Aplicações de Derivadas: Problemas e Cálculos de Raízes

146 APPLICATIONS OF DERIVATIVES PROBLEMS 1 By sketching the graph of y fx x3 3x 5 show 8 Suppose that by good luck our first approximation x that equation 1 has only one real root Hint Use the happens to be the root of fx 0 that we are seeking derivative fx 3x2 3 3x2 1 to locate the What does this imply about x2 x3 etc maxima and minima of the function and to learn where 9 Show that the function y fx defined by it is increasing and decreasing a 2 2 a Show that x 3x2 6 0 has only one real root fw Ve a and calculate it to six decimal places of accuracy or b Show that x7 3x 8 has only one real root and has the property illustrated in Fig 439 that is for any calculate it to six decimal places of accuracy positive number a if x r a then x r and if 3 Use Newtons method to calculate the positive root of x ra thn xrta x x 1 O to six decimal places of accuracy 10 Show that Newtons method applied to the function y 4 Calculate V5 to six decimal places of accuracy by solv fx Wx leads to x2 2x and is therefore useless ing the equation x2 5 0 and use this result in the for finding where fx 0 Sketch the situation quadratic formula to check the answer to Problem 3 11 In Example of Section 41 we saw from its graph that 5 Use Newtons method to calculate V10 to six decimal the function y fx 2x3 3x2 12x 12 has pos places of accuracy itive zeros close to x 09 and x 29 Use Newtons 6 Consider a spherical shell ft thick whose volume equals method to calculate these zeros to six decimal places of the volume of the hollow space inside it Use Newtons accuracy method to calculate the shells outer radius to six deci 12 Find a solution of 2x cos x correct to six decimal mal places of accuracy places 7 Ahollow spherical buoy of radius 2 ft has specific grav 13 Find the smallest positive solution of each of the fol ity so it floats on water in such a way as to displace lowing equations correct to six decimal places its own volume Show that the depth x to which it is sub a 4x 1 sinx merged is a root of the equation x 6x2 8 0 and b x sin x use Newtons method to calculate this root to six deci 14 How many solutions does the equation mal places of accuracy Hint The volume of a spherical segment of height A cut from a sphere of radius a is we ah2a h3 have Why Ever since its beginning calculus has served primarily as a tool for the physical 4 sciences The uses of mathematics in the social sciences have arisen more re OPTIONAL cently In this section we discuss several applications of calculus to microeco App een ae nomics the branch of economics that studies the economic decisions of individ ual businesses or industries More precisely we focus our attention on the ECONOMICS production and marketing of a single commodity by a single firm MARGINAL ANALYSIS The most important management decisions in a particular firm usually depend on the costs and revenues involved We shall examine applications of derivatives to the cost and revenue functions COST MARGINAL COST AND AVERAGE COST The total cost to a firm of producing x units of a given commodity is a certain function of x called the cost function and denoted by Cx Here x can be the number of pieces produced or the number of pounds or the number of bushels and so on The cost Cx can be measured in dollars in thousands of dollars in French francs or in any other monetary unit To determine the cost function Cx is a difficult task for experts in book keeping and accounting Here however we take this function as given We shall 47 OPTIONAL APPLICATIONS TO ECONOMICS MARGINAL ANALYSIS 147 CQ Figure 440 assume for the sake of definiteness that x is the number of pieces or units pro duced and therefore a nonnegative integer and also that the cost Cx is mea sured in dollars For most commodities such as TV sets or calculators x can only be a nonnegative integer so the graph of Cx might look like the sequence of dots in Fig 440 However economists usually assume that these dots are con nected by asmooth curve as shown in the figure Accordingly Cx is understood to be defined for all nonnegative values of x not just for nonnegative integers Many components make up the total cost Some like capital expenditures for buildings and machinery are fixed and do not depend on x Others like wages and the cost of raw materials are roughly proportional to the amount x produced If this were all then the cost function would have the very simple form Cx a bx where a is the fixed cost and b is the constant running cost per unit But this is not all and most cost functions are not as simple as this The es sential point is the fact that a time restriction is present and that Cx is the cost of producing x units of the product in a given time interval say week There will then be a fixed cost of a dollars per week as before but the variable part of the cost will probably increase more than proportionally to x as the weekly production x increases because of overtime wages the need to use older ma chinery that breaks down more frequently and other inefficiencies that arise from forcing production to higher and higher levels The cost function Cx might then have the form at bxtcx or a bx t cx dx or it might be a function even more complicated than these The general nature of such a cost function is suggested in Fig 440 The derivative Cx of the cost function is called the marginal cost This de rivative is or course the rate of change of cost with respect to the