Calculus_for_Bus_and_Econ_Sample.pdf

8/9/2019 Calculus_for_Bus_and_Econ_Sample.pdf

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Calculus Applied to Business and EconomicsSolve the following problems completely. Provide complete, clear and coherent solutions. Work independently. Useonly blue or black ink for your solutions.

I. Equations, Inequalities, Functions and Graphs

(1) Profit-and-Loss Analysis. The band Freestyle has fixed costs of P40,000 for producing a new CD. Thereafter,the variable costs are P5 per CD, and the CD will sell for P100.

(a) Find and graphC(x), the total cost of producing x CDs.(b) Find and graphR(x), the total revenue from the sale ofx CDs. Use the same axes as in part (a).

(c) Find and graphP(x), the total profit from the production and sale of x CDs. Use the same axes as inpart (a) and (b).

(d) How many CDs must the band sell in order to break even?

(2) Profit-and-Loss Analysis. Market research indicates that consumers will buy x thousand units of a particularkind of coffee maker when the unit price is

p(x) = 27x + 5, 100

pesos. The cost of producing x thousand units is

C(x) = 223x2 + 350x + 8, 500

thousand pesos.

(a) What are the revenue and profit functions,R(x) and P(x), for this production process?

(b) How many coffee makers must they sell in order to break even?

(c) For what values ofx is production of the coffee makers profitable?

(d) Graph the total cost, total revenue and total profit function on the same axes.

(3) Profit-and-Loss Analysis. The demand for a particular product is given in terms of the level of productionx by

D(x) = 0.02x + 29units and total cost in dollars by

C(x) = 1.43x2 + 18.3x + 15.6

(a) What are the revenue and profit functions,R(x) and P(x), for this production process?

(b) How many units must they sell in order to break even?

(c) For what values ofx is production of this product profitable?

(d) Graph the total cost, total revenue and total profit function on the same axes.

(4) Consumer ExpenditureSuppose that x thousand units of a particular commodity are sold each month whenthe price is p euros per unit, where

p= p(x) = 5(24 x)The total monthly consumer expenditure E is the total amount of money consumers spend during eachmonth.

(a) Express total monthly expenditure Eas a function of the unit price p and sketch the graph of E(p).(Expenditure=unit price

quantity sold; derive an expression for x in terms ofp.)

(b) Discuss the economic significance of the p intercepts of the expenditure function E(p).

(c) Use the graph in part (a) to determine the market price that generates the greatest total monthlyconsumer expenditure. How many units will be sold during each month at the optimal price?

(5) Supply and DemandSuppose it is known that producers will supply x units of a certain commodity to themarket when the price is p = S(x) dollars per unit and that the same number of units will be demanded(bought) by consumers when the price is p = D(x) dollars per unit, where

p= S(x) = x2 + A and p= D(x) = Bx + 59

for constants A and B. It is also known that no units will be supplied until the unit price is $3 and thatmarket equilibrium occurs when x = 7 units.


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(a) Use this information to findA and B and the equilibrium unit price.

(b) Sketch the supply and demand curves on the same graph.

(c) What is the difference between the supply price and the demand price when 5 units are produced?When 10 units are produced?

II. Basic Concepts of Derivatives

(1) Rate of Change of ProfitA manufacturer can produce digital recorders at a cost of $50 apiece. It is estimated

that if the recorders are sold for p dollars apiece, consumers will buy q= 120p recorders each month.(a) Express the manufacturers profitPas a function ofq.

(b) What is the average rate of profit obtained as the level of production increases from q= 0 to q= 20?

(c) At what rate is profit changing when q = 20 recorders are produced? Is the profit increasing ordecreasing at this level of production?

(2) Cost ManagementA company uses a truck to deliver its products. To estimate costs, the manager modelsgas consumption by the funciton

G(x) = 1

250

1, 200

x + x

gal/mile, assuming that the truck is driven at a constant speed ofx miles per hour, for x . The driver ispaid P200 per hour to drive the truck 250 miles, and gasoline costs P40 per gallon.

(a) Find an expression for the total costC(x) of the trip.

(b) Find C(x).

