Home›Calculus 1›Linear approximation›Problem 3.116

Linear approximation

Problem 3.116 · easy

Use the linear approximation of \( \displaystyle f(x) = \ln{\left(x \right)} \) at \( \displaystyle x = 1 \) to estimate \( \displaystyle f(\frac{19}{20}) \).

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
\[ - \frac{1}{20} \]
Evaluate at x = \frac{19}{20}.Reviewed

Answer \( L(\frac{19}{20}) = - \frac{1}{20} \approx -0.05 \)

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 applies the linearization formula, computes the derivative and function value at the point of tangency, and evaluates the linear approximation at the target point.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the linearization formula, computes the derivative and function value at the point of tangency, and evaluates the linear approximation at the target point.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — Line 4 claims the linearization is 'x - 1', but the linearization formula L(x) = f(a) + f'(a)(x-a) with f(1)=0 and f'(1)=1 yields L(x) = x - 1. However, the label in line 4 says '-- The linearization' but the equation shown is just 'x - 1, x - 1'. While algebraically L(x) = x-1 is correct, the solution skips explicitly stating L(x) = 0 + 1*(x-1) = x-1, which is fine. Wait, let's look closer. Line 4 equation is 'x - 1, x - 1'. This is an identity, not a definition. It doesn't explicitly state L(x) = x - 1. But more importantly, Line 5 evaluates this at x = 19/20. 19/20 - 1 = -1/20. The math is correct. Why would it fail? Let's re-read the prompt constraints. 'Check that... each conclusion follows from the lines before it'. Line 4 is just an identity 'x-1 = x-1'. It does not define L(x). The previous lines calculated f(a) and f'(a). There is no line that actually constructs L(x) = f(a) + f'(a)(x-a). Line 4 just asserts x-1 equals x-1. It fails to link the previous calculations to the expression x-1. A student might not see how 0 and 1 become x-1 without the explicit substitution step L(x) = 0 + 1(x-1). This is a missing logical step in the derivation, making the conclusion in line 5 not strictly follow from the previous lines as presented.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the linear approximation formula, computes the derivative and function value at the point of tangency, and evaluates the linearization at the target point. The steps are logically sound and algebraically correct.
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.