Home›Calculus 2›Horizontal and vertical tangents of parametric curves›Problem 8.247

Horizontal and vertical tangents of parametric curves

Problem 8.247 · medium

Find the points where \( \displaystyle x = t^{3} - 3 t,\ y = t^{2} - 4 t \) 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
\[ \frac{d}{d t} \left(t^{2} - 4 t\right) = 2 t - 4 \]
dy/dt.✓ Proved
Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
Reviewed
\[ \left. 2 t - 4 \right|_{\substack{ t=2 }} = 0 \]
t = 2: horizontal, at (2, -4).✓ Proved
\[ \left. 3 t^{2} - 3 \right|_{\substack{ t=-1 }} = 0 \]
t = -1: vertical, at (2, 5).✓ Proved
\[ \left. 3 t^{2} - 3 \right|_{\substack{ t=1 }} = 0 \]
t = 1: vertical, at (-2, -3).✓ Proved

Answer \( \text{horizontal at } (2, -4);\ \text{vertical at } (2, 5), (-2, -3) \)

Lines: 5 proved, 1 reviewed. 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	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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the conditions for horizontal and vertical tangents, solves for the critical parameter values, and accurately computes the corresponding Cartesian coordinates.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the conditions for horizontal and vertical tangents, solves for the critical parameter values, and accurately computes the corresponding Cartesian coordinates.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly identifies the points for vertical tangents. At t = -1, x = (-1)^3 - 3(-1) = 2 and y = (-1)^2 - 4(-1) = 5, so the point is (2, 5), which matches the stated answer. However, at t = 1, x = 1^3 - 3(1) = -2 and y = 1^2 - 4(1) = -3, so the point is (-2, -3), which also matches. Wait, let me re-read the solution steps carefully. Step 5 says 't = -1: vertical, at (2, 5)'. Step 6 says 't = 1: vertical, at (-2, -3)'. The stated answer says 'vertical at (2, 5), (-2, -3)'. The calculations in the steps are correct. The points are correct. Why did I think it was wrong? Let me re-verify the coordinates. t=-1: x = -1 + 3 = 2, y = 1 + 4 = 5. Point (2,5). t=1: x = 1 - 3 = -2, y = 1 - 4 = -3. Point (-2,-3). The solution steps are correct. The stated answer is correct. Is there a subtle error? Step 3 states the conditions correctly. Step 4 checks dy/dt=0 at t=2. dx/dt at t=2 is 3(4)-3=9 != 0. Correct. Step 5 checks dx/dt=0 at t=-1. dy/dt at t=-1 is 2(-1)-4 = -6 != 0. Correct. Step 6 checks dx/dt=0 at t=1. dy/dt at t=1 is 2(1)-4 = -2 != 0. Correct. The solution seems perfectly sound. Why would it fail? Perhaps the prompt implies I should find an error if one exists, but if none exists, I pass. Let me look closer at the text. 'Subs(3*t**2 - 3, t, -1) = 0 -- t = -1: vertical, at (2, 5).' This is a comment, not an equation. The equation is proved. The comment is correct. 'Subs(3*t**2 - 3, t, 1) = 0 -- t = 1: vertical, at (-2, -3).' This is also correct. The stated answer matches the derived points. There is no error.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: pass 2026-10-14 — The solution correctly identifies the conditions for horizontal and vertical tangents, solves for the parameter values, and computes the corresponding coordinates accurately.
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.