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Linear approximation

Problem 3.16 · easy

Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 2 \) to estimate \( \displaystyle f(\frac{201}{100}) \).

The linearization at a is L(x) = f(a) + f'(a)(x − a).
Reviewed
\[ \left. 1 \frac{1}{x} \right|_{\substack{ x=2 }} = \frac{1}{2} \]
f(a).✓ Proved
\[ \left. \frac{d}{d x} 1 \frac{1}{x} \right|_{\substack{ x=2 }} = - \frac{1}{4} \]
f'(a).✓ Proved
\[ 1 - \frac{x}{4} \]
The linearization.Reviewed
\[ \frac{199}{400} \]
Evaluate at x = \frac{201}{100}.Reviewed

Answer \( L(\frac{201}{100}) = \frac{199}{400} \approx 0.4975 \)

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 values accurately, and evaluates the linear approximation at the specified 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 values accurately, and evaluates the linear approximation at the specified point.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The linearization formula in step 4 is incorrect. It uses 1 - x/4, which corresponds to f(a) + f'(a)x, missing the subtraction of 'a' in the term (x-a). The correct linearization is 1/2 - 1/4(x-2) = 1 - x/4 is actually correct algebraically? Let's recheck. L(x) = 1/2 + (-1/4)(x-2) = 1/2 - x/4 + 1/2 = 1 - x/4. Wait, the algebra is correct. Let me re-read carefully. L(x) = f(a) + f'(a)(x-a). f(2)=1/2. f'(2)=-1/4. L(x) = 1/2 - 1/4(x-2) = 1/2 - x/4 + 2/4 = 1 - x/4. The equation in step 4 says '1 - x/4'. This is correct. Step 5 evaluates 1 - (201/100)/4 = 1 - 201/400 = 199/400. This is also correct. Why did I think it was wrong? Let me check the prompt again. '1 - x/4' is the simplified linearization. Is there a subtle error? No. The solution is sound.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-26 — The linearization formula in step 4 is incorrect. It should be L(x) = 1/2 - 1/4(x - 2) = 1 - x/4, but the derivation shown as '1 - x/4' implies the constant term was calculated incorrectly or the expansion was skipped without showing the correct intermediate step L(x) = 1/2 - 1/4(x-2). Wait, let's re-evaluate: L(x) = f(2) + f'(2)(x-2) = 1/2 + (-1/4)(x-2) = 1/2 - x/4 + 2/4 = 1/2 - x/4 + 1/2 = 1 - x/4. The equation in step 4 is actually correct algebraically. However, the label says 'The linearization' and the equation is '1 - x/4, 1 - x/4'. This is correct. Let's check the evaluation. f(201/100) approx L(201/100) = 1 - (201/100)/4 = 1 - 201/400 = (400-201)/400 = 199/400. The math is correct. Why would it fail? Let's look closer at the prompt's constraints. 'Check that... each conclusion follows from the lines before it'. Step 4 jumps from f(a) and f'(a) to the final simplified form 1-x/4. While algebraically correct, is it misleading? No. Is there an error? Let's re-read carefully. Step 4: '1 - x/4, 1 - x/4'. This is the linearization. Step 5 evaluates it. The solution is actually correct. Did I miss something? Ah, look at step 4 again. The previous steps calculated f(a)=1/2 and f'(a)=-1/4. The linearization is L(x) = 1/2 - 1/4(x-2). Simplifying: 1/2 - x/4 + 1/2 = 1 - x/4. The step is correct. Is there a subtle issue? Maybe the variable? No. Maybe the bounds? No. Let's look at the 'Stated answer'. 199/400. The calculation yields 199/400. The solution is sound. Wait, is there a 'style' issue? The prompt asks for 'error', 'misleading', or 'style'. If it's sound, 'pass'. Let me double check the arithmetic one more time. f(x) = 1/x. a=2. f(2)=1/2. f'(x)=-1/x^2. f'(2)=-1/4. L(x) = 1/2 - 1/4(x-2). L(2.01) = 1/2 - 1/4(0.01) = 0.5 - 0.0025 = 0.4975. 199/400 = 0.4975. Correct. Why did I think it might fail? Perhaps the jump in step 4 is considered 'style' if it skips the unsimplified form? But it's not wrong. I will mark it pass.
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.