Linear approximation
Problem 3.269 · easy
Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 5 \) to estimate \( \displaystyle f(\frac{501}{100}) \).
- The linearization at a is L(x) = f(a) + f'(a)(x − a).Reviewed
- \[ \left. \frac{1}{x} \right|_{\substack{ x=5 }} = \frac{1}{5} \]f(a).✓ Proved
- \[ \left. \frac{d}{d x} \frac{1}{x} \right|_{\substack{ x=5 }} = - \frac{1}{25} \]f'(a).✓ Proved
- \[ - \frac{x}{25} + \frac{2}{5} = \frac{2}{5} - \frac{x}{25} \]The linearization.✓ Proved
- \[ \frac{499}{2500} \]Evaluate at x = \frac{501}{100}.✓ Proved
Answer \( L(\frac{501}{100}) = \frac{499}{2500} \approx 0.1996 \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 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 |
| 5 | ✓ 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 | the estimate is within Taylor's error bound of the true value |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the linearization formula, computes the derivative and function value at x=5, constructs the linear approximation L(x), and evaluates it at the specified point. The arithmetic is correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the linearization formula, computes the derivative and function value at x=5, constructs the linear approximation L(x), and evaluates it at the specified point. The arithmetic is correct.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the linearization formula, computes the derivative and function value at x=5, and evaluates the resulting linear approximation at the specified point. The arithmetic is correct.gpt-oss:20b: pass 2026-09-28
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_approximation, checked 2026-09-28 with SymPy 1.14.0.