Linear approximation in two variables
Problem 10.319 · 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 \( \displaystyle f\left(\frac{19}{10}, \frac{21}{20}\right) \).
- \[ \left[\begin{matrix}\frac{\partial}{\partial x} \sqrt{- x^{2} - 7 y^{2} + 20}\\\frac{\partial}{\partial y} \sqrt{- x^{2} - 7 y^{2} + 20}\end{matrix}\right] = \left[\begin{matrix}- \frac{x}{\sqrt{- x^{2} - 7 y^{2} + 20}}\\- \frac{7 y}{\sqrt{- x^{2} - 7 y^{2} + 20}}\end{matrix}\right] \]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
- \[ \frac{59}{20} \]The estimate; the true value is about 2.94491.✓ Proved
Answer \( L(x, y) = - \frac{2 x}{3} - \frac{7 y}{3} + \frac{20}{3},\ f \approx \frac{59}{20} = 2.95000 \)
✓ 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
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-04gpt-oss:20b: pass 2026-10-04qwen3.6:27b-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. All steps are mathematically 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.