Linear approximation in two variables
Problem 10.328 · 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 \( \displaystyle f\left(\frac{99}{50}, \frac{51}{50}\right) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{- x^{2} - 7 y^{2} + 20}\\\frac{\partial}{\partial y} \sqrt{- x^{2} - 7 y^{2} + 20}\end{matrix}\right] = \left[\begin{matrix}- \frac{x}{\sqrt{- x^{2} - 7 y^{2} + 20}}\\- \frac{7 y}{\sqrt{- x^{2} - 7 y^{2} + 20}}\end{matrix}\right] \]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
- \[ \frac{89}{30} \]The estimate; the true value is about 2.96594.✓ Proved
Answer \( L(x, y) = - \frac{2 x}{3} - \frac{7 y}{3} + \frac{20}{3},\ f \approx \frac{89}{30} = 2.96667 \)
✓ 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 given point, constructs the linear approximation formula, and calculates the estimate. All steps are mathematically 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 formula, and calculates the estimate. All steps are mathematically sound.gpt-oss:20b: pass 2026-10-04qwen3.6:27b-mlx: pass 2026-10-04gpt-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.