Home›Calculus 2›Tangent lines to parametric curves›Problem 8.75

Tangent lines to parametric curves

Problem 8.75 · hard

Find the equation of the tangent line to \( \displaystyle x = t^{3} - t,\ y = t^{2} - 3 t \) at \( \displaystyle t = -1 \).

dy/dx = (dy/dt)/(dx/dt).
Reviewed
\[ \frac{d}{d t} \left(t^{3} - t\right) = 3 t^{2} - 1 \]
dx/dt.✓ Proved
\[ \frac{d}{d t} \left(t^{2} - 3 t\right) = 2 t - 3 \]
dy/dt.✓ Proved
\[ - \frac{5}{2} \]
The slope at t = -1.Reviewed
\[ \left[\begin{matrix}0\\4\end{matrix}\right] \]
The point.Reviewed
\[ 4 - \frac{5 x}{2} \]
Point-slope form.Reviewed

Answer \( y = 4 - \frac{5 x}{2} \)

✓ Nihil obstat Lines: 2 proved, 4 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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
5	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	both sides are the same expression: the line states a value, and the working that produced it is not on the line
6	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass — The solution correctly applies the chain rule for parametric curves, computes the derivatives and slope accurately, identifies the correct point, and derives the correct tangent line equation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-04 — The solution correctly applies the chain rule for parametric curves, computes the derivatives and slope accurately, identifies the correct point, and derives the correct tangent line equation.
gpt-oss:20b: pass 2026-10-04
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the chain rule for parametric derivatives, evaluates the slope and point at t = -1, and derives the correct tangent line equation.
gpt-oss:20b: pass 2026-09-29
qwen3.6:27b-mlx: pass 2026-09-29 — The solution correctly applies the chain rule for parametric derivatives, calculates the slope and point at t = -1, and derives the correct tangent line equation.
gpt-oss:20b: pass 2026-09-29

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.