production level x The economic meaning of this important concept will become clearer as we proceed As a first step in this direction we point out that it is a good approximation to think of the marginal cost Cx at a given production level x as the extra cost of producing one more unit To see this we recall the definition of the derivative Ce RiAn GG CO fig 148 APPLICATIONS OF DERIVATIVES We therefore have the approximation Cl fo Oy where this approximation is good if Ax is suitably small It is customary in economics to assume that Ax 1 meets the requirement of being suitably small Therefore we have Cx Cx 1 Cx approximately In words the marginal cost at each level of production x is the extra cost required to produce the next unit of output the x 1st unit Example 1 Suppose a company has estimated that the cost in dollars of pro ducing x units is Cx 5000 7x 002x Then the marginal cost is Cx 7 004x The marginal cost at the production level of 1000 units is C1000 7 0041000 47unit The exact cost of producing the 1001st unit is C1001 C1000 5000 71001 00210012 5000 71000 0021000 4702 The difference between the marginal cost for x 1000 and the exact cost of pro ducing the 1001st unit is clearly negligible The graph of a typical cost function is shown in Fig 441 This cost function is increasing because it costs more to produce more The marginal cost Cx is the slope of the tangent to the cost curve The cost curve is initially concave down the marginal cost is decreasing because it costs more to produce the first piece than to produce one more piece when many are being produced this reflects the more efficient use of the fixed costs of production At a certain production level Xo there is a point of inflection Po and the cost curve becomes concave up the marginal cost is increasing because when we produce almost as much as we can it becomes more expensive to increase production by even a small amount As we suggested earlier reasons for this might include greater overtime costs or more frequent breakdowns of the equipment as we strain our productive capac ity It is a reasonable view that the most efficient production level for a manufac turer is that which minimizes the average cost Cx which of course is the cost per unit when x units are produced We sketch a typ ical average cost curve in Fig 442 by noticing that Cxx is the slope of the line joining the origin to the point P in Fig 441 We know that some cost is un 47 OPTIONAL APPLICATIONS TO ECONOMICS MARGINAL ANALYSIS Cx CO x x Figure 441 A cost function Figure 442 Marginal cost and average cost avoidable even before a single unit is producedfor instance the capital ex penditures mentioned earlier utilities insurance and so onso CO 0 This shows that Cxx has the limit oo as x 7 0 We are particularly interested in the minimum that Cxx appears to have To locate this minimum we find the critical point x1 of the function Cxx by calculating the derivative by means of the quotient rule Cx xC x Cx dx x x2 This derivative must be zero at the critical point so xC x Cx 0 or Cx Cx x We therefore have the following basic law of economics If the average cost is a minimum then marginal cost average cost I In other words at the peak of operating efficiency the marginal cost equals the average cost We see from this that the graphs of marginal cost and average cost intersect at the point of minimum average cost as shown in Fig 442 Like other principles of economics this is usually established by extensive verbal discus sions supported by tables and graphs However the calculus derivation is brief and clear Equation 1 has an interesting geometric interpretation for the cost function shown in Fig 441 At the production level x x1 where Cxx has a minimum the line from the origin to the point Pi is tangent to the graph We can see the reason for this by noticing that the average cost Cxlx decreases as P moves to the right along the curve toward Pi and then increases as P moves beyond Pi 149 150 APPLICATIONS OF DERIVATIVES Example 2 A firm estimates that the cost in dollars of producing x units is Cx 3400 4x 0002x a Find the cost marginal cost and average cost of producing 500 units 1000 units 1500 units and 2000 units b What is the minimum average cost and at what production level is this achieved Solution a The marginal cost is Cx 4 0004x The average cost is 4 EY 220 codon Xx Xx We use these formulas to calculate the entries in the following table giving all amounts in dollars or dollars per unit rounded to the nearest cent 2G Cx Ci Cxx 500 5900 6 1180 1000 9400 8 940 1500 13900 10 927 2000 19400 12 970 b When average cost is a minimum we must have marginal cost average cost cw 2 Xx 4 0004x 4 0002x This equation simplifies to 0002x a x so x 1700000 and x V1700000 1304 To verify that this production level actually gives a minimum for Cxx we ob serve that the second derivative Cxx 6800x3 0 so the graph of Cxx is concave up for all x 0 and we have a minimum Finally the minimum av erage cost is C1304 3400 1304 1304 4 00021304 922 REVENUE PROFIT AND DEMAND It is clearly important for a manager to know all about the cost function but this is not enough The overall purpose of the firm is to make a profit and for this it 47 OPTIONAL APPLICATIONS TO ECONOMICS M ARGINAL ANALYSIS is essential to consider the income from sales or the revenue as economists call it And this requires us to bring the consumers buyers of the product into the picture The revenue function Rx is the