(c) At what rate is the cost C(x) changing with respect tox when the truck is driven at 40 miles per hour?Is the cost increasing or decreasing at that speed?

(3) Demand and RevenueThe manager of a company that produces graphing calculators determines that whenx thousand calculators are produced, they will all be sold when the price is

p(x) = 1, 000

0.3x2 + 8

dollars per calculator.

(a) At what rate is demandp(x) changing with respect to the level of productionx when 3, 000 calculators

are produced?(b) At what rate is revenue changing when 3,000 calculators are produced? Is revenue increasing or de-

creasing at this level of production?

(4) Average Revenue, Cost and ProfitGiven revenue and cost functions

R(x) = 400x and C(x) = 50

x + 1, 000

find each of the following. Assume R(x) and C(x) are in pesos and x is the number of items produced.

(a) The average cost, the average revenue, and the average profit when items are produced and sold

(b) The rate at which average profit is changing when 9 items are produced

(5) Consumer Demand and PriceAn importer of Brazilian coffee estimates that local consumers will buy

approximatelyQ(P) =437.4

p2 kilograms of the coffee per week when the price isp pesos per kilogram. It is

estimated that t weeks from now the price of this coffee will be

p(t) = 0.4t2 + 2t + 20 pesos per kilogram.

(a) Express the weekly demand (kilograms sold) for the coffee as a function oft.

(b) How many kilograms of the coffee will consumers be buying from the importer 10 weeks from now?

(c) At what rate is the demand changing with respect to time 10 weeks from now? Is it increasing ordecreasing at this time?


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(6) Production The number of units Q of a particular commodity that will be produced with K thousanddollars of capital expenditure is modeled by

Q(K) = 500K2

3 .

Suppose that capital expenditure varies with time in such a way that t months from now there will be K(t)thousand dollars of capital expenditure, where

K(t) = 2t4

+ 3t + 149t + 2

.

(a) What will be the capital expenditure 3 months from now? How many units will be produced at thistime?

(b) At what rate will production be changing with respect to time 5 months from now? Will productionbe increasing or decreasing at this time?

(7) marginal analysisA manufacturer estimates that when x units of a particular commodity are produced,the total cost will be C(x) = 1

8x2 + 3x + 98 euros, and furthermore, that all x units will be sold when the

price is p(x) = 13

(75 x) dollars per unit.(a) Find the marginal cost and the marginal revenue.

(b) Use marginal cost to estimate the cost of producing the ninth unit.

(c) What is the actual cost of producing the ninth unit?(d) Use marginal revenue to estimate the revenue derived from the sale of the ninth unit.

(e) What is the actual revenue derived from the sale of the ninth unit?

(f) Derive the profit functionP(x) and differentiate to obtain marginal profit P(x).

(g) Use marginal profit to estimate the profit derived from the sale of the ninth unit.

(h) What is the actual profit derived from the sale of the ninth unit?

(8) Isoquant and Marginal Rate of Technical SubstitutionSuppose the output at a certain factory is

Q= 2x3 + x2y+ y3 units,

where x is the number of hours of skilled labor used and y is the number of hours of unskilled labor.The current labor force consists of 30 hours of skilled labor and 20 hours of unskilled labor. Use implicit

differentiation to estimate the change in unskilled labor y that should be made to offset a1-hour increasein skilled labor x so that output will be maintained at its current level.

(9) Related RatesThe manager of a company determines that when qhundred units of a particular commodityare produced, the total cost of production is Cthousand pesos, where C2 3q3 = 4, 275. When 1,500 unitsare being produced, the level of production is increasing at the rate of 20 units per week. What is the totalcost at this time and at what rate is it changing? (Differentiate both side with respect to time t.)

(10) Related Rates When the price of a certain commodity is p pounds per unit, the manufacturer is willing tosupply x thousand units, where

x2 2xpp2 = 31How fast is the supply changing when the price is 9 per unit and is increasing at the rate of 20 cents perweek?

III. Derivatives and Optimization

(1) Marginal AnalysisA manufacturer estimates that when q thousand units of a particular commodity areproduced each month, the total cost will be

C(q) = 0.4q2 + 3q+ 40

thousand pesos, and all qunits can be sold at a price ofp(q) = 22.2 1.2qpesos per unit.(a) Determine the level of production that results in maximum profit. What is the maximum profit?

