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Linear approximation

Problem 3.266 · easy

Use the linear approximation of \( \displaystyle f(x) = \sqrt[3]{x} \) at \( \displaystyle x = 64 \) to estimate \( \displaystyle f(\frac{322}{5}) \).

The linearization at a is L(x) = f(a) + f'(a)(x − a).
\[ \left. x^{1 \cdot \frac{1}{3}} \right|_{\substack{ x=64 }} = 4 \]
f(a).✓ Proved
\[ \left. \frac{d}{d x} x^{1 \cdot \frac{1}{3}} \right|_{\substack{ x=64 }} = \frac{1}{48} \]
f'(a).✓ Proved
\[ \frac{x}{48} + \frac{8}{3} \]
The linearization.
\[ \frac{481}{120} \]
Evaluate at x = \frac{322}{5}.

Answer \( L(\frac{322}{5}) = \frac{481}{120} \approx 4.00833 \)

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. The correct linearization is L(x) = 4 + (1/48)(x - 64) = x/48 + 8/3, but step 4 presents x/48 + 8/3 without showing the derivation from f(a) + f'(a)(x-a), and more critically, the constant term 8/3 is correct (4 - 64/48 = 4 - 4/3 = 8/3), so the equation itself is algebraically correct. However, the sentence in step 1 is generic, and step 4 is just an identity. The real issue is that the solution skips the explicit construction of L(x) from steps 2 and 3, making it unclear if the student knows how to combine them. But wait, step 4 is marked 'proved' by CAS, so the algebra is fine. Let's look closer. L(x) = 4 + (1/48)(x-64) = 4 + x/48 - 64/48 = 4 + x/48 - 4/3 = 8/3 + x/48. Step 4 says x/48 + 8/3. This is correct. Step 5 evaluates this at 322/5: (322/5)/48 + 8/3 = 322/240 + 8/3 = 161/120 + 320/120 = 481/120. This is also correct. Why would it fail? The prompt asks to review sentences. Step 1 is a standard definition. There are no other sentences. The equations are proved. Is there a subtle error? No. The solution is sound. Wait, let me re-read the prompt constraints. 'Check that... each conclusion follows from the lines before it'. Step 4 is an equation, not a sentence. Step 5 is an equation. The only sentence is Step 1. Step 1 is correct. So it should pass? Let me double check the arithmetic. f(64) = 4. f'(x) = 1/3 x^(-2/3). f'(64) = 1/3 * 1/16 = 1/48. L(x) = 4 + 1/48(x-64). L(322/5) = 4 + 1/48(322/5 - 320/5) = 4 + 1/48(2/5) = 4 + 2/240 = 4 + 1/120 = 481/120. The solution in step 4 gives x/48 + 8/3. At x=322/5, this is 322/240 + 8/3 = 161/120 + 320/120 = 481/120. The math is correct. The sentences are minimal but correct. I will mark it as pass.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The linearization formula in step 4 is incorrect. The correct linearization is L(x) = 4 + (1/48)(x - 64) = x/48 + 8/3, but step 4 presents x/48 + 8/3 without showing the derivation from f(a) + f'(a)(x-a), and more critically, the constant term 8/3 is correct (4 - 64/48 = 4 - 4/3 = 8/3), so the equation itself is algebraically correct. However, the sentence in step 1 is generic, and step 4 is just an identity. The real issue is that the solution skips the explicit construction of L(x) from steps 2 and 3, making it unclear if the student knows how to combine them. But wait, step 4 is marked 'proved' by CAS, so the algebra is fine. Let's look closer. L(x) = 4 + (1/48)(x-64) = 4 + x/48 - 64/48 = 4 + x/48 - 4/3 = 8/3 + x/48. Step 4 says x/48 + 8/3. This is correct. Step 5 evaluates this at 322/5: (322/5)/48 + 8/3 = 322/240 + 8/3 = 161/120 + 320/120 = 481/120. This is also correct. Why would it fail? The prompt asks to review sentences. Step 1 is a standard definition. There are no other sentences. The equations are proved. Is there a subtle error? No. The solution is sound. Wait, let me re-read the prompt constraints. 'Check that... each conclusion follows from the lines before it'. Step 4 is an equation, not a sentence. Step 5 is an equation. The only sentence is Step 1. Step 1 is correct. So it should pass? Let me double check the arithmetic. f(64) = 4. f'(x) = 1/3 x^(-2/3). f'(64) = 1/3 * 1/16 = 1/48. L(x) = 4 + 1/48(x-64). L(322/5) = 4 + 1/48(322/5 - 320/5) = 4 + 1/48(2/5) = 4 + 2/240 = 4 + 1/120 = 481/120. The solution in step 4 gives x/48 + 8/3. At x=322/5, this is 322/240 + 8/3 = 161/120 + 320/120 = 481/120. The math is correct. The sentences are minimal but correct. I will mark it as pass.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the linear approximation formula, computes the derivative and function value at the point of tangency, and evaluates the linearization at the specified point. The arithmetic is correct.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the linearization formula, computes the derivative and function value at the point of tangency, constructs the linear approximation, and evaluates it at the specified point. All steps are mathematically sound.
gpt-oss:20b: pass 2026-09-28

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.