Linear approximation
Problem 3.106 · easy
Use the linear approximation of \( \displaystyle f(x) = \sin{\left(x \right)} \) at \( \displaystyle x = 0 \) to estimate \( \displaystyle f(\frac{1}{100}) \).
- The linearization at a is L(x) = f(a) + f'(a)(x − a).
- \[ \left. \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 0 \]f(a).✓ Proved
- \[ \left. \frac{d}{d x} \sin{\left(x \right)} \right|_{\substack{ x=0 }} = 1 \]f'(a).✓ Proved
- \[ x \]The linearization.
- \[ 1 \cdot \frac{1}{100} = \frac{1}{100} \]Evaluate at x = \frac{1}{100}.
Answer \( L(\frac{1}{100}) = \frac{1}{100} \approx 0.01 \)
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: passqwen3.6:27b-mlx: fail (error) — Line 4 is labeled as 'The linearization' but contains the identity 'x, x', which is mathematically incorrect and fails to define L(x) = x. The solution skips the crucial step of constructing the linearization function from the computed values.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-09-28 — Line 4 is labeled as 'The linearization' but contains the identity 'x, x', which is mathematically incorrect and fails to define L(x) = x. The solution skips the crucial step of constructing the linearization function from the computed values.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-27 — Line 4 claims the linearization is x, but fails to explicitly state the formula L(x) = x derived from the previous steps. While the result is correct, the solution skips the crucial step of assembling L(x) = 0 + 1*(x-0) = x, making the transition from derivatives to the function L(x) opaque and logically incomplete.gpt-oss:20b: pass 2026-09-27qwen3.6:27b-mlx: fail (error) 2026-09-26 — Line 4 is labeled as 'The linearization' but contains the equation 'x, x', which is an identity rather than the derived linearization function L(x) = x. The solution fails to explicitly construct the linearization formula L(x) = f(a) + f'(a)(x-a) using the computed values before evaluating it.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.