Linear approximation
Problem 3.185 · easy
Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{1}{20}) \).
- 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.✓ Proved
- \[ 1 \cdot \frac{1}{20} = \frac{1}{20} \]Evaluate at x = \frac{1}{20}.✓ Proved
Answer \( L(\frac{1}{20}) = \frac{1}{20} \approx 0.05 \)
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) — Step 4 is labeled as 'The linearization' but contains the equation 'x, x', which is not the linearization function L(x) = x. The linearization is derived from steps 2 and 3 as L(x) = 0 + 1*(x-0) = x, but the solution fails to explicitly state this function or the substitution step, jumping directly to the evaluation. While the final answer is correct, the solution omits the crucial step of constructing the linear approximation function L(x) before evaluating it, making the logic incomplete and potentially confusing for a student learning the method.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Step 4 is labeled as 'The linearization' but contains the equation 'x, x', which is not the linearization function L(x) = x. The linearization is derived from steps 2 and 3 as L(x) = 0 + 1*(x-0) = x, but the solution fails to explicitly state this function or the substitution step, jumping directly to the evaluation. While the final answer is correct, the solution omits the crucial step of constructing the linear approximation function L(x) before evaluating it, making the logic incomplete and potentially confusing for a student learning the method.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: fail (error) 2026-09-26 — The solution fails to explicitly construct the linearization function L(x) = x before evaluating it. Step 4 is nonsensical ('x, x') and does not show the application of the formula from Step 1, leaving the student without the crucial intermediate step of forming the approximation function.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 2026-09-26 with SymPy 1.14.0.