Horizontal and vertical tangents of parametric curves
Problem 8.377 · medium
Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = 3 t^{2} - 9 \) 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
- \[ \frac{d}{d t} \left(3 t^{2} - 9\right) = 6 t \]dy/dt.✓ Proved
- Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
- \[ \left. 6 t \right|_{\substack{ t=0 }} = 0 \]t = 0: horizontal, at (0, -9).✓ Proved
- \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=-1 }} = 0 \]t = -1: vertical, at (2, -6).✓ Proved
- \[ \left. 3 t^{2} - 3 \right|_{\substack{ t=1 }} = 0 \]t = 1: vertical, at (-2, -6).✓ Proved
Answer \( \text{horizontal at } (0, -9);\ \text{vertical at } (2, -6), (-2, -6) \)
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 for vertical tangents. It claims t = -1 and t = 1 correspond to points (2, -6) and (-2, -6), but substituting t = -1 into the parametric equations yields (-2, -6) and t = 1 yields (2, -6). Additionally, the logic for vertical tangents is flawed: it checks where dx/dt = 0 (which is correct for vertical tangents) but labels the resulting points incorrectly based on wrong coordinate substitutions.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly identifies the points for vertical tangents. It claims t = -1 and t = 1 correspond to points (2, -6) and (-2, -6), but substituting t = -1 into the parametric equations yields (-2, -6) and t = 1 yields (2, -6). Additionally, the logic for vertical tangents is flawed: it checks where dx/dt = 0 (which is correct for vertical tangents) but labels the resulting points incorrectly based on wrong coordinate substitutions.gpt-oss:20b: pass 2026-10-18qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies points with vertical tangents. At t = ±1, dx/dt = 0 but dy/dt = ±6 ≠ 0, so the tangents are indeed vertical. Howegpt-oss:20b: pass 2026-10-16qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly identifies the points for vertical tangents. It claims t = -1 and t = 1 yield vertical tangents at (2, -6) and (-2, -6), but substituting these t-values into the parametric equations gives (-2, -6) and (2, -6) respectively. Furthermore, the coordinates listed in the stated answer for vertical tangents are swapped relative to the t-values used in the derivation, or the t-values are wrong for the coordinates provided. Specifically, t=1 gives x=-2, y=-6, and t=-1 gives x=2, y=-6. The stated answer lists vertical at (2, -6) and (-2, -6), which are the correct points, but the derivation in lines 5 and 6 has the mapping wrong (t=-1 -> (2,-6) is correct, t=1 -> (-2,-6) is correct, but the text says t=-1 is vertical at (2,-6) and t=1 is vertical at (-2,-6). Wait, let's re-read carefully. Line 5 says t=-1 is vertical at (2, -6). x(-1) = -1 - (-3) = 2. y(-1) = 3-9 = -6. This is correct. Line 6 says t=1 is vertical at (-2, -6). x(1) = 1-3 = -2. y(1) = 3-9 = -6. This is correct. So the points are correct. However, the stated answer says vertical at (2, -6) and (-2, -6). The solution derives them correctly. Is there an error? Let's check the horizontal tangent. t=0, x=0, y=-9. Correct. Why would this fail? Ah, look at the stated answer: "vertical at (2, -6), (-2, -6)". The solution finds t=-1 gives (2, -6) and t=1 gives (-2, -6). This matches. Is there a subtle error? The problem asks for points. The solution provides them. Let's look closer at the derivatives. dx/dt = 3t^2 - 3. dy/dt = 6t. Vertical when dx/dt=0, dy/dt!=0. 3t^2-3=0 => t=1, -1. At t=1, dy/dt=6!=0. At t=-1, dy/dt=-6!=0. So vertical tangents exist at t=1, -1. Points are (-2, -6) and (2, -6). The solution says: Line 5: t=-1 vertical at (2, -6). Correct. Line 6: t=1 vertical at (-2, -6). Correct. Horizontal when dy/dt=0, dx/dt!=0. 6t=0 => t=0. At t=0, dx/dt=-3!=0. Point (0, -9). Correct. The solution seems correct. Why did I think it failed? Maybe I misread the coordinates in the stated answer vs the solution. Stated: vertical at (2, -6), (-2, -6). Solution: t=-1 -> (2, -6), t=1 -> (-2, -6). This is consistent. Is there a style issue? No. Is there a misleading issue? No. Wait, look at line 5 and 6 again. Line 5: "Subs(3*t**2 - 3, t, -1) = 0 -- t = -1: vertical, at (2, -6)." Line 6: "Subs(3*t**2 - 3, t, 1) = 0 -- t = 1: vertical, at (-2, -6)." The logic is sound. The algebra is checked. The sentences are correct. Verdict: pass.gpt-oss:20b: pass 2026-10-15
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.