Home›Calculus 2›Horizontal and vertical tangents of parametric curves›Problem 8.503

Horizontal and vertical tangents of parametric curves

Problem 8.503 · medium

Find the points where \( \displaystyle x = t^{2} - 2 t,\ y = t^{2} - 4 t - 3 \) has a horizontal or a vertical tangent line.

\[ \frac{d}{d t} \left(t^{2} - 2 t\right) = 2 t - 2 \]
dx/dt.✓ Proved
\[ \frac{d}{d t} \left(t^{2} - 4 t - 3\right) = 2 t - 4 \]
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 (0, -7).✓ Proved
\[ \left. 2 t - 2 \right|_{\substack{ t=1 }} = 0 \]
t = 1: vertical, at (-1, -6).✓ Proved

Answer \( \text{horizontal at } (0, -7);\ \text{vertical at } (-1, -6) \)

✓ Nihil obstat Lines: 4 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. 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
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 correct parameter values, and maps them to the correct coordinate points.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-19 — The solution correctly identifies the conditions for horizontal and vertical tangents, solves for the correct parameter values, and maps them to the correct coordinate points.
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, solves for the parameter values, and verifies that the other derivative is non-zero at those points. The final coordinates match the stated answer.
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 and solves for the correct parameter values and points.
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.