1.4 Linear Functions of Several Variables

Transcription

1 .4 Linear Functions of Several Variables Question : What is a linear function of several independent variables? Question : What do the coefficients of the variables tell us? Question : How do you find cost, revenue and profit functions with several variables? In sections. through., we considered functions of one independent variable. The cost, revenue, and profit, were each a function of some quantity Q. If we knew the quantity Q produced and sold, we could use these functions to compute the corresponding cost, revenue and profit at those production levels. As long as we deal solely with one product and the cost, revenue, and profit involved with that product, a function of one variable is adequate. But the total cost, revenue, and profit for a business may come from many products. Each of these products may have its own costs and prices. Using this information we can formulate cost, revenue and profit functions of several variables. Each of these variables represents a quantity of a different product produced and sold by the business. In this section we ll learn how to extend what we have learned about a function of a single independent variable to functions of several independent variables.

2 Question : What is a linear function of several independent variables? In section., we introduced a linear function of one variable, y mx b. There was nothing special about the names of the variables, x and y, or the names of the constants, m and b. Another possible form for a linear function of x and y is y a0 ax. In this format, a0is the vertical intercept and a is the slope. When several independent variables are introduced, it is prudent to use names for the variables that make sense. If one variable is named x, we can extend this to n variables using subscripts. Subscripts are numbers that appear to the right of the variable and slightly lowered. The subscript is a part of the variable s name and is useful to show generically that there are many variables. For instance, if we wanted to define a function with three independent variables that describe the quantities of three different products, we might use Q, Q, and Q. In general, let x, x,, xn be the names of n independent variables. A linear function of n independent variables x, x, xn is any equation that can be written in the form y a0 axax anxn In this form, we say that y is a linear function of x, x,, xn. The letters a0, a,, an are real numbers corresponding to constants. Function notation applies to functions of several independent variables as well as functions of one independent variable. Recall that a linear function of one variable x named f would be written as f ( x) a0 ax. The independent variable for the function is placed in parentheses after the name to distinguish the variables from the constants. For a linear function of n independent variables, the n independent variables are placed in the parentheses after the name to give

3 f x, x,, xn a0 axax anxn Example Find Function Values If f ( x, x, x) 0 x x x, find the value of f (6,,). Solution Substitute x 6, x and x into the function to yield f (6,,) 0 6

4 Question : What do the coefficients of the variables tell us? For a linear function of one independent variable, the coefficient on the variable is the slope or rate of change. We can generalize this idea to linear functions of several variables. Let s consider the function gx (, x) x x. As we saw in Example, we can substitute values for the variables to obtain a value for the function. For instance, if we want to substitute 0 for x and for x we get g(0, ) 0 4 Now compare this value to the value obtained when we increase the value for x by unit, g(, ) 6 The difference between these values is g g, 0, 6 4. Since the coefficient on x is a, increasing the value for x increases the value from g by units. The coefficient on x tells us the amount the function will change when x increases by unit. In general, this applies for any value of x. If we fix x at some value and find the difference between the function at x and at x we get,, g x g x x x x 6x 6 We can also apply this reasoning to the coefficient of x. If we fix x at some value and find the difference between the function at x and x we get 4

5 g 0, x g 0, x 0 x 0 x 0 x 0 x The coefficient on x, -, indicates the amount the function will change when x grows by unit. This leads to the following generalization. If f x, x,, xn a0 axax anxn is a linear function of n independent variables, the coefficient a i for values of i from to n indicates the amount the function will change when the variable x i is increased by one unit and all other variables are fixed. Example Interpret Coefficients The function S( A, E) 05A 4E describes the monthly sales, in thousands of dollars, at a large dairy distributor with E employees and A thousand dollars of monthly advertising costs. a. What does the coefficient of E tell you about the monthly sales? Solution The coefficient of E, 4, relates the change in sales to the variable E when it is increased by unit. An increase of unit in E means an increase of employee. This leads to an increase in sales of 4 thousand dollars or $,4,000. b. What does the coefficient of A tell you about monthly sales? Solution The coefficient of A, 05, relates the change in sales to the variable A when it increases by unit. An increase of unit in A means 5

