Calculus of vector-valued functions
Problem 9.337 · medium
Find the unit tangent vector \( \displaystyle \mathbf T \) of \( \displaystyle \mathbf r(t) = \left\langle \cos{\left(4 t \right)}, \sin{\left(4 t \right)}, t \right\rangle \) at \( \displaystyle t = 1 \).
- \[ \left[\begin{matrix}\frac{d}{d t} \cos{\left(4 t \right)}\\\frac{d}{d t} \sin{\left(4 t \right)}\\\frac{d}{d t} t\end{matrix}\right] = \left[\begin{matrix}- 4 \sin{\left(4 t \right)}\\4 \cos{\left(4 t \right)}\\1\end{matrix}\right] \]r′(t).✓ Proved
- \[ \sqrt{1 + 16 \cos^{2}{\left(4 \right)} + 16 \sin^{2}{\left(4 \right)}} = \sqrt{17} \]‖r′(1)‖.✓ Proved
- \[ \left[\begin{matrix}- \frac{4 \sin{\left(4 \right)}}{\sqrt{1 + 16 \cos^{2}{\left(4 \right)} + 16 \sin^{2}{\left(4 \right)}}}\\\frac{4 \cos{\left(4 \right)}}{\sqrt{1 + 16 \cos^{2}{\left(4 \right)} + 16 \sin^{2}{\left(4 \right)}}}\\\frac{1}{\sqrt{1 + 16 \cos^{2}{\left(4 \right)} + 16 \sin^{2}{\left(4 \right)}}}\end{matrix}\right] = \left[\begin{matrix}- \frac{4 \sqrt{17} \sin{\left(4 \right)}}{17}\\\frac{4 \sqrt{17} \cos{\left(4 \right)}}{17}\\\frac{\sqrt{17}}{17}\end{matrix}\right] \]T = r′/‖r′‖.✓ Proved
Answer \( \mathbf T(1) = \left\langle - \frac{4 \sqrt{17} \sin{\left(4 \right)}}{17}, \frac{4 \sqrt{17} \cos{\left(4 \right)}}{17}, \frac{\sqrt{17}}{17} \right\rangle \)
✓ 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 |
| 3 | ✓ 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 numerical velocity, normalised |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: pass
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07qwen3.6:27b-mlx: pass 2026-10-07gpt-oss:20b: pass 2026-10-07
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 2026-10-07 with SymPy 1.14.0.