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Linear approximation

Problem 3.309 · easy

Use the linear approximation of \( \displaystyle f(x) = \sqrt[3]{x} \) at \( \displaystyle x = 64 \) to estimate \( \displaystyle f(\frac{129}{2}) \).

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.
\[ \frac{385}{96} \]
Evaluate at x = \frac{129}{2}.

Answer \( L(\frac{129}{2}) = \frac{385}{96} \approx 4.01042 \)

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 L(x) = 4 + (1/48)(x - 64), which simplifies to x/48 + 5/3, not x/48 + 8/3. The constant term 8/3 corresponds to evaluating at x=0 instead of accounting for the shift from a=64.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-10-03 — The linearization formula in line 4 is incorrect. It should be L(x) = 4 + (1/48)(x - 64), which simplifies to x/48 + 5/3, not x/48 + 8/3. The constant term 8/3 corresponds to evaluating at x=0 instead of accounting for the shift from a=64.
gpt-oss:20b: pass 2026-10-03
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the linear approximation formula, computes the derivative and function value at the specified point, and evaluates the linearization at the target x-value. The arithmetic is correct.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the linearization formula, computes the function value and derivative at x=64, constructs the linear approximation L(x), and evaluates it at the specified point. All steps are mathematically sound.
gpt-oss:20b: pass 2026-09-29

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.