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Linear approximation

Problem 3.108 · easy

Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 9 \) to estimate \( \displaystyle f(\frac{48}{5}) \).

The linearization at a is L(x) = f(a) + f'(a)(x − a).
Reviewed
\[ \left. \sqrt{x} \right|_{\substack{ x=9 }} = 3 \]
f(a).✓ Proved
\[ \left. \frac{d}{d x} \sqrt{x} \right|_{\substack{ x=9 }} = \frac{1}{6} \]
f'(a).✓ Proved
\[ \frac{x}{6} + \frac{3}{2} \]
The linearization.Reviewed
\[ \frac{31}{10} \]
Evaluate at x = \frac{48}{5}.Reviewed

Answer \( L(\frac{48}{5}) = \frac{31}{10} \approx 3.1 \)

✓ Nihil obstat Lines: 2 proved, 3 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
5	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass — 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.

Every verdict on record (6)

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
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly identifies the linearization formula, computes the derivative and function value at x=9, constructs the linear approximation L(x), and evaluates it at x=48/5 to obtain the correct result.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the linear approximation formula, computes the derivative and function value at x=9, and evaluates the resulting linearization at x=48/5 to obtain the correct estimate.
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.