ࡱ>  jH5@ ҁbjbj22 +:XXyjjjjjjj~ZZZZD\~2]|mmmmmmm$OR€jinmminin€jjmm׀uuuinjmjmuinu.u'vV\~@jj m] `?.gZ/p6~ p,0~xYetY ~~jjjjYj PumT=nYn€€~~,B.d,u ~~B.Chapter 10 Multi-Variable Functions 10.1 Functions of Several Independent Variables In the previous chapters we worked with functions containing one independent variable. Now we will expand this concept and include more than one variable in a function. Let's look at this in the form of an example. Scoot-It is a motorized scooter company that produces one type of scooter called the explorer. If each explorer scooter sells for $7500 then the company's weekly revenue function, in hundreds of dollars, can be modeled by EMBED Equation.DSMT4where x represents the number of explorer scooters sold each week. This revenue function is a function of one independent variable. Suppose, however, the company decides to expand its line of scooters and begins to produce the discovery scooter, a model the company sells for $5000. If we let y be the number of discovery scooters sold each week then the company's weekly revenue function, in hundreds of dollars, for selling both types of scooters is EMBED Equation.DSMT4. This second revenue function is a function of two independent variables, namely x and y, and R(x,y) is called the dependent variable. We can use this revenue function to determine the amount of revenue generated when 4 explorer scooters and 6 discovery scooters are sold by evaluating R(4,6) as shown below. EMBED Equation.DSMT4 Thus, Scoot-It will make $60,000 in revenue when 4 explorer scooters and 6 discovery scooters are sold. Example 10-____: A local music hall is selling ticket to Fridays concert. The matinee show costs $15 per person and the evening show costs $25 per person. If x is the number of matinee tickets purchased and y is the number of evening tickets purchased for the Friday concert, find the revenue function R(x,y) for Friday's concert then evaluate and interpret R(125, 200). Solution: Since revenue is the selling price times the number of tickets sold, the revenue function for Friday's concert is EMBED Equation.DSMT4and EMBED Equation.DSMT4. This means that when 125 tickets are sold for the matinee show and 200 tickets are sold for the evening show, the music hall generates a revenue of $6,875.END OF SOLUTION Example 10-____: The Scoot-It motorized scooter company has determined that its demand equation for the two types of scooters are given by EMBED Equation.DSMT4 where x is the number of explorer scooters sold per week and y is the number of discovery scooters sold per week. Find the weekly revenue function R(x,y) and then evaluate and interpret R(10, 18). Solution: Since revenue is the price times the number of items sold the revenue function for Scoot-It is EMBED Equation.DSMT4 So EMBED Equation.DSMT4 This means that when 10 explorer scooters and 18 discovery scooters are produced and sold Scoot-It earns $109,400 in revenue. END OF SOLUTION Example 10-____: If the Scoot-It motorized scooter company's weekly cost function, in hundreds of dollars, is given by EMBED Equation.DSMT4 and the revenue function is the same from example 10___ above, determine the company's profit function P(x,y) then evaluate and interpret P(10,18). Solution: Since profit = revenue - cost we can obtain the following profit function for Scoot-It. EMBED Equation.DSMT4 So EMBED Equation.DSMT4 Thus, when 10 explorer scooters and 18 discovery scooters are produced and sold, Scoot-It earns $50,100 in profit.END OF SOLUTION 10.2 CobbDouglas Production Function Production functions help companies estimate how many goods can be produced with a given number of inputs. A commonly used production function is one of the form  EMBED Equation.DSMT4  and is called the CobbDouglas production function, named after the two founders. In the Cobb-Douglas production function, L is the number of units of labor, K is the number of unit of capital, and a, b, c are constants. Example 10-____: Suppose a metal manufacturing company has a Cobb-Douglas production function, in hundreds of pounds, of  EMBED Equation.DSMT4  where L is the number of hours of labor and K is the dollar amount of capital invested. If the company uses 2,000 hours of labor and $1500 in capital, how many pounds of metal will be produced? Solution: To find the number of pounds of metal produced we need to evaluate  EMBED Equation.DSMT4  Thus, the manufacturing company will produce approximately 1,611,900 pounds of metal when 2000 hours of labor and $1500 in capital is invested. End of Solution The advantages of using the Cobb-Douglas production function for estimating production are many and are widely used in empirical work. One advantage of the Cobb-Douglas function is that returns to scale can be determined by simply summing the exponents on the two inputs, labor and capital. The chart below shows how to