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Linear approximation

Problem 3.27 · easy

Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{1}{25}) \).

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.
\[ 1 \cdot \frac{1}{25} = \frac{1}{25} \]
Evaluate at x = \frac{1}{25}.

Answer \( L(\frac{1}{25}) = \frac{1}{25} \approx 0.04 \)

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: pass
qwen3.6:27b-mlx: fail (error) — Line 4 is labeled as 'The linearization' but contains the variable 'x' instead of the derived linear function L(x) = x. This fails to explicitly construct the approximation formula before evaluating it.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — Line 4 is labeled as 'The linearization' but contains the variable 'x' instead of the derived linear function L(x) = x. This fails to explicitly construct the approximation formula before evaluating it.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Line 4 is labeled as 'The linearization' but contains the identity 'x, x', which is not the linearization L(x) = x derived from lines 2 and 3. The solution fails to explicitly state the formula L(x) = x before evaluating it.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Line 4 is labeled as 'The linearization' but contains the equation 'x, x', which is not the linearization L(x) = x. The solution fails to explicitly state the formula for L(x) derived from steps 2 and 3, jumping directly to the evaluation. While the final answer is correct, the intermediate step is malformed and does not represent the linearization function itself.
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.