6 an increase of thousand dollars in advertising. This leads to an increase in sales of 05 thousand dollars or $05,000. We can now interpret what coefficients a through a n mean, but what about the constant a 0? The constant a0 is similar to the vertical intercept b for a linear function of one independent variable f ( x) mx b. In that case, b is the value of the function when the variable is set equal to zero, 0 0 f m b b. For a linear function of n independent variables, the corresponding process is to set all independent variables equal to zero. If f ( x, x,, x ) a0 ax a x, we can set each variable equal to zero to obtain n n n f (0,0,,0) a a 0 a 0 a 0 a 0 n 0 The only term that remains is the term containing the constant a 0. If f ( x, x,, xn) a0 ax anxn is a linear function of n independent variables, a 0 is the value of the function when all independent variables are set equal to zero. Example Fixed Costs for an Organic Dairy The daily cost function (in dollars) for an organic dairy is given by the linear function CQ (, Q, Q) Q.Q.5Q 0,000 where Q is the number of gallons of whole milk produced, Q is the number of gallons of % milk produced, and Q is the number of gallons of % milk produced. a. Find the fixed costs for the organic dairy based on this function. Solution The fixed costs are costs incurred when nothing is produced. We can find this cost by finding C (0,0,0) or by noting that the constant 6

7 gives the function s value when all variables are zero. Either way, the fixed costs are $0,000. b. How much does each additional gallon of % milk cost to produce? Solution The coefficient of Q gives the change in the cost when Q is increased by unit. In this case, the coefficient is.5 which tells us that another gallon of % milk will cost an additional $.5. We can calculate the amount it would cost to change production in Example by any amount by adding the amount to the appropriate independent variable and utilizing the cost function. If we want to find the additional cost incurred from increasing the production of % milk by 000 gallons, calculate,, 000,, C Q Q Q C Q Q Q Q Q Q Q Q Q Q.Q.5Q 50 Q.Q.5Q 50 The extra 000 gallons cost $50. This number is independent of the initial production level. If we start with no % milk and increase production to 000 gallons or increase production of % milk from 0,000 gallons to,000 gallons, the additional cost will be $50. For linear functions like this one, the increase is always independent of the production level. Another way of thinking about this is that the variable costs for % milk are $.5 per gallon. This is precisely the coefficient on the variable representing % milk, Q. 7

8 Question : How do you find cost, revenue and profit functions with several variables? In Example, we examined a cost function of several variables. This proved useful for modeling the total cost in a business that produces several different goods like milk. If the company receives revenue from several different products, we can utilize the fact that the revenue from each individual product by multiplying the price per unit times the number of units sold to get the revenue for that product. By summing the revenue for each of the products, we can form a model for the total revenue. In the next example, we ll continue working with the organic dairy from Example to model the total revenue from selling three types of milk. Example 4 Organic Dairy Revenue The organic dairy in Example charges $.49 per gallon for whole milk, $4.9 per gallon for % milk, and $4.59 per gallon for % milk. If the variables Q, Q, and Q represent the number of gallons of whole, % and % milk produced and sold respectively, find a linear function R( Q, Q, Q ) to model the total revenue. Solution We ll find a linear function of the variables Q, Q, and Q. The coefficients in this function correspond to the increase in revenue from increasing the number of gallons of each type by unit. These coefficients are simply the prices the dairy charges for one gallon of each type of milk. The linear function will have the form R( Q, Q, Q ) pq p Q p Q where p, p, and p are the price per gallon for each type of milk. Notice that there is no constant term in the sum since producing and selling no milk leads to no revenue. If we had included a positive constant, this would have been interpreted as revenue corresponding to 8

9 ,, 0,0,0 Q Q Q. With the prices in place, we get a linear model for total revenue in dollars, R Q, Q, Q.49Q 4.9Q 4.59Q It is very common for cost and revenue functions to have many variables. Each variable corresponds to a unique product or service that a business provides to consumers. The coefficients of the cost function represent the variable costs per unit for each product or service and the constant represents the fixed costs of production. The coefficients of the revenue function represent the unit price for each product or service. Example 5 Profit The cost function for the organic dairy is CQ (, Q, Q) Q.Q.5Q 0,000 and the revenue function is RQQ,, Q.49Q 4.9Q 4.59Q where Q is the number of gallons of whole milk, Q is the number of gallons of % milk, Q is the number of gallons of % milk that the dairy produces and sells. a. Find the profit function Profit( Q, Q, Q ). Solution Profit is the difference between revenue and costs, Profit( Q, Q, Q).49Q4.9Q 4.59Q Q.Q.5Q 0,000 R( Q, Q, Q) C( Q, Q, Q).49Q.09Q.9Q 0,000 9