determine the returns to scale. Properties of the Cobb-Douglas Production Function Given a Cobb-Douglas production function  EMBED Equation.DSMT4  if b + c = 1 then the function displays constant return to scale. That is, increasing labor and capital by a factor of p causes production to increase by the same factor p. b + c > 1 then the function displays increasing returns to scale. That is an increase in labor and capital by a factor p causes production to increase by more than the factor p. b + c < 1 then the function displays decreasing returns to scale. That is an increase in labor and capital by a given factor p causes production to increase by less than the factor p. Lets look at this in the form of an example. If a companys Cobb-Douglas production function is represented by  EMBED Equation.DSMT4  then EMBED Equation.DSMT4 . If we increase labor and capital by a factor of 3 then we would expect production to increase by a factor of 3 because 0.3 + 0.7 = 1.  EMBED Equation.DSMT4 . We notice that 3(3.249) = 9.747. Thus, we have confirmed the production function displays constant returns to scale. Now suppose a company has a Cobb-Douglas production function of  EMBED Equation.DSMT4  then  EMBED Equation.DSMT4 . Once again, if we increase labor and capital by a factor of 3 then we would expect production to increase by more than a factor of 3 because 0.4 + 0.8 = 1.2.  EMBED Equation.DSMT4  Since 14.949 > (4)(3) = 12 the production function displays increasing returns to scale. Lastly, suppose a company has a Cobb-Douglas production function of  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4 . If labor and capital are increased by a factor of 3, then we would expect production to increase by less than a factor of 3 because 0.2 + 0.5 = 0.7 < 1.  EMBED Equation.DSMT4  Since 4.957 < (2.297)(3) = 6.891 the production function displays decreasing returns to scale. Example 10-______: A sporting goods manufacturing company has a Cobb-Douglas production function of EMBED Equation.DSMT4 , where  EMBED Equation.DSMT4  is the number of units produced. If the company invests 800 hours of labor and $3000 in capital, how many sporting good units will be produced? What type of returns to scale does the production function display? Solution: Evaluating  EMBED Equation.DSMT4  we get  EMBED Equation.DSMT4 . Thus, the sporting goods manufacturing company can expect to produce 694 units of sporting goods when 800 hours of labor and $3000 in capital are invested. Since the sum of the exponents on the inputs are less than 1, that is 0.36 + 0.43 = 0.79, the production function displays a decreasing returns to scale. End of solution. Returns to scale are important for determining how many firms will populate an industry. When increasing returns to scale exist, one large firm will produce more cheaply than two small firms. Small firms will thus have a tendency to merge to increase profits, and those that do no merge will eventually fail. On the other hand, if an industry has decreasing returns to scale, a merger of two small firms to create a large firm will cut output, raise average costs, and lower profits. In such industries, many small firms should exist rather than a few large firms. ( HYPERLINK "http://ingrimayne.saintjoe.edu/econ/TheFirm/ProductionFunct.html" http://ingrimayne.saintjoe.edu/econ/TheFirm/ProductionFunct.html) 10.3 The 3-Dimensional Coordinate Plane Now that we are familiar with functions of two variables it is time to look at the graphs created by these functions. Since we have two independent variables and one dependent variable, the 3-dimensional coordinate plane consists of three axes, all of which are perpendicular to one another. The points in a 3-dimensional coordinate system are represented with an ordered triple (x, y, z). The point (1, 2, 3) is graphed below. Figure 10-_____ A plot of the point (1, 2, 3).  The graph of a function f(x,y) is called a surface and are typically very difficult to graph by hand. Thus, in these modules we will use a 3-dimensional graphing applet to graph any function of two variables. Some examples of surfaces are shown in figure 10-___ below. Figure 10-_____ Graphs of various functions of two variables.   Example 10-___: Find the domains of the following functions a) EMBED Equation.DSMT4 b)EMBED Equation.DSMT4 c) EMBED Equation.DSMT4 Solution: a) For the function EMBED Equation.DSMT4 the inequality EMBED Equation.DSMT4 must hold true. Therefore the domain of EMBED Equation.DSMT4 is all x such that EMBED Equation.DSMT4. b) Since the denominator of a rational function can not be zero, the domain of EMBED Equation.DSMT4 is all x such that EMBED Equation.DSMT4or EMBED Equation.DSMT4. c) Since the arguement of a logarithmic function must always be greater than zero, that is EMBED Equation.DSMT4, the domain of EMBED Equation.DSMT4 is all x such that EMBED Equation.DSMT4. 