Calculus of vector-valued functions
Problem 9.259 · medium
For \( \displaystyle \mathbf r(t) = \left\langle t \cos{\left(t \right)}, t \sin{\left(t \right)}, t \right\rangle \), find \( \displaystyle \mathbf r'(t) \) and the tangent vector at \( \displaystyle t = 2 \).
- \[ \left[\begin{matrix}\frac{d}{d t} t \cos{\left(t \right)}\\\frac{d}{d t} t \sin{\left(t \right)}\\\frac{d}{d t} t\end{matrix}\right] = \left[\begin{matrix}- t \sin{\left(t \right)} + \cos{\left(t \right)}\\t \cos{\left(t \right)} + \sin{\left(t \right)}\\1\end{matrix}\right] \]Differentiate each component.✓ Proved
- \[ \left[\begin{matrix}- 2 \sin{\left(2 \right)} + \cos{\left(2 \right)}\\2 \cos{\left(2 \right)} + \sin{\left(2 \right)}\\1\end{matrix}\right] \]At t = 2.✓ Proved
Answer \( \mathbf r'(t) = \left\langle - t \sin{\left(t \right)} + \cos{\left(t \right)}, t \cos{\left(t \right)} + \sin{\left(t \right)}, 1 \right\rangle,\ \mathbf r'(2) = \left\langle - 2 \sin{\left(2 \right)} + \cos{\left(2 \right)}, 2 \cos{\left(2 \right)} + \sin{\left(2 \right)}, 1 \right\rangle \)
✓ 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 each component of the vector function and evaluates the result at t=2. The steps are logically sound and the final answer matches the stated answer.
Every verdict on record (4)
qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly differentiates each component of the vector function and evaluates the result at t=2. The steps are logically sound and the final answer matches the stated answer.gpt-oss:20b: pass 2026-10-05qwen3.6:27b-mlx: pass 2026-10-05 — The solution correctly differentiates each component of the vector function and evaluates the derivative at t=2. The steps are logically sound and the final answer matches the stated answer.gpt-oss:20b: pass 2026-10-05
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 2026-10-05 with SymPy 1.14.0.