Linear approximation
Problem 3.230 · easy
Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 4 \) to estimate \( \displaystyle f(\frac{201}{50}) \).
- The linearization at a is L(x) = f(a) + f'(a)(x − a).
- \[ \left. \frac{1}{x} \right|_{\substack{ x=4 }} = \frac{1}{4} \]f(a).✓ Proved
- \[ \left. \frac{d}{d x} \frac{1}{x} \right|_{\substack{ x=4 }} = - \frac{1}{16} \]f'(a).✓ Proved
- \[ - \frac{x}{16} + 1 \cdot \frac{1}{2} = \frac{1}{2} - \frac{x}{16} \]The linearization.✓ Proved
- \[ \frac{199}{800} \]Evaluate at x = \frac{201}{50}.✓ Proved
Answer \( L(\frac{201}{50}) = \frac{199}{800} \approx 0.24875 \)
Lines: 4 proved, 1 not checked. 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 | Not checked | — | 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: fail (error) — The linearization formula in step 4 is incorrect; it uses x instead of (x-4), resulting in L(x) = 1/2 - x/16 instead of the correct L(x) = 1/2 - (x-4)/16. Although the final numerical answer is coincidentally correct due to an arithmetic error in the evaluation step canceling out the formula error, the derivation is fundamentally flawed.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The linearization formula in step 4 is incorrect; it uses x instead of (x-4), resulting in L(x) = 1/2 - x/16 instead of the correct L(x) = 1/2 - (x-4)/16. Although the final numerical answer is coincidentally correct due to an arithmetic error in the evaluation step canceling out the formula error, the derivation is fundamentally flawed.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-27 — The linearization formula in step 4 is incorrect; it uses (x - 0) instead of (x - 4), resulting in L(x) = 1/2 - x/16 instead of the correct L(x) = 1/4 - (x - 4)/16. Consequently, the final estimate is wrong.gpt-oss:20b: pass 2026-09-27
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 2026-09-27 with SymPy 1.14.0.