Linear approximation
Problem 3.184 · easy
Use the linear approximation of \( \displaystyle f(x) = \sqrt{x} \) at \( \displaystyle x = 9 \) to estimate \( \displaystyle f(\frac{19}{2}) \).
- 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.✓ Proved
- \[ \frac{37}{12} \]Evaluate at x = \frac{19}{2}.✓ Proved
Answer \( L(\frac{19}{2}) = \frac{37}{12} \approx 3.08333 \)
✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 | ✓ 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: pass — The solution correctly applies the linearization formula, computes the derivative and function value at x=9, and evaluates the resulting linear approximation at x=19/2. The algebraic steps are verified as correct.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the linearization formula, computes the derivative and function value at x=9, and evaluates the resulting linear approximation at x=19/2. The algebraic steps are verified as correct.gpt-oss:20b: pass 2026-09-26qwen3.6:27b-mlx: pass 2026-09-26 — The solution correctly applies the linear approximation formula, computes the derivative and function value accurately, and evaluates the result at the specified point.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 2026-09-26 with SymPy 1.14.0.