Tangent lines
Problem 2.1146 · easy
Find an equation of the tangent line to \( \displaystyle y = - 2 x^{3} + x^{2} - 3 x - 2 \) at \( \displaystyle x = 2 \).
- The tangent line passes through the point (a, f(a)) and has slope f'(a).Reviewed
- \[ \left. - 2 x^{3} + x^{2} - 3 x - 2 \right|_{\substack{ x=2 }} = -20 \]The point of tangency.✓ Proved
- \[ \frac{d}{d x} \left(- 2 x^{3} + x^{2} - 3 x - 2\right) = - 6 x^{2} + 2 x - 3 \]Differentiate.✓ Proved
- \[ \left. - 6 x^{2} + 2 x - 3 \right|_{\substack{ x=2 }} = -23 \]The slope at the point.✓ Proved
- \[ 26 - 23 x \]Point-slope form, then simplify.✓ Proved
Answer \( y = 26 - 23 x \)
✓ 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 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 derives the correct linear equation. The steps are logically sound and algebraically verified.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the point of tangency and the slope, and derives the correct linear equation. The steps are logically sound and algebraically verified.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly identifies the point of tangency and the slope, and derives the correct linear equation for the tangent line.gpt-oss:20b: pass 2026-09-28
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 2026-09-28 with SymPy 1.14.0.