Tangent lines to parametric curves
Problem 8.67 · medium
Find the equation of the tangent line to \( \displaystyle x = 2 t,\ y = t^{3} - 3 t \) at \( \displaystyle t = -1 \).
- dy/dx = (dy/dt)/(dx/dt).
- \[ \frac{d}{d t} 2 t = 2 \]dx/dt.✓ Proved
- \[ \frac{d}{d t} \left(t^{3} - 3 t\right) = 3 t^{2} - 3 \]dy/dt.✓ Proved
- \[ 0 \]The slope at t = -1.
- \[ \left[\begin{matrix}-2\\2\end{matrix}\right] \]The point.
- \[ 2 \]Point-slope form.
Answer \( y = 2 \)
Lines: 2 proved, 4 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 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 5 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| 6 | Not checked | — | both sides are the same expression: the line states a value, and the working that produced it is not on the line |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | a secant through nearby points of the curve has the same slope; the line passes through the point |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — The solution fails to explicitly state the point-slope equation or the final linear equation, merely asserting 'Point-slope form' without showing the derivation. Additionally, the final answer 'y = 2' is presented without showing the substitution of the point and slope into the equation, making the logical gap between the components and the final result opaque.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (error) 2026-10-03 — The solution fails to explicitly state the point-slope equation or the final linear equation, merely asserting 'Point-slope form' without showing the derivation. Additionally, the final answer 'y = 2' is presented without showing the substitution of the point and slope into the equation, making the logical gap between the components and the final result opaque.gpt-oss:20b: pass 2026-10-03qwen3.6:27b-mlx: fail (style) 2026-09-28 — [domain objection, downgraded to style] The slope calculation is incorrect; dy/dt at t=-1 is 0, but dx/dt is 2, so the slope is 0, not undefined or handled incorrectly. However, the final answer y=2 is correct for a horizontal tangent. The main issue is that line 4 claims the slope is 0 (which is correct) but the notation '0, 0' is ambiguous and line 6 '2, 2' is not a valid derivation step. More critically, the solution fails to explicitly state the point-slope equation y - y1 = m(x - x1) leading to y - 2 = 0(x + 2) => y = 2. The provided 'equations' are just values, not the logical steps to derive the line equation. Specifically, line 6 '2, 2' does not represent the equation y=2. It's a non-sequitur.gpt-oss:20b: pass 2026-09-28qwen3.6:27b-mlx: fail (error) 2026-09-28 — The slope calculation is incorrect; dy/dt at t=-1 is 0, but dx/dt is 2, so the slope dy/dx should be 0, not 2. The final equation y=2 is correct, but the derivation claims a slope of 2.gpt-oss:20b: fail (error) 2026-09-28 — The solution incorrectly states the slope as "0, 0" and later lists the point as "2, 2" in the point‑slope form. The correct slope is 0 and the correct point is (-2, 2). The erroneous values would mislead a student.
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, checked 2026-10-13 with SymPy 1.14.0.