Schaum's Outline of Theory and Problems of Beginning Calculus

144 INSTANTANEOUS RATE OF CHANGE[CHAP. 19then x + Ax, and the new value of y will bef(x + Ax). Hence, the change Ay in the value of the functionisAy =f(x + Ax) - f(x)The ratio of the change in the function value to the change in the argument isThe instantaneous rate of change of y with respect to x is defined to beAYlim -= limAx-OAx Ax-0 AxThe instantaneous rate of change is evaluated by the derivative. It follows that, for Ax close to 0, Ay/Axwill be close tof’(x), so thatAy x f’(x) Ax (1 9.1)Solved Problems19.1 The weekly profit P, in dollars, of a corporation is determined by the number x of radios producedper week, according to the formulaP = 7 5 - ~ 0 .03~~ - 15000(a) Find the rate at which the profit is changing when the production level x is loo0 radios perweek. (6) Find the change in weekly profit when the production level x is increased to 1001radios per week.(a) The rate of change of the profit P with respect to the production level x is dP/dx = 75 - 0.06~. Whenx = 1000,-= dPdx75 - O.O6(1ooO) = 75 - 60 = 15 dollars per radio(b) In economics, the rate of change of profit with respect to the production level is called the marginalpro& According to (29.2), the marginal profit is an approximate measure of how much the profit willchange when the production level is increased by one unit. In the present case, we haveP(1000) = 75(1OOO) - 0.03(1000)2 - 15000= 75000 - 30000 - 15000 = 30000P(1001) = 75(1001) - 0.03(1001)2 - 15000= 75075 - 30060.03 - 15000 30014.97AP = P(1001) - P(1OOO) = 14.97 dollars per weekwhich is very closely approximated by the marginal profit, 15 dollars, as computed in part (a).19.2 The volume V of a sphere of radius r is given by the formula V = 4nr3/3. (a) How fast is thevolume changing relative to the radius when the radius is 10 millimeters? (b) What is the changein volume when the radius changes from 10 to 10.1 millimeters?