(b) At what level of production is the average cost per unit minimized?


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(2) Marginal AnalysisA manufacturer estimates that if x units of a particular commodity are produced, thetotal cost will be C(x) pounds, where

C(x) = x3 24x2 + 350x + 338

(a) Determine the level of production that results in maximum profit. What is the maximum profit?

(b) At what level of production is the average cost per unit minimized?

(3) Elasticity of Demand Suppose the demand qand price p for a certain commodity are related by the linearequationq= 240 2p (for 0 p 120).(a) Express the elasticity of demand as a function ofp.

(b) Calculate the elasticity of demand when the price is p = 100 euros. Interpret your answer.

(c) Calculate the elasticity of demand when the price is p = 50 euros. Interpret your answer.

(d) At what price is the elasticity of demand equal to 1? What is the economic significance of this price?

(4) Elasticity of DemandThe manager of a bookstore determines that when a certain new paperback novel ispriced at p dollars per copy, the daily demand will be q= 300p2 where 0 p 300.(a) Determine where the demand is elastic, inelastic, and of unit elasticity with respect to price.

(b) Interpret the results of part (a) in terms of the behavior of total revenue as a function of price.

(5) Minimizing Inventory Cost A store in California sells 360 hybrid bicycles per year. It costs $8 to store one

bicycle for a year. To reorder, there is a fixed cost of $10, plus $2 for each bicycle.(a) How many times per year should the store order bicycles to minimize costs?

(b) What lot size will minimize costs?

(6) Minimizing Installation CostA cable is to be run from a power plant on one side of a river 1,200 meterswide to a factory on the other side, 1,500 meters downstream. The cost of running the cable under thewater is P250 per meter, while the cost over land is P200 per meter. What is the most economical routeover which to run the cable?

(7) Interest Compounding Continuously Suppose that is invested in a savings account for which interest iscompounded continuously at 4.3% per year. That is, the balance Pgrows at the rate given by

dP

dt = 0.043P.

(a) Find the function that satisfies the equation. Write it in terms ofP0 and 0.043.

(b) Suppose that P200,000 is invested. What is the balance after 1 yr? After 2 yr?

(c) When will an investment of 200,000 double itself?

(d) How much should have been invested if an accumulated value of P300,000 is expected after 3 years?

(8) Business: An Advertising Model A company begins a radio advertising campaign in New York City tomarket a new product. The percentage of the target market that buys a product is normally a functionof the duration of the advertising campaign. The radio station estimates this percentage, as a decimal, byusing

f(t) = 1 e0.08t, t 0for this type of product, where t is the number of days of the campaign. The target market is approximately1,000,000 people and the price per unit is $0.50. If the campaign costs $2,000 per day, how long should it

last in order to maximize profit?

IV. Integrals and Applications

(1) Total Cost from Marginal CostAn air conditioning company determines that the marginal cost, in pesos,for the xth air conditioner is given by

C(x) = 8x + 20, 000, C(0) = 0

Find the total cost of producing 100 air conditioners.


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(2) Total Cost from Marginal CostA company finds that the rate at which the quantity of a product thatconsumers demand changes with respect to price is given by the marginal-demand function

D(x) = 4, 000x2

where x is the price per unit, in dollars. Find the demand function if it is known that 1003 units of theproduct are demanded by consumers when the price is $4 per unit.

(3) Consumer and Producer SurplusFind (a) the equilibrium point, (b) the consumer surplus at the equilibriumpoint, and (c) the producer surplus at the equilibrium point when D(x) is the price, in pesos per unit, thatconsumers are willing to pay for x units of an item, and S(x) is the price, in pesos per unit, that producersare willing to accept forx units. Be sure to graph the supply and demand functions and label the area thatcorresponds to the measures of consumer and producer surplus.

(i) D(x) = (x 4)2; S(x) = x2 + 2x + 6(ii) D(x) = (x 3)2; S(x) = x2 + 2x + 1

Calculus_for_Bus_and_Econ_Sample.pdf

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