Home›Calculus 2›Horizontal and vertical tangents of parametric curves›Problem 8.498

Horizontal and vertical tangents of parametric curves

Problem 8.498 · medium

Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} \) 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
\[ \frac{d}{d t} t^{2} = 2 t \]
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, 0).✓ Proved
\[ \left. 3 t^{2} - 3 \right|_{\substack{ t=-1 }} = 0 \]
t = -1: vertical, at (2, 1).✓ Proved
\[ \left. 3 t^{2} - 3 \right|_{\substack{ t=1 }} = 0 \]
t = 1: vertical, at (-2, 1).✓ Proved

Answer \( \text{horizontal at } (0, 0);\ \text{vertical at } (2, 1), (-2, 1) \)

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: pass
qwen3.6:27b-mlx: inconclusive — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the points for vertical tangents. It claims t = -1 corresponds to (2, 1) and t = 1 corresponds to (-2, 1), but su

Every verdict on record (6)

qwen3.6:27b-mlx: inconclusive 2026-10-19 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the points for vertical tangents. It claims t = -1 corresponds to (2, 1) and t = 1 corresponds to (-2, 1), but su
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: fail (error) 2026-10-18 — The solution incorrectly identifies the points for vertical tangents. At t = 1, x = 1^3 - 3(1) = -2, so the point is (-2, 1), which matches the stated answer, but the comment in line 6 says 'vertical at (-2, 1)' while line 5 says 'vertical at (2, 1)' for t = -1. Let's check t = -1: x = (-1)^3 - 3(-1) = -1 + 3 = 2, y = (-1)^2 = 1. So (2, 1) is correct for t = -1. Let's check t = 1: x = 1^3 - 3(1) = -2, y = 1^2 = 1. So (-2, 1) is correct for t = 1. The coordinates are actually correct in the comments. However, the logic in line 3 is incomplete. It states conditions for horizontal/vertical tangents but fails to check the case where both dx/dt and dy/dt are zero (singular points). At t = 0, dy/dt = 0 and dx/dt = -3 != 0, so horizontal is fine. At t = 1, dx/dt = 0 and dy/dt = 2 != 0, so vertical is fine. At t = -1, dx/dt = 0 and dy/dt = -2 != 0, so vertical is fine. The setup is correct. The calculations are correct. The comments map t to points correctly. Wait, let me re-read line 5 and 6 comments. Line 5: t=-1 -> (2,1). Line 6: t=1 -> (-2,1). The stated answer is vertical at (2,1) and (-2,1). This matches. Is there any error? The problem asks for points where there is a horizontal OR vertical tangent. The solution finds them. The logic holds. Why would it fail? Ah, look at line 3. It defines horizontal as dy/dt=0, dx/dt!=0. Vertical as dx/dt=0, dy/dt!=0. This is the standard definition for parametric curves. The solution applies this correctly. The points are calculated correctly. The verdict should be pass? Let me double check the 'adversarial' nature. Is there a subtle error? Let's re-verify the coordinates. t=0: x=0, y=0. Horizontal. Correct. t=1: x=-2, y=1. Vertical. Correct. t=-1: x=2, y=1. Vertical. Correct. The solution seems perfectly sound. Why did I think it might fail? Maybe the prompt implies I should find something wrong? No, my job is to find what is wrong. If nothing is wrong, I pass. Wait, look at the stated answer in the problem description: "vertical at (2, 1), (-2, 1)". The solution comments say: "t = -1: vertical, at (2, 1)" and "t = 1: vertical, at (-2, 1)". This is consistent. Is there a style issue? "Subs(2*t, t, 0) = 0" is a bit weird notation for a sentence, but it's marked as an equation/proved. The sentences are just line 3. Line 3 is correct. I will mark it as pass.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-18 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the points for vertical tangents. It claims vertical tangents occur at t = -1 and t = 1, but at these values dy/d
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.