10.4 First Order Partial Derivatives In section 10-1 we were given the cost, revenue, and profit functions for the Scoot-It motorized scooter company. In addition, we found P(10,18) =501 and interpreted this to mean that when 10 explorer scooters and 18 discovery scooters are sold the company earns $50,100 in profit. Now supose the company's owners need to determine if the production of more scooters will result in more profit, and if more profit is obtained, which type of scooter would produce more profit. To do this type of analysis, we must take first order partial derivatives. That is, we need to fix the production of one type of scooter and determine the amount of addition profit obtained when the production of the other scooter increases by one. So if we take the profit function EMBED Equation.DSMT4 and fix the number of discover scooters produced at 18 but let the production of explorer scooters vary the following profit function is formed. EMBED Equation.DSMT4 Since P(x) contains a single variable, we can find the marginal profit function by taking the derivative of EMBED Equation.DSMT4with respect to x as shown below. EMBED Equation.DSMT4 Substituting x = 10 into the marginal profit function will approximate the amount of profit that is obtained when one additional explorer scooter is produced. EMBED Equation.DSMT4 Thus, at a production level of 10 explorer and 18 discovery scooters, Scoot-It's profit will increase by approximately $250 when one additional explorer scooter is produced. We can do the same analysis if we fix the number of explorer scooters produced to 10 and allow the number of discovery scooters to vary. This gives us a profit function in terms of the single variable y. EMBED Equation.DSMT4 Now we can find the marginal profit function for the discovery scooters. EMBED Equation.DSMT4 Substituting y = 18 into the marginal profit function will approximate the amount of profit that is obtained when one additional discover scooter is produced. EMBED Equation.DSMT4 Thus, at a production level of 10 explorer and 18 discovery scooters, Scoot-It's profit will increase by approximately $7,000 when the 19th discovery scooter is produced. From our analysis, we conclude that Scoot-It's marginal profit is higher for the discovery soooters when the production level is 10 explorer and 18 discovery scooters. It is important to notice that when the partial derivative of P(x,y) was taken with respect to x it was denoted by EMBED Equation.DSMT4 and when the partial derivative of P(x,y) was taken with respect to y it was denoted by EMBED Equation.DSMT4. Example 10-___: Given EMBED Equation.DSMT4 find EMBED Equation.DSMT4and EMBED Equation.DSMT4. Solution: When finding EMBED Equation.DSMT4we need to substitute y = 7 into f(x,y), take a partial derivative with respect to x and then substitute x = 4 into the resulting function. EMBED Equation.DSMT4 Now to find EMBED Equation.DSMT4we need to substitute x = 4 into f(x,y), take a partial derivative with respect to y and then substitute y = 7 into the resulting function. EMBED Equation.DSMT4 END OF SOLUTION Lets look at the previous example geometrically. When y is fixed at 7, that is when you substitute y = 7 into the function, the surface EMBED Equation.DSMT4 is cut parallel to the x-axis 7 units from the origin. The cross-section that is formed by this cut is a curve as shown in the figure below. Figure 10-_____ Use the applet to draw EMBED Equation.DSMT4 and the plane y = 7 When we take a partial derivative with respect to x we are finding the slope of the lines tangent to this cross-section. If we were to draw a tangent line at x = 4, we know from our calculations above the tangent line would have a slope of 518. A different curve is formed when EMBED Equation.DSMT4 is cut by the plane x = 4. The intersection of x = 4 and the surface, along with the resulting cross-section is shown in the figure below. Figure 10-_____ Use the applet to draw EMBED Equation.DSMT4 and the plane x = 4 If we were to draw a tangent line on these curve where y = 7 we know that the slope of that tangent line would be 1128. We are now ready to find partial derivatives of EMBED Equation.DSMT4 without reducing it to a function of one variable. Example 10-___: Find EMBED Equation.DSMT4 and EMBED Equation.DSMT4 for each of the following functions a) EMBED Equation.DSMT4 b) EMBED Equation.DSMT4 c) EMBED Equation.DSMT4 Solution: a) To find EMBED Equation.DSMT4 we need to hold y constant and take the derivative of f(x,y) with respect to x, thus we need to find EMBED Equation.DSMT4 Since the derivative of a constant times a function is the constant times the derivative of the function we obtain the following derivatives for each of the terms. EMBED Equation.DSMT4 To find EMBED Equation.DSMT4 we need to treat x as a constant and take the derivative