Final answer:
To calculate the velocity and acceleration vectors and the speed at t=0 for the particle with position function r(t), we must differentiate r(t) to find v(t) and a(t), then evaluate these derivatives at t=0. For speed, we find the magnitude of the velocity vector at t=0.
Step-by-step explanation:
The student is asking for the velocity and acceleration vectors and the speed of a particle at a specific time, given its position function. In this case, the position function is r(t) = e⁵ᵗj - 9 cos(6t)k, and the time indicated is t = 0. To find the velocity v(t), we need to take the derivative of the position function with respect to t.
Similarly, to find the acceleration a(t), we need to take the derivative of the velocity function with respect to t. Speed is the magnitude of the velocity vector and can be calculated by taking the square root of the sum of the squares of the velocity vector components.
Firstly, we find v(0), by differentiating r(t) with respect to t and evaluating it at t = 0.
Secondly, we find a(0), by differentiating v(t) with respect to t and evaluating it at t = 0. Lastly, we calculate the speed at t = 0 by finding the magnitude of the v(0) vector.