Linear approximation
Problem 3.348 · easy
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 81 \) to estimate \( \displaystyle f(\frac{403}{5}) \).
- The linearization at a is L(x) = f(a) + f'(a)(x − a).
- \[ \left. \sqrt{x} \right|_{\substack{ x=81 }} = 9 \]f(a).✓ Proved
- \[ \left. \frac{d}{d x} \sqrt{x} \right|_{\substack{ x=81 }} = \frac{1}{18} \]f'(a).✓ Proved
- \[ \frac{x}{18} + \frac{9}{2} \]The linearization.✓ Proved
- \[ \frac{404}{45} \]Evaluate at x = \frac{403}{5}.✓ Proved
Answer \( L(\frac{403}{5}) = \frac{404}{45} \approx 8.97778 \)
Lines: 4 proved, 1 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.
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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 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: passqwen3.6:27b-mlx: fail (error) — The linearization formula in line 4 is incorrect. It should be L(x) = 9 + (1/18)(x - 81), which simplifies to x/18 + 9/2. However, the provided equation x/18 + 9/2 = x/18 + 9/2 is a tautology that does not explicitly show the derivation from the point-slope form, but more critically, the evaluation in line 5 is wrong. L(403/5) = (403/5)/18 + 9/2 = 403/90 + 405/90 = 808/90 = 404/45. Wait, the calculation 404/45 is actually correct. Let me re-read the prompt's constraint. 'Each line... is either an EQUATION... or a SENTENCE'. Line 4 is marked [equation, proved]. The equation x/18 + 9/2 = x/18 + 9/2 is trivially true but fails to represent the linearization L(x) derived from steps 2 and 3. A student might think the linearization is just 'x/18 + 9/2' without understanding it comes from f(a) + f'(a)(x-a). However, the bigger issue is usually whether the setup is clear. Let's look closer. L(x) = 9 + 1/18(x-81) = 9 + x/18 - 81/18 = 9 + x/18 - 9/2 = x/18 + 9/2. So the expression is correct. The evaluation 403/5 * 1/18 + 9/2 = 403/90 + 405/90 = 808/90 = 404/45. The math is correct. Is there a sentence error? Line 1 is a sentence. It states the formula. It is correct. The prompt asks to review sentences. Line 1 is the only sentence. It is correct. The equations are marked 'proved'. If the equations are correct, and the sentence is correct, it should pass. But wait, line 4 is labeled [equation, proved]. The content is 'x/18 + 9/2 = x/18 + 9/2'. This is a tautology, not a derivation. It doesn't link back to f(a) and f'(a). A student reading this might not see how the linearization was constructed. However, the instructions say 'do not re-check the algebra of a line marked proved'. If the algebra is correct, I shouldn't fail it for that. But is the *sentence* part of the solution flawed? There are no other sentences. The problem is likely that the solution is too sparse or the 'equation' line is misleading because it doesn't show the structure L(x) = ... . But strictly, if the algebra is correct, it passes? No, 'misleading' is a severity. A tautology as a step is misleading. It implies no work was done to get the linearization. Also, the prompt says 'Check that... each conclusion follows from the lines before it'. Line 4 does not follow from 2 and 3 logically in the text provided, it just asserts an identity. It skips the step of constructing L(x). This is a logical gap. Verdict: fail, severity: misleading.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The linearization formula in line 4 is incorrect. It should be L(x) = 9 + (1/18)(x - 81), which simplifies to x/18 + 9/2. However, the provided equation x/18 + 9/2 = x/18 + 9/2 is a tautology that does not explicitly show the derivation from the point-slope form, but more critically, the evaluation in line 5 is wrong. L(403/5) = (403/5)/18 + 9/2 = 403/90 + 405/90 = 808/90 = 404/45. Wait, the calculation 404/45 is actually correct. Let me re-read the prompt's constraint. 'Each line... is either an EQUATION... or a SENTENCE'. Line 4 is marked [equation, proved]. The equation x/18 + 9/2 = x/18 + 9/2 is trivially true but fails to represent the linearization L(x) derived from steps 2 and 3. A student might think the linearization is just 'x/18 + 9/2' without understanding it comes from f(a) + f'(a)(x-a). However, the bigger issue is usually whether the setup is clear. Let's look closer. L(x) = 9 + 1/18(x-81) = 9 + x/18 - 81/18 = 9 + x/18 - 9/2 = x/18 + 9/2. So the expression is correct. The evaluation 403/5 * 1/18 + 9/2 = 403/90 + 405/90 = 808/90 = 404/45. The math is correct. Is there a sentence error? Line 1 is a sentence. It states the formula. It is correct. The prompt asks to review sentences. Line 1 is the only sentence. It is correct. The equations are marked 'proved'. If the equations are correct, and the sentence is correct, it should pass. But wait, line 4 is labeled [equation, proved]. The content is 'x/18 + 9/2 = x/18 + 9/2'. This is a tautology, not a derivation. It doesn't link back to f(a) and f'(a). A student reading this might not see how the linearization was constructed. However, the instructions say 'do not re-check the algebra of a line marked proved'. If the algebra is correct, I shouldn't fail it for that. But is the *sentence* part of the solution flawed? There are no other sentences. The problem is likely that the solution is too sparse or the 'equation' line is misleading because it doesn't show the structure L(x) = ... . But strictly, if the algebra is correct, it passes? No, 'misleading' is a severity. A tautology as a step is misleading. It implies no work was done to get the linearization. Also, the prompt says 'Check that... each conclusion follows from the lines before it'. Line 4 does not follow from 2 and 3 logically in the text provided, it just asserts an identity. It skips the step of constructing L(x). This is a logical gap. Verdict: fail, severity: misleading.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (error) 2026-10-03 — The linearization formula in line 4 is incorrect. It should be L(x) = 9 + (1/18)(x - 81), but the solution provides x/18 + 9/2, which corresponds to a different constant term and fails to represent the tangent line at x=81.gpt-oss:20b: pass 2026-10-03
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 2026-10-03 with SymPy 1.14.0.