13The Marginal Cost, Marginal Revenue, and Marginal ProfitThe marginal cost, marginal revenue, and marginal profit mean the rate of change,or slope, of linear revenue, cost, and profit functions, respectively.Example.Using the previous example’s cost, revenue, and profit functions for producing and sellingG-pods, determine the marginal cost, marginal revenue, and marginal profit.The Depreciation FunctionExample.A flying car that costs $12,500 depreciates linearly to zero in 10 years. Find a formula forthe flying car’s value as a function of time.How much is the flying car worth three years after it ispurchased?
14CHAPTER 1.FUNCTIONS AND CHANGESupply and Demand Curves•Thesupply curve, for a given item, relates the quantity,q, of the item thatmanufacturers are willing to make per unit time to the price,p, for which theitem can be sold.•Thedemand curverelates the quantity,q, of an item demanded by consumersper unit time to the price,p, of the item.•Theequilibrium pointis the price and quantity of an item in which the supplyis equal to the demand, i.e., supply = demand.Example.The demand curve for filet mignon steaks is given byq= 20000-50pand the supply curveis given byq= 450pfor 0≤q≤20000, where price is in dollars.a)If the price of a filet mignon steak was $65, what quantity are consumers willing to buy and whatquantity are producers willing to supply?b)Find the equilibrium price and quantity. Interpret your results in context of filet mignon steaks.
1.5.EXPONENTIAL FUNCTIONS151.5Exponential FunctionsExponential FunctionsLetP(t) be an exponential function such thatP(t) =P0at,whereP0is the initial quantity andais a factor by whichP(t) changes.•Ifa >1, thenP(t) has exponential growth. In this case the growth factor isgiven bya= 1 +r, whereris the growth rate.•If 0< a <1, thenP(t) has exponential decay. In this case the decay factor isgiven bya= 1-r, whereris the decay rate.Example.The caffeine in the body from a soy vanilla latte,C, changes with time. Find a formula forCin mg at timetminutes later ifCis:a)increasing by 0.25 mg per minute.b)decreasing by 0.25 mg per minute.c)increasing by 3.65% per minute.d)decreasing by 3.65% per minute.A linear function hasconstant average rate of change.Anexponentialfunctionhasconstantsuccessiveratio,i.e.,constantpercentrate of change.