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Unformatted text preview: 7 Functions of Several Variables Suppose you run a company which produces two types of television. Let x 1 be the quantity of type 1 and x 2 be the quantity of type 2. Then the usual functions in which we are interested (profit, revenue, and cost) will depend both on x 1 and x 2 . Typically we write P ( x 1 ,x 2 ), R ( x 1 ,x 2 ), and C ( x 1 ,x 2 ). These are examples of functions of two variables. More generally, profit, revenue and cost could be functions of many variables, such as number of employees, cost of materials, and so on. So typically we will get P ( x 1 ,x 2 ,...,x n ) and so on. This is a much more realistic picture of how these functions work. The problem is that we still want to optimize them, ie, find their maxes and mins. But the calculus we know so far is useless for functions of many variables. In this chapter we develop the tools to find the maxes and mins of these functions. 7.1 The Three-Dimensional Coordinate System We will focus on functions of two variables. These will be written as z = f ( x,y ). For example f ( x,y ) = x 2 y + 3 y 2 x g ( x,y ) = ln( xy ) h ( x,y ) = e x 2 + y 2 x and so on. The graphs of one-variable functions y = f ( x ) were drawn in the xy-plane because for each value of x the function gave you a corresponding value of y . Two-variable functions like z = f ( x,y ) give us a z value for each point ( x,y ) in the xy-plane we input, and so therefore the graph it we will need a third axis. We call this the z-axis and view it as coming up perpendicular to the x- and y-axis. These three axes together form the three-dimensional coordinate system (also called 3-space or R 3 ). Each point in 3-space is determined uniquely by its three coordinates ( x,y,z ). Many of the same formulas from the xy-plane still apply: Midpoint between ( x 1 ,y 1 ,z 1 ) and ( x 2 ,y 2 ,z 2 ): x 1 + x 2 2 , y 1 + y 2 2 , z 1 + z 2 2 Distance between ( x 1 ,y 1 ,z 1 ) and ( x 2 ,y 2 ,z 2 ): p ( x 1- x 2 ) 2 + ( y 1- y 2 ) 2 + ( z 1- z 2 ) 2 In R 2 , the equation x 2 + y 2 = r 2 corresponds to a circle with radius r centered at the origin. In 3-space, the equation x 2 + y 2 + z 2 = r 2 corresponds to a sphere of radius r centered at the origin. 7.2 Surfaces For functions of one variable in R 2 , the most basic objects are lines. Lines are still important in R 3 , but more important are planes , by which I mean infinite flat surfaces. In addition to the basic planes based on the axes (the xy-, xz- and yz-planes) there are many more at varying heights and angles. In R 3 , instead of tangent lines we will have tangent planes....
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- Fall '09