Chapter 3 | Derivatives
227
Example 3.10 Rate of Change of Profit
A toy company can sell x electronic gaming systems at a price of p =−0.01 x +400 dollars per gaming system. The cost of manufacturing x systems is given by C ( x ) =100 x +10,000 dollars. Find the rate of change of profit when 10,000 games are produced. Should the toy company increase or decrease production? Solution The profit P ( x ) earned by producing x gaming systems is R ( x )− C ( x ), where R ( x ) is the revenue obtained from the sale of x games. Since the company can sell x games at p =−0.01 x +400 per game, R ( x ) = xp = x (−0.01 x + 400) = −0.01 x 2 +400 x . Consequently, P ( x ) =−0.01 x 2 +300 x −10,000. Therefore, evaluating the rate of change of profit gives P ′ (10000) = lim x →10000 P ( x )− P (10000) x −10000 = lim x →10000 −0.01 x 2 +300 x − 10000 − 1990000 x −10000 = lim x →10000 −0.01 x 2 +300 x −2000000 x −10000 =100. Since the rate of change of profit P ′ (10,000) > 0 and P (10,000) > 0, the company should increase production.
3.5 A coffee shop determines that the daily profit on scones obtained by charging s dollars per scone is P ( s ) =−20 s 2 +150 s −10. The coffee shop currently charges $3.25 per scone. Find P ′(3.25), the rate of change of profit when the price is $3.25 and decide whether or not the coffee shop should consider raising or lowering its prices on scones.
Made with FlippingBook - professional solution for displaying marketing and sales documents online