Home›Calculus 3›Linear approximation in two variables›Problem 10.469

Linear approximation in two variables

Problem 10.469 · 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}{10}, \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
\[ \frac{9}{10} \]
The estimate; the true value is about 0.90032.✓ Proved

Answer $ L(x, y) = x + 1,\ f \approx \frac{9}{10} = 0.90000 $

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: pass
qwen3.6:27b-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 jump from the gradient values to the final linear function unjustified. Additionally, the intermediate equation in step 2 is nonsensical and does not clearly represent the evaluation of the function and its derivatives at the point $(0,0)$.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-10
qwen3.6:27b-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 jump from the gradient values to the final linear function unjustified. Additionally, the intermediate equation in step 2 is nonsensical and does not clearly represent the evaluation of the function and its derivatives at the point $(0,0)$.
qwen3.6:27b-mlx: inconclusive 2026-10-10 — reviewer returned a non-object
gpt-oss:20b: pass 2026-10-10

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.