Linear approximation
Problem 3.343 · easy
Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{97}{100}) \).
- The linearization at a is L(x) = f(a) + f'(a)(x − a).
- \[ \left. \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 0 \]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
- \[ - \frac{3}{100} \]Evaluate at x = \frac{97}{100}.✓ Proved
Answer \( L(\frac{97}{100}) = - \frac{3}{100} \approx -0.03 \)
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 solution fails to explicitly construct the linearization function L(x) using the formula and computed values before evaluating it. Step 4 is a tautology that does not represent the linearization, and Step 5 jumps to the final numerical result without showing the substitution into L(x) = 0 + 1*(x-1), making the logical connection between the derivative calculation and the final estimate opaque.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to explicitly construct the linearization function L(x) using the formula and computed values before evaluating it. Step 4 is a tautology that does not represent the linearization, and Step 5 jumps to the final numerical result without showing the substitution into L(x) = 0 + 1*(x-1), making the logical connection between the derivative calculation and the final estimate opaque.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: pass 2026-10-03 — The solution correctly applies the linearization formula and computes the derivative and function value at the point of tangency. The final arithmetic is correct.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.