Horizontal and vertical tangents of parametric curves
Problem 8.236 · medium
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 3 \) has a horizontal or a vertical tangent line.
- \[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \left(t^{2} - 3\right) = 2 t \]dy/dt.✓ Proved
- Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
- \[ \left. 2 t \right|_{\substack{ t=0 }} = 0 \]t = 0: horizontal, at (0, -3).✓ Proved
- \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=-1 }} = 0 \]t = -1: vertical, at (2, -2).✓ Proved
- \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=1 }} = 0 \]t = 1: vertical, at (-2, -2).✓ Proved
Answer \( \text{horizontal at } (0, -3);\ \text{vertical at } (2, -2), (-2, -2) \)
Lines: 5 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 |
| 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: fail (error) — The solution incorrectly identifies the points with vertical tangents. It claims t = -1 and t = 1 yield vertical tangents, but at these values dy/dt is non-zero (2 and -2 respectively), satisfying the condition for horizontal tangents (since dx/dt is 0). The points (2, -2) and (-2, -2) actually have horizontal tangents, not vertical ones. There are no vertical tangents for this curve.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-17 — The solution incorrectly identifies the points with vertical tangents. It claims t = -1 and t = 1 yield vertical tangents, but at these values dy/dt is non-zero (2 and -2 respectively), satisfying the condition for horizontal tangents (since dx/dt is 0). The points (2, -2) and (-2, -2) actually have horizontal tangents, not vertical ones. There are no vertical tangents for this curve.gpt-oss:20b: pass 2026-10-17qwen3.6:27b-mlx: inconclusive 2026-10-15 — reviewer response could not be parsed: { "verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the points for vertical tangents. It claims t = -1 and t = 1 yield vertical tangents at (2, -2) and (-2, -2), bugpt-oss:20b: pass 2026-10-15qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution incorrectly identifies the points for vertical tangents. It claims t = -1 and t = 1 yield vertical tangents, but at these values dx/dt = 0 while dy/dt is non-zero, which is correct for vertical tangents. However, the coordinates calculated are wrong: at t = -1, x = (-1)^3 - 3(-1) = 2 and y = (-1)^2 - 3 = -2, so (2, -2) is correct. At t = 1, x = 1^3 - 3(1) = -2 and y = 1^2 - 3 = -2, so (-2, -2) is correct. Wait, let me re-read the solution's claims. Line 5 says 't = -1: vertical, at (2, -2)'. Line 6 says 't = 1: vertical, at (-2, -2)'. The stated answer says 'vertical at (2, -2), (-2, -2)'. The solution's logic for finding t is correct (dx/dt = 0 => t = +/- 1). The coordinates are correct. Why did I think it was wrong? Let me re-read the prompt's stated answer. Stated answer: horizontal at (0, -3); vertical at (2, -2), (-2, -2). Solution line 4: t=0, horizontal, at (0, -3). Correct. Solution line 5: t=-1, vertical, at (2, -2). Correct. Solution line 6: t=1, vertical, at (-2, -2). Correct. Is there an error? Ah, look at line 5 and 6 comments. Line 5 says 'Subs(3*t**2 - 3, t, -1) = 0 -- t = -1: vertical...'. This is correct. Line 6 says 'Subs(3*t**2 - 3, t, 1) = 0 -- t = 1: vertical...'. This is correct. Wait, is there a subtle error? Let's check the horizontal tangent condition. dy/dt = 2t = 0 => t=0. dx/dt at t=0 is -3 != 0. So horizontal at t=0. Point: x=0, y=-3. Correct. Vertical tangent condition. dx/dt = 3t^2 - 3 = 0 => t = +/- 1. dy/dt at t=1 is 2 != 0. dy/dt at t=-1 is -2 != 0. So vertical at t=1 and t=-1. Points: t=1 => x=-2, y=-2. t=-1 => x=2, y=-2. The solution matches the stated answer. Is there a style issue? The comments in lines 4, 5, 6 mix the substitution result with the conclusion. But the prompt asks to check sentences. Lines 4, 5, 6 are marked [equation, unchecked]. The text after the double dash is likely a comment or part of the equation line in this format. If they are treated as sentences, they are correct. If they are just comments, they are fine. Let me look closer. Is there a missing case? What if dx/dt = 0 and dy/dt = 0 simultaneously? 3t^2-3=0 and 2t=0. No common solution. So no cusps or singular points. The solution seems correct. Why would it fail? Maybe the order? No. Maybe the phrasing? 'Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0.' This is a standard sufficient condition. Is it necessary? If dy/dt=0 and dx/dt=0, it's a singular point, not necessarily a horizontal tangent (could be a cusp). The problem asks for points with horizontal/vertical tangents. Usually, this implies non-singular points. The solution correctly excludes singular points by checking dx/dt != 0. It seems sound. Let me re-read the prompt constraints. 'Find what is wrong'. If nothing is wrong, pass. But usually these have a trick. Let's check the coordinates again. t=1: x = 1-3 = -2. y = 1-3 = -2. Point (-2, -2). Solution line 6 says 'at (-2, -2)'. Correct. t=-1: x = -1+3 = 2. y = 1-3 = -2. Point (2, -2). Solution line 5 says 'at (2, -2)'. Correct. t=0: x=0, y=-3. Solution line 4 says 'at (0, -3)'. Correct. The logic is sound. The algebra is marked unchecked but presumably correct. The sentences are correct. I will pass it.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 2026-10-17 with SymPy 1.14.0.