Horizontal and vertical tangents of parametric curves
Problem 8.495 · medium
Find the points where \( \displaystyle x = t^{3} - 9 t,\ y = t^{2} - 2 \) has a horizontal or a vertical tangent line.
- \[ \frac{d}{d t} \left(t^{3} - 9 t\right) = 3 t^{2} - 9 \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \left(t^{2} - 2\right) = 2 t \]dy/dt.✓ Proved
- Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.Reviewed
- \[ \left. 2 t \right|_{\substack{ t=0 }} = 0 \]t = 0: horizontal, at (0, -2).✓ Proved
- \[ \left. 3 t^{2} - 9 \right|_{\substack{ t=- \sqrt{3} }} = 0 \]t = -sqrt(3): vertical, at (6*sqrt(3), 1).✓ Proved
- \[ \left. 3 t^{2} - 9 \right|_{\substack{ t=\sqrt{3} }} = 0 \]t = sqrt(3): vertical, at (-6*sqrt(3), 1).✓ Proved
Answer \( \text{horizontal at } (0, -2);\ \text{vertical at } (6*sqrt(3), 1), (-6*sqrt(3), 1) \)
✓ Nihil obstat Lines: 5 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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 | Reviewed | reviewers: gpt-oss:20b, qwen3.6:27b-mlx | a sentence; read, not computed |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 5 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 6 | ✓ 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: pass — The solution correctly identifies the conditions for horizontal and vertical tangents, solves for the parameter values, and correctly maps them to the Cartesian coordinates.
Every verdict on record (6)
qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the conditions for horizontal and vertical tangents, solves for the parameter values, and correctly maps them to the Cartesian coordinates.gpt-oss:20b: pass 2026-10-19qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the conditions for horizontal and vertical tangents, solves for the parameter t, and computes the corresponding coordinates accurately.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the conditions for horizontal and vertical tangents, solves for the parameter t, and correctly computes the corresponding coordinates. The logic is sound and the arithmetic is correct.gpt-oss:20b: pass 2026-10-18
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 2026-10-19 with SymPy 1.14.0.