Linear approximation in two variables
Problem 10.322 · hard
Find the linear approximation of \( \displaystyle f(x, y) = e^{x} \cos{\left(y \right)} \) at \( \displaystyle (0, 0) \), and use it to estimate \( \displaystyle f\left(\frac{1}{20}, \frac{1}{10}\right) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} e^{x} \cos{\left(y \right)}\\\frac{\partial}{\partial y} e^{x} \cos{\left(y \right)}\end{matrix}\right] = \left[\begin{matrix}e^{x} \cos{\left(y \right)}\\- e^{x} \sin{\left(y \right)}\end{matrix}\right] \]f_x and f_y.✓ Proved
- \[ \left[\begin{matrix}1\\1\\0\end{matrix}\right] \]f, f_x and f_y at the point.✓ Proved
- \[ x + 1 \]The linearization.✓ Proved
- \[ \frac{21}{20} \]The estimate; the true value is about 1.04602.✓ Proved
Answer \( L(x, y) = x + 1,\ f \approx \frac{21}{20} = 1.05000 \)
✓ Nihil obstat 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. Reviewers found nothing wrong with the explanation.
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: pass — The solution correctly computes the partial derivatives, evaluates them at the point (0,0), constructs the linear approximation L(x,y) = 1 + x, and uses it to estimate the function value. The algebraic steps are verified and the logic is sound.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the partial derivatives, evaluates them at the point (0,0), constructs the linear approximation L(x,y) = 1 + x, and uses it to estimate the function value. The algebraic steps are verified and the logic is sound.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the partial derivatives, evaluates them at (0,0), constructs the linear approximation L(x,y) = 1 + x, and uses it to estimate the function value. The algebraic steps are verified and the logic is sound.gpt-oss:20b: pass 2026-10-04
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 2026-10-04 with SymPy 1.14.0.