Linear approximation in two variables
Problem 10.524 · 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 \( \displaystyle f\left(\frac{69}{10}, \frac{41}{20}\right) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \ln{\left(x - 3 y \right)}\\\frac{\partial}{\partial y} \ln{\left(x - 3 y \right)}\end{matrix}\right] = \left[\begin{matrix}\frac{1}{x - 3 y}\\- \frac{3}{x - 3 y}\end{matrix}\right] \]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
- \[ - \frac{1}{4} \]The estimate; the true value is about -0.28768.✓ Proved
Answer \( L(x, y) = x - 3 y - 1,\ f \approx - \frac{1}{4} = -0.25000 \)
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 linear approximation formula $L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$, jumping directly to the result. While the algebraic result is correct, the missing logical step of substituting the point $(7,2)$ and derivatives into the standard formula makes the derivation incomplete and potentially confusing for a student learning the method.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to explicitly state the linear approximation formula $L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$, jumping directly to the result. While the algebraic result is correct, the missing logical step of substituting the point $(7,2)$ and derivatives into the standard formula makes the derivation incomplete and potentially confusing for a student learning the method.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to evaluate the partial derivatives at the point (7, 2) before constructing the linearization. It incorrectly uses the symbolic expressions for f_x and f_y in the linearization formula instead of their numerical values (1/1 and -3/1), leading to an incorrect linear approximation L(x, y) = x - 3y - 1 instead of the correct L(x, y) = 1/2 + (x-7) - 3(y-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/linear_approx_2var, checked 2026-10-12 with SymPy 1.14.0.