Tangent lines
Problem 2.1268 · medium
Find an equation of the tangent line to \( \displaystyle y = - x - 3 \sin{\left(x \right)} + 1 \) at \( \displaystyle x = \pi \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).Reviewed
- \[ \left. - x - 3 \sin{\left(x \right)} + 1 \right|_{\substack{ x=\pi }} = 1 - \pi \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(- x - 3 \sin{\left(x \right)} + 1\right) = - 3 \cos{\left(x \right)} - 1 \]Differentiate.✓ Proved
- \[ \left. - 3 \cos{\left(x \right)} - 1 \right|_{\substack{ x=\pi }} = 2 \]The slope at the point.✓ Proved
- \[ 2 x - 3 \pi + 1 \]Point-slope form, then simplify.✓ Proved
Answer \( y = 2 x - 3 \pi + 1 \)
Lines: 4 proved, 1 reviewed. 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.
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 line meets the curve at x = a, and its slope matches a central difference quotient of f there |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly identifies the point of tangency and the slope, and applies the point-slope formula accurately to derive the final equation.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly identifies the point of tangency and the slope, and applies the point-slope formula accurately to derive the final equation.gpt-oss:20b: pass 2026-09-29qwen3.6:27b-mlx: inconclusive 2026-09-29 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The derivative in step 3 is incorrect; the derivative of -x is -1, so f'(x) should be -1 - 3*cos(x), not -3*cos(x) - 1 (which implies the derivativegpt-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/tangent_line, checked 2026-09-29 with SymPy 1.14.0.