Calculus of vector-valued functions
Problem 9.419 · medium
Find the unit tangent vector \( \displaystyle \mathbf T \) of \( \displaystyle \mathbf r(t) = \left\langle 3 t, 4 t^{2}, 2 t^{3} \right\rangle \) at \( \displaystyle t = 2 \).
- \[ \left[\begin{matrix}\frac{d}{d t} 3 t\\\frac{d}{d t} 4 t^{2}\\\frac{d}{d t} 2 t^{3}\end{matrix}\right] = \left[\begin{matrix}3\\8 t\\6 t^{2}\end{matrix}\right] \]r′(t).✓ Proved
- \[ 29 \]‖r′(2)‖.✓ Proved
- \[ \left[\begin{matrix}\frac{3}{29}\\\frac{16}{29}\\\frac{24}{29}\end{matrix}\right] \]T = r′/‖r′‖.✓ Proved
Answer \( \mathbf T(2) = \left\langle \frac{3}{29}, \frac{16}{29}, \frac{24}{29} \right\rangle \)
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. The reviewers disagree about how one step is explained; every verdict is in the receipt.
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 — The solution correctly computes the derivative, evaluates its magnitude at t=2, and normalizes the vector. The steps are logically sound and the final answer is correct.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-08qwen3.6:27b-mlx: pass 2026-10-08 — The solution correctly computes the derivative, evaluates its magnitude at t=2, and normalizes the vector. The steps are logically sound and the final answer is correct.qwen3.6:27b-mlx: fail (error) 2026-10-08 — The solution fails to explicitly evaluate the derivative r'(t) at t=2 before normalizing. While the final numerical answer is correct, the logical step from r'(t) = <3, 8t, 6t^2> to the normalized vector <3/29, 16/29, 24/29> skips the crucial evaluation r'(2) = <3, 16, 24>, making the derivation opaque and potentially misleading regarding the order of operations.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.