Linear approximation in two variables
Problem 10.520 · hard
Find the linear approximation of \( \displaystyle f(x, y) = \frac{x}{x + y} \) at \( \displaystyle (2, 1) \), and use it to estimate \( \displaystyle f\left(\frac{21}{10}, \frac{21}{20}\right) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \frac{x}{x + y}\\\frac{\partial}{\partial y} \frac{x}{x + y}\end{matrix}\right] = \left[\begin{matrix}- \frac{x}{\left(x + y\right)^{2}} + \frac{1}{x + y}\\- \frac{x}{\left(x + y\right)^{2}}\end{matrix}\right] \]f_x and f_y.✓ Proved
- \[ \left[\begin{matrix}\frac{2}{3}\\1 \cdot \frac{1}{9}\\- \frac{2}{9}\end{matrix}\right] = \left[\begin{matrix}\frac{2}{3}\\\frac{1}{9}\\- \frac{2}{9}\end{matrix}\right] \]f, f_x and f_y at the point.✓ Proved
- \[ \frac{x}{9} - \frac{2 y}{9} + \frac{2}{3} \]The linearization.✓ Proved
- \[ \frac{2}{3} \]The estimate; the true value is about 0.66667.✓ Proved
Answer \( L(x, y) = \frac{x}{9} - \frac{2 y}{9} + \frac{2}{3},\ f \approx \frac{2}{3} = 0.66667 \)
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 substitute the specific point (21/10, 21/20) into the linearization formula L(x, y). Instead, it incorrectly states the estimate is 2/3, which is the value of the function at the center of approximation (2, 1), not at the target point.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to substitute the specific point (21/10, 21/20) into the linearization formula L(x, y). Instead, it incorrectly states the estimate is 2/3, which is the value of the function at the center of approximation (2, 1), not at the target point.gpt-oss:20b: pass 2026-10-12qwen3.6:27b-mlx: fail (error) 2026-10-12 — The solution fails to actually evaluate the linear approximation at the specified point (2.1, 1.05). Instead, it incorrectly claims the estimate is 2/3, which is just the value of the function at the center point (2, 1), ignoring the linear terms.
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.