of f(x,y) with respect to y, thus we need to find EMBED Equation.DSMT4 Taking each of these derivatives we get the following. EMBED Equation.DSMT4 b) To find EMBED Equation.DSMT4we need to treat y as a constant and take the derivative of each term with respect to x. To do this we will need to apply the derivative rule for logarithmic functions. EMBED Equation.DSMT4 Now let's find EMBED Equation.DSMT4. To take this partial derivative we will need to apply the product rule to find EMBED Equation.DSMT4as well as use derivative rules for exponential and logarithmic functions. . EMBED Equation.DSMT4 c) To find EMBED Equation.DSMT4we will need to use the chain rule and as well as the derivative rules for exponential functions. EMBED Equation.DSMT4 Now let's find EMBED Equation.DSMT4 EMBED Equation.DSMT4 10.4 Competitive vs. Complementary Commodities Two commodities are said to be competitive, or substitute, if a decrease in the demand of one commodity produces an increase in the demand of the other commodity. Suppose two brands of coffee, Jitters and Shakers, are competitive commodities. Let EMBED Equation.DSMT4  represent the number of 20oz bags of Jitters brand coffee demanded and  EMBED Equation.DSMT4  represent the number of 20oz bags of Shakers brand coffee demanded where p is the price per bag of Jitters brand coffee and q is the price per bag of Shakers brand coffee. If the price of Jitters brand coffee increases while the price of Shakers brand coffee is held constant then the demand for Jitters brand coffee will typically decrease, that is EMBED Equation.DSMT4 , and since the commodities are competitive, the demand for Shakers brand coffee will increase, that is  EMBED Equation.DSMT4 . Likewise, if the price of Shakers brand coffee increases while the price for Jitters brand coffee is held constant, then the demand for Shakers brand coffee will typically decrease, that is  EMBED Equation.DSMT4 , and since the commodities are competitive, the demand for Jitters brand coffee will increase, that is  EMBED Equation.DSMT4 . Two commodities are said to be complementary if a decrease in the demand of one commodity produces a decrease in the demand of the other commodity. Suppose digital video disk (DVD) players and digital video disks (DVDs) are complementary commodities. Let  EMBED Equation.DSMT4  represent then number of DVD players demanded and  EMBED Equation.DSMT4  represent the number of DVDs demanded where p is the price of each DVD player and q is the price of each DVD. If the price of a DVD player increases and the price of a DVD is held constant, then the demand for DVD players will decrease,  EMBED Equation.DSMT4  and since these two commodities are complementary, the demand for DVDs will also decrease,  EMBED Equation.DSMT4 . Similarly, if the price for DVDs increases and the price for DVD players is held constant, then the demand for DVDs will decrease,  EMBED Equation.DSMT4 , and since the commodities are complementary, the demand for DVD players will decrease,  EMBED Equation.DSMT4 . This leads us to the following test for complementary and competitive commodities. Test for Competitive and Complementary Commodities Let EMBED Equation.DSMT4  represent then number of commodity A demanded and  EMBED Equation.DSMT4  represent the number of commodity B demanded where p is the unit price commodity A and q is the unit price of commodity B. Partial Derivatives Commodities A and B  EMBED Equation.DSMT4  Competitive  EMBED Equation.DSMT4  Complementary  EMBED Equation.DSMT4  Neither  EMBED Equation.DSMT4  Neither Example 10-_____: If  EMBED Equation.DSMT4  is the demand function for commodity A and  EMBED Equation.DSMT4  is the demand function for commodity B, determine if the two commodities are complementary, competitive, or neither. Solution: We need to find  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  Thus, these commodities are neither competitive or complementary. 10.5 Second Order Partial Derivatives A second order partial derivative is the derivative of the first order partial derivative and they are typically denoted in the following ways. EMBED Equation.DSMT4 Example 10-___: Given EMBED Equation.DSMT4 find all four second order partial derivatives. Solution: To find the second order partial derivatives we first need to find EMBED Equation.DSMT4 and EMBED Equation.DSMT4. EMBED Equation.DSMT4 EMBED Equation.DSMT4 To find EMBED Equation.DSMT4 we will take the partial derivative of EMBED Equation.DSMT4with respect to x. EMBED Equation.DSMT4 To find EMBED Equation.DSMT4 we will take the partial derivative of EMBED Equation.DSMT4with respect to y. EMBED Equation.DSMT4 To find EMBED Equation.DSMT4 we will take the partial derivative of EMBED Equation.DSMT4with respect to y. EMBED Equation.DSMT4 To find EMBED Equation.DSMT4 we will take the partial derivative of EMBED