total revenue or income derived from pro ducing and selling x units of the product The marginal revenue is the derivative R x of this function By the same type of interpretation as used above the mar ginal revenue can be thought of as the extra revenue received from the sale of one more unit Rx Rx 1 Rx approximately Example 3 Many business decisions are based on an analysis of the costs and revenues at the margin or at the edgehence the expression marginal analy sis for this kind of thinking To understand this let us suppose we are running a taxi company in New York City and are trying to decide whether to add one more cab to our large fleet If it will make money for the company then we add it otherwise not Clearly we need to consider the costs and revenues involved Since the choice is between adding this cab and leaving the fleet the same size the crucial question is whether the additional revenue generated by one more cab is greater or smaller than the additional cost incurred This additional revenue and cost are precisely the mar ginal revenue and marginal cost Therefore if the marginal revenue is greater than the marginal cost then we should clearly add the cab and increase our profit This is nothing but simple common sense expressed in the economists language of marginal this and marginal that The total profit derived from producing and selling x units is Px Rx Cx This is called the profit function it is what is left over from the revenue after the cost is deducted A firm will lose money when production is too low because of fixed costs and also when production is too high because of high marginal costs Unless the firm can operate profitably at some inbetween level of production the business will fail so we can assume that the profit curve looks something like Fig 443 Px The marginal profit is the derivative P x of the profit function In order to maximize profit we look for the critical points of Px that is the points where the marginal profit is zero But if Px Rx Cx 0 then Rx Cx This gives another basic law of economics If the profit is a maximum then marginal revenue marginal cost To satisfy ourselves that this condition gives a maximum and not a minimum we can use the second derivative test Figure 443 A profit function 1 5 1 x 152 APPLICATIONS OF DERIVATIVES x p p x Figure 444 Demand curve Figure 445 Price function Px Rx Cx 0 or Rx Cx Thus the profit will be a maximum if Rx Cx and Rx Cx In order to put teeth into these generalities we must consider the nature of the consumers who constitute the market Normally the higher the price of a com modity the smaller the number x that will be sold Thus x the number de manded is a decreasing function of the price p of a unit and this function is usually determined by market research The demand curve Fig 444 displays this dependence and under these circumstances the variable x is called the de mand and the function x xp is the demand function For the sake of conve nience in comparing the demand curve with the cost function economists usu ally interchange the axes and consider p as a function of x p px as shown in Fig 445 This function is called the price function When x units of a commodity are sold at a price of px dollars per unit then the revenue Rx is evidently the product of the price per unit and the number of units sold Rx xpx and the profit is Px xpx Cx If both the price function px and the cost function Cx are known then the law stated above can be used to find the value of x that maximizes profit It is clear that this value of x need not be the one that minimizes the average cost for the latter depends only on the cost function Cx That is profit depends on the whims of the marketplace while efficiency is an internal matter Example 4 What production level will maximize profit for a firm with cost func tion Cx 2400 9x 0002x2 and demand function x 12000 500p 47 OPTIONAL APPLICATIONS TO ECONOMICS MARGINAL ANALYSIS Solution First we point out that the economic meaning of the demand function is that no units will be sold x 0 at a price of 24 per unit but for every dol lar decrease in price 500 more units will be sold If we solve for p then we ob tain the price function I px 24 500 x The revenue function is therefore I Rx xpx 24x 500 x2 so the marginal revenue is and the marginal cost is Rx 24 20 x Cx 9 0004x 9 20x When the profit is a maximum then marginal revenue equals marginal cost that is and solving yields I I 24 250 X g 250 X or x 1875 To verify that this gives a maximum we calculate the second derivatives II I R x 250 Cx 20 Since Rx Cx for all x the production level x 1875 maximizes profit The corresponding price is pl875 2025 ELASTICITY OF DEMAND The nature of the demand curve in Fig 444 depends on the particular product under consideration It is relatively flat or inelastic for bread and motor oil since people tend to buy what they need without much regard for the price and relatively steep or elastic for candy since no one really needs it but more peo ple buy more of it when the price is low The elasticity of demand is an important concept of quantitative economics To introduce it in a precise way let p x be an arbitrary point on the demand curve in Fig 444 If p increases by a small amount Jp and fu is the corre sponding decrease in x then the ratio of the percentage decrease in x to the per centage increase in p is 100iitx 100iplp The elasticity of demand Ep at the price level p is now defined by 1 53 1 54 APPLICATIONS OF