For r(t) = ⟨t, t^2, t^3⟩, which expression represents the speed |v(t)|?

For r(t) = ⟨t, t^2, t^3⟩, which expression represents the speed |v(t)|?

• A. sqrt(65)
• B. sqrt(100)
• C. sqrt(85)
• D. sqrt(161)

Answer: (D)

Speed is the magnitude of the velocity vector, so first find the velocity by differentiating the position function: r′(t) = ⟨1, 2t, 3t^2⟩. The speed is the length of this vector: |v(t)| = sqrt(1^2 + (2t)^2 + (3t^2)^2) = sqrt(1 + 4t^2 + 9t^4). If we evaluate at t = 2, we get |v(2)| = sqrt(1 + 4·4 + 9·16) = sqrt(1 + 16 + 144) = sqrt(161). That matches the given choice. In general, the speed is the function sqrt(9t^4 + 4t^2 + 1); the listed constants correspond to a specific t, namely t = 2.

Speed is the magnitude of the velocity vector, so first find the velocity by differentiating the position function: r′(t) = ⟨1, 2t, 3t^2⟩. The speed is the length of this vector: |v(t)| = sqrt(1^2 + (2t)^2 + (3t^2)^2) = sqrt(1 + 4t^2 + 9t^4).

If we evaluate at t = 2, we get |v(2)| = sqrt(1 + 4·4 + 9·16) = sqrt(1 + 16 + 144) = sqrt(161). That matches the given choice. In general, the speed is the function sqrt(9t^4 + 4t^2 + 1); the listed constants correspond to a specific t, namely t = 2.