Calculus of vector-valued functions
Problem 9.415 · medium
For \( \displaystyle \mathbf r(t) = \left\langle \cos{\left(3 t \right)}, \sin{\left(3 t \right)}, t \right\rangle \), find \( \displaystyle \mathbf r'(t) \) and the tangent vector at \( \displaystyle t = \pi \).
- \[ \left[\begin{matrix}\frac{d}{d t} \cos{\left(3 t \right)}\\\frac{d}{d t} \sin{\left(3 t \right)}\\\frac{d}{d t} t\end{matrix}\right] = \left[\begin{matrix}- 3 \sin{\left(3 t \right)}\\3 \cos{\left(3 t \right)}\\1\end{matrix}\right] \]Differentiate each component.✓ Proved
- \[ \left[\begin{matrix}0\\-3\\1\end{matrix}\right] \]At t = pi.✓ Proved
Answer \( \mathbf r'(t) = \left\langle - 3 \sin{\left(3 t \right)}, 3 \cos{\left(3 t \right)}, 1 \right\rangle,\ \mathbf r'(\pi) = \left\langle 0, -3, 1 \right\rangle \)
✓ Nihil obstat Every line of this solution was proved by the computer algebra system SymPy. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation.
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 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | central difference quotients of each component |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass — The solution correctly differentiates the vector function and evaluates it at t = pi. The steps are logically sound and the final answer matches the stated answer.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly differentiates the vector function and evaluates it at t = pi. The steps are logically sound and the final answer matches the stated answer.qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly differentiates the vector function and evaluates it at t = pi. The steps are logically sound and the final answer is correct.gpt-oss:20b: pass 2026-10-08
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/vector_function_calculus, checked 2026-10-08 with SymPy 1.14.0.