Linear approximation
Problem 3.265 · easy
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{101}{100}) \).
- The linearization at a is L(x) = f(a) + f'(a)(x − a).Reviewed
- \[ \left. \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 0 \]f(a).✓ Proved
- \[ \left. \frac{d}{d x} \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 1 \]f'(a).✓ Proved
- \[ x - 1 \]The linearization.✓ Proved
- \[ 1 \cdot \frac{1}{100} = \frac{1}{100} \]Evaluate at x = \frac{101}{100}.✓ Proved
Answer \( L(\frac{101}{100}) = \frac{1}{100} \approx 0.01 \)
Lines: 4 proved, 1 reviewed. 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 | 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 applies the linear approximation formula, computes the derivative and function value at the point of tangency, and evaluates the resulting linear function at the target point.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the linear approximation formula, computes the derivative and function value at the point of tangency, and evaluates the resulting linear function at the target point.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly construct the linearization function L(x) using the formula in step 1 and the values in steps 2 and 3. Step 4 is labeled 'The linearization' but only shows the term (x-1), omitting the f(a) and f'(a) components, making the derivation logically incomplete.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.