Once upon a time a brave and mathematically-inclined warrior named Al Jebra set out on a quest to rescue the kidnapped Princess Polly Nomial. First, Al needed to get to the room of the stolen Princess to read the ransom note left behind. The Princess lived in the tower with no stairs (for security purposes, ironically). Al plans to use his 17 ft ladder to climb in a window 15 ft up the tower wall.

How far from the wall should Al position the bottom of the ladder so that the top reaches the window ledge? The ladder is 17 ft long.

If p(x) is the total value of the production when there are x workers in a plant, then the average productivity of the workforce at the plant is a(x) = p(x) x . (a) find a'(x). a'(x) = xp'(x) − p(x) x a'(x) = xp'(x) − p(x) x2 a'(x) = p'(x) − p(x) x a'(x) = xp'(x) − p'(x) x2 a'(x) = p'(x) − xp(x) x2 why does the company want to hire more workers if a'(x) > 0? a'(x) > 0 ⇒ a(x) is ; that is, the average productivity as the size of the workforce increases. (b) if p'(x) is greater than the average productivity, which of the following must be true? p'(x) − xp(x) > 0 p'(x) − xp(x) < 0 xp'(x) − p'(x) > 0 xp'(x) − p(x) < 0 xp'(x) − p(x) > 0