Home›Calculus 2›Horizontal and vertical tangents of parametric curves›Problem 8.502

Horizontal and vertical tangents of parametric curves

Problem 8.502 · medium

Find the points where \( \displaystyle x = t^{3} - 9 t,\ y = t^{2} - 4 t - 2 \) 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
\[ \frac{d}{d t} \left(t^{2} - 4 t - 2\right) = 2 t - 4 \]
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 (-10, -6).✓ Proved
\[ \left. 3 t^{2} - 9 \right|_{\substack{ t=- \sqrt{3} }} = 0 \]
t = -sqrt(3): vertical, at (6*sqrt(3), 1 + 4*sqrt(3)).✓ Proved
\[ \left. 3 t^{2} - 9 \right|_{\substack{ t=\sqrt{3} }} = 0 \]
t = sqrt(3): vertical, at (-6*sqrt(3), 1 - 4*sqrt(3)).✓ Proved

Answer \( \text{horizontal at } (-10, -6);\ \text{vertical at } (6*sqrt(3), 1 + 4*sqrt(3)), (-6*sqrt(3), 1 - 4*sqrt(3)) \)

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

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19
gpt-oss:20b: pass 2026-10-19
qwen3.6:27b-mlx: pass 2026-10-18 — The solution correctly identifies the conditions for horizontal and vertical tangents for parametric curves and computes the correct points.
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: pass 2026-10-18 — 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: fail (error) 2026-10-18 — The vertical tangent points are listed with the x‑coordinates swapped. For t=√3, x=−6√3, y=1−4√3; for t=−√3, x=6√3, y=1+4√3. The solution lists the opposite pairs.

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.