INSTRUCTOR: A particle moves in a path defined by the vector valued function r of t equals parentheses t squared minus 3t close parentheses i plus open parentheses 2t minus 4 close parentheses j plus open parentheses t plus 2 close parentheses k, where t measures time in seconds and where distance is measured in feet. Find the velocity, acceleration, and speed as functions of time.

Well, velocity, so v of t, velocity is equal to the derivative of our position function. So that would mean our velocity is equal to parentheses 2t minus 3 close parentheses i plus 2j plus k, and that is in feet per second, if we had a value for it.

Acceleration, a of t, is equal to the derivative of velocity. The derivative of velocity here is 2i plus-- well, actually, that's it. So 2i and that would be feet per second squared.

Finally, speed is equal to the magnitude of our velocity function. That would be equal to the square root of parentheses 2t minus 3 close parentheses squared plus 2 squared plus 1 squared, square root of all of that. That would be equal to, if we expanded this, the square root of the quantity 4t squared minus 12t plus 14. And those units would be feet per second for the speed.