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Linear approximation

Problem 3.30 · easy

Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{27}{25}) \).

The linearization at a is L(x) = f(a) + f'(a)(x − a).
Reviewed
\[ \left. \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 0 \]
f(a).✓ Proved
\[ \left. \frac{d}{d x} \ln{\left(x \right)} \right|_{\substack{ x=1 }} = 1 \]
f'(a).✓ Proved
\[ x - 1 \]
The linearization.Reviewed
\[ \frac{2}{25} \]
Evaluate at x = \frac{27}{25}.Reviewed

Answer \( L(\frac{27}{25}) = \frac{2}{25} \approx 0.08 \)

Lines: 2 proved, 3 reviewed. 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
5	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — The solution correctly identifies the linearization formula, computes the derivative and function value at x=1, constructs the linear approximation L(x) = x-1, and evaluates it at x=27/25 to get 2/25. All steps are logically sound and algebraically correct.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the linearization formula, computes the derivative and function value at x=1, constructs the linear approximation L(x) = x-1, and evaluates it at x=27/25 to get 2/25. All steps are logically sound and algebraically correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The solution fails to explicitly construct the linearization formula L(x) = f(a) + f'(a)(x-a) using the computed values f(1)=0 and f'(1)=1. It jumps from the derivatives to the final evaluation without showing the intermediate step L(x) = x - 1, making the logic incomplete.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — Line 4 is labeled 'The linearization' but contains only the term (x-1), omitting the f(a) term. Although f(a)=0 makes the final result correct, the statement that L(x) = x-1 is technically incorrect notation for the linearization function, which should be explicitly written as L(x) = 0 + 1(x-1) or simply L(x) = x-1 with the derivation shown. More critically, line 4 is marked as an equation 'x - 1, x - 1' which is a tautology, not the definition of L(x). The sentence in line 1 defines L(x), but line 4 fails to state L(x) = x - 1, instead just presenting the expression. This is a structural error in the solution presentation.
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.