Home›Calculus 2›Horizontal and vertical tangents of parametric curves›Problem 8.379

Horizontal and vertical tangents of parametric curves

Problem 8.379 · medium

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

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

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. At t = 1, x = 1^3 - 3(1) = -2, so the point is (-2, 2). At t = -1, x = (-1)^3 -

Every verdict on record (6)

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. At t = 1, x = 1^3 - 3(1) = -2, so the point is (-2, 2). At t = -1, x = (-1)^3 -
gpt-oss:20b: pass 2026-10-18
qwen3.6:27b-mlx: inconclusive 2026-10-16 — reviewer response could not be parsed: {"verdict": "fail", "severity": "error", "notes": "The solution incorrectly identifies the points for vertical tangents. At t = -1, x = 2, y = 2, but dx/dt = 0 and dy/dt = -2, so it is a vertical tang
gpt-oss:20b: pass 2026-10-16
qwen3.6:27b-mlx: fail (error) 2026-10-15 — The solution incorrectly identifies the points for vertical tangents. It claims vertical tangents occur at t = -1 and t = 1, but at t = 1, x = -2, not 2. Furthermore, the stated answer lists vertical tangents at (2, 2) and (-2, 2), which correspond to t = -1 and t = 1 respectively, but the coordinates for t=1 are (-2, 2) and for t=-1 are (2, 2). Wait, let's recheck the coordinates. x(-1) = (-1)^3 - 3(-1) = -1 + 3 = 2. y(-1) = (-1)^2 + 1 = 2. So (2,2) is correct for t=-1. x(1) = 1^3 - 3(1) = -2. y(1) = 1^2 + 1 = 2. So (-2,2) is correct for t=1. The coordinates in the stated answer are correct. However, line 5 says 't = -1: vertical, at (2, 2)' and line 6 says 't = 1: vertical, at (-2, 2)'. This seems consistent. Let me re-read the prompt's stated answer: 'vertical at (2, 2), (-2, 2)'. This matches. Is there an error? Ah, look at line 5 and 6. They are marked [equation, unchecked]. The text says 'Subs(3*t**2 - 3, t, -1) = 0 -- t = -1: vertical, at (2, 2)'. This is a sentence/comment attached to an equation. The logic is: dx/dt = 0 at t = +/- 1. dy/dt at t = +/- 1 is +/- 2, which is not 0. So vertical tangents exist. The coordinates are calculated correctly. Why would this fail? Let me check the horizontal tangent. dy/dt = 0 at t=0. dx/dt at t=0 is -3 != 0. So horizontal at t=0. x(0)=0, y(0)=1. Point (0,1). This is correct. Is there a subtle error? Maybe the order? Or maybe the prompt implies I should find errors in the *sentences*. Sentence 3 is correct. Sentences in 4, 5, 6 are comments. They seem correct. Wait, look at the stated answer again: 'horizontal at (0, 1); vertical at (2, 2), (-2, 2)'. The solution derives this. Is it possible the question considers the cusp or something? No, it's a standard parametric curve. Let me look closer at the coordinates. t=1 -> x=-2, y=2. t=-1 -> x=2, y=2. The solution says t=-1 is at (2,2) and t=1 is at (-2,2). This is correct. Why would it be wrong? Perhaps the 'unchecked' equations are not the issue, but the text. 't = 0: horizontal, at (0, 1)'. Correct. 't = -1: vertical, at (2, 2)'. Correct. 't = 1: vertical, at (-2, 2)'. Correct. Is there a missing case? No. Is the derivative wrong? No. Maybe the severity is 'style' because it's too brief? No, the instructions say 'find what is wrong'. If it's correct, I should pass. But usually these adversarial reviews have a trap. Let's re-read carefully. 'Find the points where ... has a horizontal or a vertical tangent line.' The solution finds them. Is it possible that at t=1 or t=-1, the curve has a cusp? dx/dt = 3t^2-3, dy/dt = 2t. At t=1, dx/dt=0, dy/dt=2. Slope is infinite. Vertical tangent. At t=-1, dx/dt=0, dy/dt=-2. Slope is infinite. Vertical tangent. At t=0, dy/dt=0, dx/dt=-3. Slope is 0. Horizontal tangent. Everything seems correct. Wait, did I miss something about the region or bounds? No bounds given. Did I miss a point? No. Is the notation 'Subs' standard? It's pseudo-code. The prompt says 'Each line ... is either an EQUATION ... or a SENTENCE'. Lines 4, 5, 6 are marked [equation, unchecked]. The text after '--' is part of the line. If the line is an equation, the text is a comment. The comment is correct. I will pass it. Wait, looking at similar problems, sometimes students confuse x and y coordinates. x(t) = t^3 - 3t. y(t) = t^2 + 1. t=1: x = 1-3 = -2. y = 1+1 = 2. Point (-2, 2). t=-1: x = -1+3 = 2. y = 1+1 = 2. Point (2, 2). The solution states: t=-1 at (2,2) and t=1 at (-2,2). This is correct. I see no error. I will pass.
gpt-oss:20b: pass 2026-10-15

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 2026-10-18 with SymPy 1.14.0.