Horizontal and vertical tangents of parametric curves
Problem 8.348 · medium
Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{3} - 3 t \) has a horizontal or a vertical tangent line.
- \[ \frac{d}{d t} \left(t^{2} - 2 t\right) = 2 t - 2 \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]dy/dt.✓ Proved
- Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
- \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=-1 }} = 0 \]t = -1: horizontal, at (3, 2).✓ Proved
Answer \( \text{horizontal at } (3, 2);\ \text{vertical at } none \)
Lines: 3 proved, 1 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 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | Not checked | — | a sentence; read, not computed |
| 4 | ✓ 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 | roots of dx/dt and dy/dt found by Newton's method from a dense grid |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution fails to check for vertical tangents. It sets dx/dt = 0 at t=1, but does not verify if dy/dt is non-zero at that point (dy/dt(1) = 0), which would indicate a cusp or singularity rather than a vertical tangent. While the final answer 'none' is correct, the reasoning is incomplete because it omits the analysis of the point where dx/dt = 0.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution fails to check for vertical tangents. It sets dx/dt = 0 at t=1, but does not verify if dy/dt is non-zero at that point (dy/dt(1) = 0), which would indicate a cusp or singularity rather than a vertical tangent. While the final answer 'none' is correct, the reasoning is incomplete because it omits the analysis of the point where dx/dt = 0.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: fail (error) 2026-10-16 — The solution fails to check for vertical tangents by solving dx/dt = 0. It incorrectly concludes there are no vertical tangents without verifying the condition dy/dt != 0 at t=1 (where dx/dt=0).gpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to check for vertical tangents. It correctly identifies the horizontal tangent at t=-1, but ignores the case where dx/dt = 0 (at t=1). At t=1, dy/dt is also 0, resulting in a cusp rather than a vertical tangent, but this analysis is missing entirely. The claim 'vertical at none' is not supported by any work shown.gpt-oss:20b: pass 2026-10-14
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/parametric_tangent_hv, checked 2026-10-18 with SymPy 1.14.0.