Home›Calculus 2›Tangent lines to parametric curves›Problem 8.45

Tangent lines to parametric curves

Problem 8.45 · hard

Find the equation of the tangent line to \( \displaystyle x = \cos{\left(t \right)},\ y = t^{3} \) at \( \displaystyle t = \frac{\pi}{4} \).

dy/dx = (dy/dt)/(dx/dt).
Reviewed
\[ \frac{d}{d t} \cos{\left(t \right)} = - \sin{\left(t \right)} \]
dx/dt.✓ Proved
\[ \frac{d}{d t} t^{3} = 3 t^{2} \]
dy/dt.✓ Proved
\[ - \frac{3 \sqrt{2} \pi^{2}}{16} \]
The slope at t = \frac{\pi}{4}.Reviewed
\[ \left[\begin{matrix}\frac{\sqrt{2}}{2}\\\frac{\pi^{3}}{64}\end{matrix}\right] \]
The point.Reviewed
\[ - \frac{3 \sqrt{2} \pi^{2} \left(x - \frac{\sqrt{2}}{2}\right)}{16} + \frac{\pi^{3}}{64} = - \frac{3 \sqrt{2} \pi^{2} x}{16} + \frac{\pi^{3}}{64} + \frac{3 \pi^{2}}{16} \]
Point-slope form.✓ Proved

Answer \( y = - \frac{3 \sqrt{2} \pi^{2} x}{16} + \frac{\pi^{3}}{64} + \frac{3 \pi^{2}}{16} \)

✓ Nihil obstat Lines: 3 proved, 3 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	✓ 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	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 equations, computes the derivatives and point coordinates accurately, and derives the correct tangent line equation.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-09-28 — The solution correctly applies the chain rule for parametric equations, computes the derivatives and point coordinates accurately, and derives the correct tangent line equation.
gpt-oss:20b: pass 2026-09-28
qwen3.6:27b-mlx: pass 2026-09-27 — The solution correctly applies the chain rule for parametric curves, computes the derivatives and point coordinates accurately, and derives the correct tangent line equation.
gpt-oss:20b: pass 2026-09-27
qwen3.6:27b-mlx: pass 2026-09-26
gpt-oss:20b: pass 2026-09-26

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.