10 b. Find the profit per unit for each type of milk. Solution The coefficients on the profit function correspond to the change in profit when the corresponding variable is increased by unit. Since the coefficient of Q is.49, the profit increases by $.49 when more gallon of whole milk is produced and sold. This is the profit per gallon for whole milk. The same reasoning for the other types of milk leads to a profit per unit for % milk of $.09 per gallon and $.9 per gallon for % milk. In real applications, the units are often modified to make the numbers more manageable. In Example, we examined the sales of a business in thousands of dollars given by S A, E 05A 4E where A is the amount spent on advertising in thousands of dollars and E is the number of employees. If we were to find a value for this function like S(0,5) We would interpret this as indicating that advertising expenses of $0,000 and using 5 employees would lead to sales of $8,0,000. The variable A is scaled in thousands of dollars. Scaling a variable means that the actual amount is divided by an amount to obtain the value for the variable. In this case, dividing $0,000 by $000 yields A 0. The dependent variable, sales, is also scaled in thousands of dollars. If we are given a value for this variable like 80, we need to multiply by $000 to get the actual amount. The value 80 thousand dollars is the same as 80 $000 or $8,0,000. When independent variables are scaled, the cost and revenue functions change. In Example, we found the revenue function for an organic dairy that charges $.49 per gallon for whole milk, $4.9 per gallon for % milk, and $4.59 per gallon for % milk. The function describing the revenue was R Q, Q, Q.49Q 4.9Q 4.59Q 0

11 where Q is the number of gallons of whole milk, Q is the number of gallons of % milk, Q is the number of gallons of % milk that the dairy produces and sells. Since an organic dairy might produce and sell thousands of gallons of each week, we could choose to scale the independent variables in thousands of gallons. Instead of inputting values like 000 into a variable, we would input instead. This introduces subtle changes into the revenue function since changes of unit in the scaled variable results in a change of 000 in gallons of milk. Let s look at how this will change the revenue function. R( Q, Q, Q ) pq p Q p Q represent the revenue for the organic dairy where Let Q, Q and Q are the amounts of whole, %, and % milk in thousands of gallons produced and sold respectively. As we found in Example, the coefficients p, p, and p are the unit prices of each type of milk. However, when the variables are scaled the meaning of one unit changes. Now the unit price corresponds to the price of thousand gallons of milk. A price of $.49 per gallon is equivalent to $490 per thousand gallons of milk. If we change each unit price to account for the scaling, we get The value R Q, Q, Q 490Q 490Q 4590Q R 0,, represents the revenue from producing and selling 000 gallons of % milk. If we were to calculate the same revenue from the function in Example, we would get R 0,000, The key part of each calculation is middle term, dollars dollars 490 thousand gallons thousand gallons versus 4.9 gallon 000 gallon

12 In each case, the units in red reduce to leading to a function value of 880 dollars. By examining the units carefully in each term, we can deduce the units on the revenue. The units on each term must be the same for the terms to add together properly. Another possibility is to scale the variables in thousands of gallons but use the unit prices in dollars per gallon. In this case the middle term would be 4.9 thousand gallon dollars gallon In this case, the numbers multiply to 8.8 and the units are thousands of dollars. To define the function with the revenue in thousands of dollars we would write the function as R Q, Q, Q.49Q 4.9Q 4.59Q with each of the variables scaled in thousands of gallons. Now the revenue from 000 gallons of % milk is calculated as R 0,, and the revenue is 8.8 thousand dollars or $880. Each of these models may be used to calculate the revenue for the organic dairy. However, the units you choose for the independent and dependent variables in the revenue and cost functions should match. If not, you might find yourself subtracting cost in dollars from revenue in thousands of dollars. This would lead to a value for profit that makes no sense.