific point.
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 2026-10-10 with SymPy 1.14.0.