Home›Calculus 2›Horizontal and vertical tangents of parametric curves›Problem 8.248

Horizontal and vertical tangents of parametric curves

Problem 8.248 · medium

Find the points where \( \displaystyle x = 3 \cos{\left(t \right)},\ y = 3 \sin{\left(2 t \right)} \) for \( \displaystyle 0 \le t < 2\pi \) has a horizontal or a vertical tangent line.

\[ \frac{d}{d t} 3 \cos{\left(t \right)} = - 3 \sin{\left(t \right)} \]
dx/dt.✓ Proved
\[ \frac{d}{d t} 3 \sin{\left(2 t \right)} = 6 \cos{\left(2 t \right)} \]
dy/dt.✓ Proved
Horizontal tangent: dy/dt = 0 with dx/dt ≠ 0. Vertical: dx/dt = 0 with dy/dt ≠ 0.
Reviewed
\[ \left. 6 \cos{\left(2 t \right)} \right|_{\substack{ t=\frac{\pi}{4} }} = 0 \]
t = pi/4: horizontal, at (3*sqrt(2)/2, 3).✓ Proved
\[ \left. 6 \cos{\left(2 t \right)} \right|_{\substack{ t=\frac{3 \pi}{4} }} = 0 \]
t = 3*pi/4: horizontal, at (-3*sqrt(2)/2, -3).✓ Proved
\[ \left. 6 \cos{\left(2 t \right)} \right|_{\substack{ t=\frac{5 \pi}{4} }} = 0 \]
t = 5*pi/4: horizontal, at (-3*sqrt(2)/2, 3).✓ Proved
\[ \left. 6 \cos{\left(2 t \right)} \right|_{\substack{ t=\frac{7 \pi}{4} }} = 0 \]
t = 7*pi/4: horizontal, at (3*sqrt(2)/2, -3).✓ Proved
\[ \left. - 3 \sin{\left(t \right)} \right|_{\substack{ t=0 }} = 0 \]
t = 0: vertical, at (3, 0).✓ Proved
\[ \left. - 3 \sin{\left(t \right)} \right|_{\substack{ t=\pi }} = 0 \]
t = pi: vertical, at (-3, 0).✓ Proved

Answer \( \text{horizontal at } (3*sqrt(2)/2, 3), (-3*sqrt(2)/2, -3), (-3*sqrt(2)/2, 3), (3*sqrt(2)/2, -3);\ \text{vertical at } (3, 0), (-3, 0) \)

Lines: 8 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
7	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
8	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
9	✓ 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 and verifies that the derivatives are non-zero at the candidate points, ensuring no singularities are misclassified.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-17 — The solution correctly identifies the conditions for horizontal and vertical tangents and verifies that the derivatives are non-zero at the candidate points, ensuring no singularities are misclassified.
gpt-oss:20b: pass 2026-10-17
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution fails to check for singular points where both dx/dt and dy/dt are zero (at t = pi/2 and 3pi/2). At these points, the tangent is neither horizontal nor vertical, but the solution omits them without justification, potentially misleading students into thinking they are not part of the analysis or that the standard criteria apply directly without checking for 0/0 indeterminate forms.
gpt-oss:20b: pass 2026-10-15
qwen3.6:27b-mlx: fail (error) 2026-10-14 — The solution fails to check for singular points where both dx/dt and dy/dt are zero. At t = pi/2 and t = 3pi/2, both derivatives vanish, creating cusps rather than standard horizontal or vertical tangents. The solution incorrectly omits these points without justification and implicitly assumes standard tangents exist at all other critical points.
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.