Find d/dt [r(t) · u(t)] and d/dt [r(t) × u(t)] in two different ways.
r(t) = ti + 2t2j + t3k, u(t) = t4k

(a) d/dt [r(t) · u(t)]
(i) Find the product first, then differentiate.
(ii) Apply the properties of the derivative.

(b) d/dt [r(t) × u(t)]
(i) Find the product first, then differentiate.
(ii) Apply the properties of the derivative.

Four identical lab carts each have a mass of 200 kg. different masses are added to the carts and the velocities are measured. all carts move to the right.

Estimate of the number of stars in our galaxy

The bubbles in a carbonated soft drink are produced when carbonic acid decomposes to form carbon dioxide and water. in a closed system, this equilibrium exists as why does a carbonated soft drink lose carbonation when the container is left open? a. water evaporates, which favors the formation of h2co3. b. pressure decreases, which favors the formation of h2co3. c. the open container warms up, which causes more gas to escape. d. co2is removed, which favors the formation of h2co3.