Linear approximation in two variables
Problem 10.421 · 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 \( \displaystyle f\left(\frac{141}{20}, \frac{21}{10}\right) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \ln{\left(x - 3 y \right)}\\\frac{\partial}{\partial y} \ln{\left(x - 3 y \right)}\end{matrix}\right] = \left[\begin{matrix}\frac{1}{x - 3 y}\\- \frac{3}{x - 3 y}\end{matrix}\right] \]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
- \[ - \frac{1}{4} \]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) and the substitution of the specific point (7,2). While the algebraic results in lines 2-4 are correct, the sentences describing them are vacuous tautologies that do not demonstrate the method or justify how the values were derived from the problem statement.
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 state the linear approximation formula L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b) and the substitution of the specific point (7,2). While the algebraic results in lines 2-4 are correct, the sentences describing them are vacuous tautologies that do not demonstrate the method or justify how the values were derived from the problem statement.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution incorrectly evaluates the function and its derivatives at the point (7, 2). f(7, 2) = ln(1) = 0, f_x(7, 2) = 1, and f_y(7, 2) = -3. The linearization should be L(x, y) = x - 3y - 7, not x - 3y - 1. Consequently, the estimate is wrong.gpt-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.