DERIVATIVES p ih p dx Ep hm tp0 x Dp x dp The demand is said to be elastic if Ep 1 and inelastic if Ep 1 The pos itive function Ep is a useful tool of economic analysis because it measures the responsiveness of the demand to changes in the price p it is small when the de mand curve is relatively flat so that changes in p induce relatively smaller changes in x and large when this curve is relatively steep It also has the merit of being independent of the units of measurement used for p and x This is a great con venience in many economic and business situations For example changing the units of p from dollars to French francs say and the units of x from pounds to kilograms would leave the value of the elasticity Ep unchanged because this quantity involves only the percentage changes in p and x Example 5 In Example 4 the demand function is x 12000 500p Find Ep At what price p is the demand elastic Inelastic Solution From the definition we have Ep dx x dp p 12000 SOOp SOO SOOp 12000 SOOp P 24 p From Example 4 we understand that 0 p 24 so the condition Ep 1 is equivalent to P I 24 p or p 24 p or 2p 24 or p 12 so the demand is elastic for p 12 Similarly the demand is inelastic for p 12 To understand what this means we observe that when the revenue is expressed as a function of p instead of x we have Rp pl2000 500p 500p 24 p It is easy to see from this that revenue is maximized for p 12 There fore to maximize revenue the price must be lowered if the demand is elastic and raised if the demand is inelastic These conclusions are valid for any decreasing demand function whether it is linear or not see Problem 26 The commonsense interpretation of all this is clear If the demand is elastic at a given price then a price decrease by a certain percentage causes a proportion ally larger increase in sales so the revenue which is the product of price and sales is increased Similarly if the demand is inelastic then a price increase by a certain percentage causes a proportionally smaller decrease in sales so again the revenue is increased The discussions of this section suggest several ways in which derivatives can be used in economics The most influential contribution to this subject in the twentieth century was perhaps Keyness General Theory of Employment Inter est and Money which has been characterized as an endless desert of econom ics algebra and abstraction with trackless wastes of differential calculus and only an oasis here and there of delightfully refreshing prose This may be some chapter IX of The Wordly Philosophers by Robert L Heilbroner 47 OPTIONAL APPLICATIONS TO ECONOMICS MARGINAL ANALYSIS 55 what exaggerated for the sake of its juicy phrases but nevertheless the general impression is validthat modern economics makes extensive use of many kinds of mathematics especially calculus PROBLEMS In Problems 1 a cost function in dollars is given for pro 22 A perfect competitor is an enterprise that has such a small ducing x units of a certain commodity In each case find the share of the market that it cannot influence the price of marginal cost at the production level of 500 units and also the its product and can sell as much as it produces at the pre actual cost of producing the 50Ist unit vailing market price p Figure 446 shows the cost and 1 Cx 15000 13x 003x2 revenue curves of a certain perfect competitor Sketch the Xx ee profit curve 400 2 Cx 400 10 100 3 Cx 5000 15x 001x2 00001 x3 4 Cx 3000 100Vx 7 px For each of the cost functions in Problems 510 find the min RO Pa imum average cost and the production level at which this is Cx achieved Z 5 Cx 8000 15x x 7 6 Cx 2400 3x 002x2 Yo 7 Cw 604x4 422 8 Cx 5000 2x 0001x3 a 2 ff aie ot 5 cee 9 Cx 2Vx aii 32 10 Cx 10000 8x 4x Figure 446 For each of the cost and price functions in Problems 1116 find the production level that maximizes profit Ii Cx 1240 8x 002x2 px 16 23 The daily cost toa stall company of producing x hand 12 Cx 1240 8x 002x2 px 16 ae calculators is CO 1560 50x 8x 5 dollars 13 Cx 900 35x 0001x2 px 65 am The market price of this calculator is 130 What is the 14 Cx 750 140x 02x2 4x3 maximum daily profit and what is the daily output x that px 300 4x yields this profit 15 Cx 4500 50x x2 00023 24 A small company with fixed costs overhead of a dol px 80 001x lars produces x units of a commodity which it sells at a fixed price of p dollars per unit If it costs b dollars to 16 Cx 6000 15x 555 0001x3 produce each unit where b p at what output level does px 120 005x the company break even and what is the graphical in terpretation of this breakeven point 17 A Broadway theater has seats for 2000 playgoers With 25 Suppose the company in Problem 24 produces trout fish the ticket price at 50 the average attendance at a mod ing instructional videotapes for 8 that it sells for 30 erately successful play has been 1200 When the ticket If the overhead is 14000 how many tapes must be sold price was lowered to 40 the average attendance rose to to break even 1400 26 Consider a demand curve x xp where xp is any de a Find the price function assuming that it is linear creasing function b What should the ticket price be to maximize revenue a If Ep 1 show that the revenue R px is in creased by lowering the price In Problems 1821 use the given demand function to find the Qs ha 1 whew nhalithe cavenup4iy Wenensod by selling price p that maximizes the revenue a 7 18 x 1200 20p raising the price 19 x 800 25p c Establish the formula 20 x 160 p aR 21 x 768 p dp all EQ