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.

A47.2 g block of copper whose temperature is 480 k is placed in an insulating box with a 91.8 g block of lead whose temperature is 200 k. (a) what is the equilibrium temperature of the two-block system? (b) what is the change in the internal energy of the two-block system between the initial state and the equilibrium state? (c) what is the change in the entropy of the two-block system? the heat capacities of copper and lead are 386 j/kg·k and 128 j/kg·k, respectively.