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

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

• A. sqrt(1 + 4 t^2 + 9 t^4)
• B. sqrt(1 + 2 t + 3 t^2)
• C. sqrt(1 + t^2 + t^4)
• D. sqrt(1 + 4 t^2 + 2 t^4)

Answer: (A)

Speed is the length of the velocity vector, which comes from differentiating each coordinate of the position function. For r(t) = ⟨t, t^2, t^3⟩, the velocity is v(t) = r′(t) = ⟨1, 2t, 3t^2⟩. The speed is the magnitude of this vector: |v(t)| = √(1^2 + (2t)^2 + (3t^2)^2) = √(1 + 4t^2 + 9t^4). That matches the expression with 1, 4t^2, and 9t^4 under the square root. The other forms would come from miscomputing the squared terms or introducing incorrect coefficients.

Speed is the length of the velocity vector, which comes from differentiating each coordinate of the position function.

For r(t) = ⟨t, t^2, t^3⟩, the velocity is v(t) = r′(t) = ⟨1, 2t, 3t^2⟩. The speed is the magnitude of this vector:

|v(t)| = √(1^2 + (2t)^2 + (3t^2)^2) = √(1 + 4t^2 + 9t^4).

That matches the expression with 1, 4t^2, and 9t^4 under the square root. The other forms would come from miscomputing the squared terms or introducing incorrect coefficients.