Home›Calculus 3›Linear approximation in two variables›Problem 10.323

Linear approximation in two variables

Problem 10.323 · hard

Find the linear approximation of \( \displaystyle f(x, y) = \sqrt{- x^{2} - 7 y^{2} + 20} \) at \( \displaystyle (2, 1) \), and use it to estimate \( \displaystyle f\left(\frac{19}{10}, \frac{51}{50}\right) \).

\[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{- x^{2} - 7 y^{2} + 20}\\\frac{\partial}{\partial y} \sqrt{- x^{2} - 7 y^{2} + 20}\end{matrix}\right] = \left[\begin{matrix}- \frac{x}{\sqrt{- x^{2} - 7 y^{2} + 20}}\\- \frac{7 y}{\sqrt{- x^{2} - 7 y^{2} + 20}}\end{matrix}\right] \]
f_x and f_y.✓ Proved
\[ \left[\begin{matrix}3\\- \frac{2}{3}\\- \frac{7}{3}\end{matrix}\right] \]
f, f_x and f_y at the point.✓ Proved
\[ - \frac{2 x}{3} - \frac{7 y}{3} + \frac{20}{3} \]
The linearization.✓ Proved
\[ \frac{151}{50} \]
The estimate; the true value is about 3.01781.✓ Proved

Answer \( L(x, y) = - \frac{2 x}{3} - \frac{7 y}{3} + \frac{20}{3},\ f \approx \frac{151}{50} = 3.02000 \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly computes the partial derivatives, evaluates them at the given point, constructs the linear approximation, 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 given point, constructs the linear approximation, 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
qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly computes the partial derivatives, evaluates them at the given point, constructs the linear approximation, and uses it to estimate the function value. The algebra and logic are 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.