Home›Calculus 1›Linear approximation›Problem 3.21

Linear approximation

Problem 3.21 · easy

Use the linear approximation of \( \displaystyle f(x) = \frac{1}{x} \) at \( \displaystyle x = 2 \) to estimate \( \displaystyle f(\frac{99}{50}) \).

The linearization at a is L(x) = f(a) + f'(a)(x − a).
\[ \left. 1 \frac{1}{x} \right|_{\substack{ x=2 }} = \frac{1}{2} \]
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.
\[ \frac{101}{200} \]
Evaluate at x = \frac{99}{50}.

Answer \( L(\frac{99}{50}) = \frac{101}{200} \approx 0.505 \)

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 omits the shift by 'a' (should be 1/2 - 1/4(x-2) or 1 - x/4 + 1/2 = 3/2 - x/4? No, L(x) = 1/2 - 1/4(x-2) = 1/2 - x/4 + 1/2 = 1 - x/4. Wait, let's re-evaluate. 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 + 1/2 = 1 - x/4. The equation in line 4 says '1 - x/4'. This is actually correct algebraically. Let's check the evaluation. x = 99/50 = 1.98. L(1.98) = 1 - 1.98/4 = 1 - 0.495 = 0.505 = 505/1000 = 101/200. The math is correct. Why did I think it was wrong? Let me re-read carefully. Line 4: '1 - x/4, 1 - x/4'. This is the simplified form of the linearization. It is correct. Line 5 evaluates it correctly. Is there a subtle error? The problem asks for linear approximation. The steps are correct. Wait, is the sentence in line 1 sufficient? Yes. Is the setup correct? Yes. Let me double check the arithmetic. 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) = 1/2 - x/4 + 1/2 = 1 - x/4. Correct. Evaluate at 99/50. 1 - (99/50)/4 = 1 - 99/200 = 200/200 - 99/200 = 101/200. Correct. The solution is sound. Why would it fail? Maybe the prompt implies I should find an error? 'Your job is to find what is wrong'. If nothing is wrong, I say pass. Let me look really closely at line 4. '1 - x/4, 1 - x/4'. The comment says '-- The linearization'. It is the linearization. It is correct. 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 omits the shift by 'a' (should be 1/2 - 1/4(x-2) or 1 - x/4 + 1/2 = 3/2 - x/4? No, L(x) = 1/2 - 1/4(x-2) = 1/2 - x/4 + 1/2 = 1 - x/4. Wait, let's re-evaluate. 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 + 1/2 = 1 - x/4. The equation in line 4 says '1 - x/4'. This is actually correct algebraically. Let's check the evaluation. x = 99/50 = 1.98. L(1.98) = 1 - 1.98/4 = 1 - 0.495 = 0.505 = 505/1000 = 101/200. The math is correct. Why did I think it was wrong? Let me re-read carefully. Line 4: '1 - x/4, 1 - x/4'. This is the simplified form of the linearization. It is correct. Line 5 evaluates it correctly. Is there a subtle error? The problem asks for linear approximation. The steps are correct. Wait, is the sentence in line 1 sufficient? Yes. Is the setup correct? Yes. Let me double check the arithmetic. 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) = 1/2 - x/4 + 1/2 = 1 - x/4. Correct. Evaluate at 99/50. 1 - (99/50)/4 = 1 - 99/200 = 200/200 - 99/200 = 101/200. Correct. The solution is sound. Why would it fail? Maybe the prompt implies I should find an error? 'Your job is to find what is wrong'. If nothing is wrong, I say pass. Let me look really closely at line 4. '1 - x/4, 1 - x/4'. The comment says '-- The linearization'. It is the linearization. It is correct. 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 should be L(x) = 1/2 - 1/4(x - 2), which simplifies to 1 - x/4 only if the constant term is handled correctly, but the expression '1 - x/4' implies L(2) = 1/2, which is correct, however, the derivation skips the explicit substitution of (x-a). More critically, step 4 presents '1 - x/4' as the linearization without showing the intermediate step L(x) = 1/2 - 1/4(x-2). While algebraically equivalent, the sentence in step 4 is missing, and step 4 is marked as an equation. Wait, step 4 is marked [equation, proved]. The algebra 1/2 - 1/4(x-2) = 1/2 - x/4 + 1/2 = 1 - x/4 is correct. So the equation is correct. Let's re-read the prompt constraints. 'Each line... is either an EQUATION... or a SENTENCE'. Step 4 is an equation. It is proved. So the algebra is fine. Step 5 is an equation. It is proved. So the evaluation is fine. Step 1 is a sentence. It states the formula. Step 2 and 3 are equations. Is there a missing sentence? The problem asks to review the solution. The solution seems mathematically sound. L(99/50) = 1 - (99/50)/4 = 1 - 99/200 = 101/200. The steps are correct. Why would it fail? Let's look closer. Step 4: '1 - x/4, 1 - x/4 -- The linearization.' This is an equation line. It is proved. Step 5: '101/200, 101/200 -- Evaluate at x = 99/50.' This is an equation line. It is proved. Step 1: 'The linearization at a is L(x) = f(a) + f'(a)(x - a).' This is a sentence. It is correct. Steps 2 and 3 calculate f(a) and f'(a). They are correct. The logic flows. Is there a subtle error? 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(99/50) = 1/2 - 1/4(99/50 - 2) = 1/2 - 1/4(99/50 - 100/50) = 1/2 - 1/4(-1/50) = 1/2 + 1/200 = 100/200 + 1/200 = 101/200. The math is perfect. The format is perfect. The verdict should be 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 - x/4 instead of the correct 1/2 - 1/4(x - 2). This error leads to the wrong final answer (101/200 instead of 99/100).
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.