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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.
\[ 1 \cdot \frac{1}{20} = \frac{1}{20} \]
Evaluate at x = \frac{1}{20}.

Answer \( L(\frac{1}{20}) = \frac{1}{20} \approx 0.05 \)

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 actual linear approximation formula L(x) = x. This fails to construct the required linear model.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-27 — Line 4 is labeled as 'The linearization' but contains the variable 'x' instead of the actual linear approximation formula L(x) = x. This fails to construct the required linear model.
gpt-oss:20b: pass 2026-09-27
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 2026-09-26
qwen3.6:27b-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 2026-10-13 with SymPy 1.14.0.