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Calculus of vector-valued functions

Problem 9.334 · medium

For \( \displaystyle \mathbf r(t) = \left\langle t, t^{2} - 1, 2 t \right\rangle \), find \( \displaystyle \mathbf r'(t) \) and the tangent vector at \( \displaystyle t = 0 \).

\[ \left[\begin{matrix}\frac{d}{d t} t\\\frac{d}{d t} \left(t^{2} - 1\right)\\\frac{d}{d t} 2 t\end{matrix}\right] = \left[\begin{matrix}1\\2 t\\2\end{matrix}\right] \]
Differentiate each component.✓ Proved
\[ \left[\begin{matrix}1\\0\\2\end{matrix}\right] \]
At t = 0.✓ Proved

Answer \( \mathbf r'(t) = \left\langle 1, 2 t, 2 \right\rangle,\ \mathbf r'(0) = \left\langle 1, 0, 2 \right\rangle \)

✓ 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: pass
qwen3.6:27b-mlx: pass — The solution correctly differentiates the vector-valued function component-wise and evaluates the derivative at t=0. The steps are logically sound and algebraically correct.

Every verdict on record (4)

gpt-oss:20b: pass 2026-10-07
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly differentiates the vector-valued function component-wise and evaluates the derivative at t=0. The steps are logically sound and algebraically correct.
qwen3.6:27b-mlx: pass 2026-10-07 — The solution correctly differentiates the vector function component-wise and evaluates the result at t=0. The algebraic steps are verified, and the logic is sound.
gpt-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.