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Linear approximation

Problem 3.230 · easy

Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 4 \) to estimate \( \displaystyle f(\frac{201}{50}) \).

The linearization at a is L(x) = f(a) + f'(a)(x − a).
\[ \left. 1 \frac{1}{x} \right|_{\substack{ x=4 }} = \frac{1}{4} \]
f(a).✓ Proved
\[ \left. \frac{d}{d x} 1 \frac{1}{x} \right|_{\substack{ x=4 }} = - \frac{1}{16} \]
f'(a).✓ Proved
\[ - \frac{x}{16} + 1 \cdot \frac{1}{2} = \frac{1}{2} - \frac{x}{16} \]
The linearization.
\[ \frac{199}{800} \]
Evaluate at x = \frac{201}{50}.

Answer \( L(\frac{201}{50}) = \frac{199}{800} \approx 0.24875 \)

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) — The linearization formula in line 4 is incorrect; it should be 1/4 - 1/16(x - 4), which simplifies to 1/2 - x/16, but the intermediate step shown (1/2 - x/16) skips the substitution logic and implies the linearization is just that expression without showing the derivation from L(x) = f(a) + f'(a)(x-a). More critically, line 4 presents '1/2 - x/16' as 'The linearization' directly, but the standard form L(x) = 1/4 - 1/16(x-4) is the direct application of the formula in line 1. While 1/2 - x/16 is algebraically equivalent, presenting it as the immediate result of the formula without the (x-4) term visible can be confusing. However, the bigger issue is that the solution jumps from the formula to the simplified version without showing the substitution of a=4, f(a)=1/4, f'(a)=-1/16 into the formula. But wait, the prompt says equations are checked. Let's look at the sentence in line 4? No, line 4 is marked [equation, proved]. The sentences are 1. Line 1 is correct. The error must be in the interpretation or the hidden steps? No, I only review sentences. Line 1 is the only sentence. Is it correct? Yes. Wait, look at line 4 again. It is an equation. Line 5 is an equation. The only sentence is line 1. Is line 1 correct? Yes. Why would it fail? Ah, look at the problem statement. Estimate f(201/50). 201/50 = 4.02. x-a = 0.02. L(4.02) = 1/4 - 1/16(0.02) = 0.25 - 0.00125 = 0.24875. 199/800 = 0.24875. The math is correct. Is there a sentence error? Line 1 is generic. Maybe the issue is that the solution doesn't explicitly state the value of 'a' or 'x' in the text? No, that's usually fine. Let me re-read the instructions. 'Check that... the setup models the problem... each theorem... each conclusion follows...'. Line 1 is a general formula. It doesn't model the specific problem. But usually, stating the general formula is acceptable. However, often 'setup' requires identifying a=4. Line 1 does not do that. But there are no other sentences. If the verdict is 'pass', notes are empty. If 'fail', there must be a sentence error. Is line 1 misleading? No. Is there a missing sentence? The prompt implies I review the provided lines. If the provided lines are sufficient and correct, it passes. But wait, line 4 is `1/2 - x/16`. This is `L(x)`. Line 5 evaluates it. The algebra is correct. Why would it fail? Perhaps the 'sentence' in line 1 is considered insufficient because it doesn't specify `a=4`? Or perhaps the error is that the linearization is usually written as `L(x) = ...` and line 4 just gives the expression? No, line 4 is an equation `1/2 - x/16, 1/2 - x/16`. This is tautological. It's not a sentence. I only review sentences. There is only one sentence: Line 1. It is correct. Therefore, it should pass. Wait, let me look closer at the prompt's example structure. Usually, these adversarial reviews catch subtle errors. Is `f'(a)` calculated correctly in the sentence? No, that's line 3 (equation). Is the formula in line 1 correct? Yes. Is it applied correctly? The equations show it is. Is there a sentence missing? No. I will mark it as pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The linearization formula in line 4 is incorrect; it should be 1/4 - 1/16(x - 4), which simplifies to 1/2 - x/16, but the intermediate step shown (1/2 - x/16) skips the substitution logic and implies the linearization is just that expression without showing the derivation from L(x) = f(a) + f'(a)(x-a). More critically, line 4 presents '1/2 - x/16' as 'The linearization' directly, but the standard form L(x) = 1/4 - 1/16(x-4) is the direct application of the formula in line 1. While 1/2 - x/16 is algebraically equivalent, presenting it as the immediate result of the formula without the (x-4) term visible can be confusing. However, the bigger issue is that the solution jumps from the formula to the simplified version without showing the substitution of a=4, f(a)=1/4, f'(a)=-1/16 into the formula. But wait, the prompt says equations are checked. Let's look at the sentence in line 4? No, line 4 is marked [equation, proved]. The sentences are 1. Line 1 is correct. The error must be in the interpretation or the hidden steps? No, I only review sentences. Line 1 is the only sentence. Is it correct? Yes. Wait, look at line 4 again. It is an equation. Line 5 is an equation. The only sentence is line 1. Is line 1 correct? Yes. Why would it fail? Ah, look at the problem statement. Estimate f(201/50). 201/50 = 4.02. x-a = 0.02. L(4.02) = 1/4 - 1/16(0.02) = 0.25 - 0.00125 = 0.24875. 199/800 = 0.24875. The math is correct. Is there a sentence error? Line 1 is generic. Maybe the issue is that the solution doesn't explicitly state the value of 'a' or 'x' in the text? No, that's usually fine. Let me re-read the instructions. 'Check that... the setup models the problem... each theorem... each conclusion follows...'. Line 1 is a general formula. It doesn't model the specific problem. But usually, stating the general formula is acceptable. However, often 'setup' requires identifying a=4. Line 1 does not do that. But there are no other sentences. If the verdict is 'pass', notes are empty. If 'fail', there must be a sentence error. Is line 1 misleading? No. Is there a missing sentence? The prompt implies I review the provided lines. If the provided lines are sufficient and correct, it passes. But wait, line 4 is `1/2 - x/16`. This is `L(x)`. Line 5 evaluates it. The algebra is correct. Why would it fail? Perhaps the 'sentence' in line 1 is considered insufficient because it doesn't specify `a=4`? Or perhaps the error is that the linearization is usually written as `L(x) = ...` and line 4 just gives the expression? No, line 4 is an equation `1/2 - x/16, 1/2 - x/16`. This is tautological. It's not a sentence. I only review sentences. There is only one sentence: Line 1. It is correct. Therefore, it should pass. Wait, let me look closer at the prompt's example structure. Usually, these adversarial reviews catch subtle errors. Is `f'(a)` calculated correctly in the sentence? No, that's line 3 (equation). Is the formula in line 1 correct? Yes. Is it applied correctly? The equations show it is. Is there a sentence missing? No. I will mark it as pass.
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 x instead of (x-4), resulting in L(x) = 1/2 - x/16 instead of the correct L(x) = 1/2 - (x-4)/16. Although the final numerical answer is coincidentally correct due to an arithmetic error in the evaluation step canceling out the formula error, the derivation is fundamentally flawed.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The linearization formula in step 4 is incorrect; it uses (x - 0) instead of (x - 4), resulting in L(x) = 1/2 - x/16 instead of the correct L(x) = 1/4 - (x - 4)/16. Consequently, the final estimate is wrong.
gpt-oss:20b: pass 2026-09-27

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.