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Linear approximation

Problem 3.120 · easy

Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 4 \) to estimate \( \displaystyle f(\frac{401}{100}) \).

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.
\[ \frac{399}{1600} \]
Evaluate at x = \frac{401}{100}.

Answer \( L(\frac{401}{100}) = \frac{399}{1600} \approx 0.249375 \)

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 step 4 is incorrect; it should be 1/4 - 1/16(x - 4), not 1/2 - x/16. Although the final numerical answer is coincidentally correct, the intermediate expression for L(x) is wrong.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-09-28 — The linearization formula in step 4 is incorrect; it should be 1/4 - 1/16(x - 4), not 1/2 - x/16. Although the final numerical answer is coincidentally correct, the intermediate expression for L(x) is wrong.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: fail (error) 2026-09-27 — The linearization formula in line 4 is incorrect; it should be L(x) = 1/4 - 1/16(x - 4), which simplifies to 1/2 - x/16. However, the evaluation in line 5 yields 399/1600, which is the correct numerical answer, but the intermediate expression '1/2 - x/16' is presented as the linearization without showing the simplification from the standard form, which might confuse a student about the structure of L(x). More critically, line 4 states '1/2 - x/16' is the linearization. Let's check: L(x) = 1/4 - 1/16(x-4) = 1/4 - x/16 + 4/16 = 1/4 - x/16 + 1/4 = 1/2 - x/16. The algebra is correct. The evaluation: L(4.01) = 1/2 - 4.01/16 = 0.5 - 0.250625 = 0.249375. 399/1600 = 0.249375. The solution is mathematically sound. Wait, is there a subtle error? The problem asks for the linear approximation. The steps are correct. Why would it fail? Let me re-read carefully. Line 4 says '1/2 - x/16, 1/2 - x/16 -- The linearization.' This is the simplified form. It is correct. Line 5 evaluates it. It is correct. Is there a style issue? The prompt asks to find what is wrong. If nothing is wrong, pass. But usually these adversarial reviews have a trap. Let's look at the bounds or setup. f(x)=1/x, a=4. f(4)=1/4. f'(x)=-1/x^2, f'(4)=-1/16. L(x) = 1/4 - 1/16(x-4). This simplifies to 1/2 - x/16. Evaluation at 401/100 = 4.01. 1/2 - 4.01/16 = 8/16 - 4.01/16 = 3.99/16 = 399/1600. Everything is correct. I will mark it as pass.
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 uses (1/2 - x/16) instead of the correct (1/4 - (x-4)/16). This algebraic error leads to the wrong final answer.
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.