Linear approximation
Problem 3.110 · easy
Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{2}{25}) \).
- The linearization at a is L(x) = f(a) + f'(a)(x − a).
- \[ \left. \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 0 \]f(a).✓ Proved
- \[ \left. \frac{d}{d x} \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 1 \]f'(a).✓ Proved
- \[ x \]The linearization.
- \[ \frac{2}{25} \]Evaluate at x = \frac{2}{25}.
Answer \( L(\frac{2}{25}) = \frac{2}{25} \approx 0.08 \)
Lines: 2 proved, 3 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. How we verify
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 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 5 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 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) — Step 4 is labeled as 'The linearization' but contains the identity 'x, x', which is not the linearization function L(x) = x. The linearization should be explicitly constructed as L(x) = 0 + 1*(x - 0) = x. While the final numerical answer is correct, the step claiming 'x' is the linearization without showing the construction from f(a) and f'(a) is logically incomplete and misleading regarding the definition of linearization.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — Step 4 is labeled as 'The linearization' but contains the identity 'x, x', which is not the linearization function L(x) = x. The linearization should be explicitly constructed as L(x) = 0 + 1*(x - 0) = x. While the final numerical answer is correct, the step claiming 'x' is the linearization without showing the construction from f(a) and f'(a) is logically incomplete and misleading regarding the definition of linearization.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — Step 4 is labeled as 'The linearization' but contains the identity 'x, x', which is not the linearization function L(x) = x. The linearization should be explicitly stated as L(x) = x before evaluation.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly construct the linearization function L(x) = x in step 4, instead just stating 'x, x' which is ambiguous and does not clearly show the substitution into the linearization formula. While the final numerical answer is correct, the logical step of forming L(x) and then evaluating it is missing or poorly represented.gpt-oss:20b: pass 2026-09-26
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 2026-10-13 with SymPy 1.14.0.