Equation.DSMT4with respect to x. EMBED Equation.DSMT4 End of Solution In the example above we notice that EMBED Equation.DSMT4. Although this does not hold true for all second order partial derivatives it is very difficult to find an application where this equation is not true. Thus, for all of the functions in these modules the equation EMBED Equation.DSMT4will hold true. 10.6 Relative Extrema and Saddle Points Just as we found relative maximum and minimum for functions of one variable, we will now investigate how relative maximum and minimum points are found for functions of two variables. First, however, we will make the assumption that all the functions we encounter will have second order partial derivatives that exist in some circular region in the xy-plane. This simply ensures that we are dealing with a smooth surface. The first step in finding relative extrema for functions of two variables is to find the critical point(s). Critical point(s) are found using the definition below. Definition of Critical Point The point (a,b) is called a critical point of f(x,y) if  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . Example 10-______: Determine the critical point for  Solution: First we need to find  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  Now we find where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4   EMBED Equation.DSMT4  So the critical point for  is  EMBED Equation.DSMT4 . Example 10-______: Determine the critical point for  EMBED Equation.DSMT4  Solution: First we need to find  EMBED Equation.DSMT4 and EMBED Equation.DSMT4  then set them equal to zero.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  After we set these two first partial derivatives equal to zero we will need to solve for x and y using the substitution method. To do this we will choose one of the first partial derivatives and solve it for either x or y. Lets choose the partial derivative with respect to x  EMBED Equation.DSMT4 and solve it for x.  EMBED Equation.DSMT4  Now we will take this expression for x and substitute it into  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Now substitute Now substitute y=1 into 2y-4=x and we get  EMBED Equation.DSMT4  Thus, the critical point for  EMBED Equation.DSMT4  is  EMBED Equation.DSMT4 . Example 10-_______: Determine the critical point for  EMBED Equation.DSMT4 . Solution: We need to find both first order partial derivatives, set them equal to zero, and then solve for x and y.  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Thus the critical points for  EMBED Equation.DSMT4  are (0, 4) and (2, 4). END Now that we know how to find critical points for functions of two variables we are now ready to find relative extrema for functions of two variables. When we found relative extrema for functions of one variable we found where the first derivative was equal to zero and then performed a first derivative test to locate relative extrema. The process for finding relative extrema for functions of two variables is very similar. We will find the critical points and then perform a second derivative test to locate any relative extrema. We will use the following second derivative test when finding relative extrema of functions of two variables. Second-Derivative Test for Relative Extrema Given a function  EMBED Equation.DSMT4  whose second-order partial derivatives exist in some circular region containing the critical point (a,b) as its center and if  EMBED Equation.DSMT4  then If  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4  is a relative maximum. If  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 , then  EMBED Equation.DSMT4  is a relative minimum. If  EMBED Equation.DSMT4 , the f(x,y) has a saddle point at (a,b). If  EMBED Equation.DSMT4  then the test fails. Example 10-______: Find all relative extrema and/or saddle points for  EMBED Equation.DSMT4 . Solution: Figure 10-____ graphs the given function. From this graph it is obvious that a relative maximum exists at the critical point. Figure 10-____: Snag a graph of  EMBED Equation.DSMT4  First we need to find the critical point(s) for  EMBED Equation.DSMT4 . That is we need to find where  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Thus, the critical point for the given function is at  EMBED Equation.DSMT4 . To determine if any relative extrema or saddle points exist we need to find the second partial derivatives.  EMBED Equation.DSMT4  Now we need to evaluate  EMBED Equation.DSMT4 . Since D > 0 and  EMBED Equation.DSMT4  we conclude that a relative maximum exists at  EMBED Equation.DSMT4 . End of Solution Example 10-______: Find all relative extrema for  EMBED Equation.DSMT4 . Solution: First we need to find all critical points for f(x,y).  