Consider the case when S[y] = [" √1+32 dx y(0) = 8, 0 y where y(v) > 0, 5 > 0 and the right-hand end point (v, y(v)) lies on the line ay +ẞx+y= 0, where a, ß, y are constants with ẞ +0. Show that the first-integral may be written as y√1+y = C for some constant c > 0, and that the solutions of the first-integral are circles centred at (-cs, 0) with radius C₁ i.e. y² + (x + cs)² = c², where c² = c² - 8². * Using the transversality condition and by differentiating implicitly the equation y²+(x + cs)² = c², show that y' = a/ẞ and cs = y/ß. Finally, show that in the limit as → 0, the stationary path becomes ß²y² + (ßx + y)² = 7².

The supply function for x thousand units of a product is p = x2 + 2x, dollars per unit. If the equilibrium price is $48 per unit, find the producer’s surplus. Set up an integral, and use your calculator to evaluate it.