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Linear approximation

Problem 3.265 · easy

Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{101}{100}) \).

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.
\[ 1 \cdot \frac{1}{100} = \frac{1}{100} \]
Evaluate at x = \frac{101}{100}.

Answer \( L(\frac{101}{100}) = \frac{1}{100} \approx 0.01 \)

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 (style) — [domain objection, downgraded to style] The solution fails to explicitly construct the linearization function L(x) = x - 1 before evaluating it. Line 4 merely states a tautology (x - 1 = x - 1) rather than defining L(x), and Line 5 evaluates the expression x - 1 without referencing the function L(x) defined in Line 1, breaking the logical chain.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (style) 2026-10-03 — [domain objection, downgraded to style] The solution fails to explicitly construct the linearization function L(x) = x - 1 before evaluating it. Line 4 merely states a tautology (x - 1 = x - 1) rather than defining L(x), and Line 5 evaluates the expression x - 1 without referencing the function L(x) defined in Line 1, breaking the logical chain.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the linear approximation formula, computes the derivative and function value at the point of tangency, and evaluates the resulting linear function at the target point.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-28 — The solution fails to explicitly construct the linearization function L(x) using the formula in step 1 and the values in steps 2 and 3. Step 4 is labeled 'The linearization' but only shows the term (x-1), omitting the f(a) and f'(a) components, making the derivation logically incomplete.
gpt-oss:20b: pass 2026-09-28

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.