For each hyperbola described below: (1) derive an equation of the form
x2/a2 − y2/b2 = 1 or y2/b2 − x2/a2 = 1.
(2) state any x- or y-intercepts. (3) find the equations for the asymptotes of the hyperbola.
a. let the foci be a(−2,0) and b(2,0), and let p be a point for which either pf − pg = 2 or pg − pf = 2.
b. let the foci be a(−5,0) and b(5,0), and let p be a point for which either pf − pg = 5 or pg − pf = 5.
c. consider a(0, −3) and b(0,3), and let p be a point for which either pf− pg = 2.5 or pg − pf = 2.5.
d. consider a(0, −√2) and b(0, √2), and let p be a point for which either pf − pg = 2 or pg − pf = 2.

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