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  Thus critical points exist at (0, 4) and (4, 4). Now to find whether these critical points are relative maximums, minimums, saddle points, or neither we need to find the second order partial derivatives.  EMBED Equation.DSMT4  Lets first investigate (4, 4) for relative extrema and saddle points. First we evaluate  EMBED Equation.DSMT4  where a = 4, b = 4.  EMBED Equation.DSMT4  Since D < 0, we conclude a saddle point exists at  EMBED Equation.DSMT4 . Now we determine if any relative extrema or saddle points exist at (0, 4).  EMBED Equation.DSMT4 . Since D > 0 and  EMBED Equation.DSMT4  we determine that a relative minimum exists at  EMBED Equation.DSMT4 . We can confirm our answer by graphing the given function as shown below in figure 10-_____. Figure 10-______: Need a shot snagged of the given function. Example 10-____: An on-line fitness company sells two types of shoes, aerobic and running. The store pays $30 for a pair of aerobic shoes and $45 for a pair of running shoes. The daily demand equations for each type of shoe are as follows:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  where p is the selling price for each pair of aerobic shoes and q is the selling price for each pair of running shoes. What price should the store sell each model of shoe if they want to maximize their profits? Solution: Since profit = revenue cost we need to formulate a cost and revenue equation. Since we want to find the selling price for each type of shoe we need to formulate functions in terms of p and q. The revenue function is  EMBED Equation.DSMT4  The cost function in terms of x and y is  EMBED Equation.DSMT4  where x is the number of aerobics shoes sold and y is the number of running shoes sold. To get the cost function in terms of p and q we need to use the demand equations as shown below.  EMBED Equation.DSMT4  Now that we have a cost and revenue function we can find the profit function as follows  EMBED Equation.DSMT4  Now to find the selling prices that will maximize profit we need to find the critical point and run the second derivative test. First we will find the critical point.  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4  To solve this we need to use the substitution method. We start by solving  EMBED Equation.DSMT4  for q.  EMBED Equation.DSMT4  Now we substitute this expression for q into  EMBED Equation.DSMT4 .  EMBED Equation.DSMT4  So  EMBED Equation.DSMT4 . Now that we have the critical point of (50.56, 70.39) we need to make sure that a relative maximum exists at this critical point. Thus we need to find the second order partial derivatives.  EMBED Equation.DSMT4  Now applying the second derivative test by evaluating  EMBED Equation.DSMT4  we get the following  EMBED Equation.DSMT4  Since D > 0 and  EMBED Equation.DSMT4  the second derivative test states that a relative maximum exists at the critical point. Thus, to maximize profit, the shoe company should sell a pair of aerobic shoes for $50.56 and a pair of running shoes for $70.39. End of Solution Sample Quiz 1. A computer company sells two types of laser printers. The demand equations for each type of printer are as follows:  EMBED Equation.DSMT4  where x is the number of type A laser printers sold and y is the number of type B laser printers sold. If it costs the company $75 for each type A laser printer and $95 for each type B laser printer, find the companys profit function,  EMBED Equation.DSMT4 , then evaluate and interpret  EMBED Equation.DSMT4  2. The productivity of a third world country is approximated by the function  EMBED Equation.DSMT4  where L is the units of labor and K is the units of capital invested in production. Evaluate and interpret  EMBED Equation.DSMT4  then determine the returns to scale for this country. 3. Find the domain of  EMBED Equation.DSMT4  4. A music production company has released two new compact discs (CDs). The companys revenue function, in dollars, is given by  EMBED Equation.DSMT4  Evaluate and interpret  EMBED Equation.DSMT4 . 5. Given  EMBED Equation.DSMT4  find  EMBED Equation.DSMT4  and  EMBED Equation.DSMT4 . 6. An electronics company has two products whose demand functions are as follows:  EMBED Equation.DSMT4  Determine whether these two products are competitive, complementary, or neither. 7. Given  EMBED Equation.DSMT4  find  EMBED Equation.DSMT4 . 8. Find the critical point(s) for  EMBED Equation.DSMT4 . 9. Find all relative extrema and/or saddle point(s) for  EMBED Equation.DSMT4 . 10. A home and garden shop sells two types of tomato plants. The shop pays $2 for each crate of cherry tomato plants and $5 for each crate of Big Boy tomato plants. The daily demand equations for each type of tomato plant are as follows:  EMBED Equation.DSMT4  where p is the selling price for each crate of cherry tomatoes and q is the selling price for each crate of Big Boy tomatoes. 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