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Unformatted text preview: (3/22/08) CHAPTER 14 Derivatives with Two or More Variables Many mathematical models involve functions of two or more variables. The elevation of a point on a mountain, for example, is a function of two horizontal coordinates; the density of the earth at points in its interior is a function of three coordinates; the pressure in a gas-filled balloon is a function of its temperature and volume; and if a store sells a thousand items, its profit might be studied as a function of the amounts of each of the thousand items that it sells. This chapter deals with the differential calculus of such functions. We study functions of two variables in Sections 14.1 through 14.6. We discuss vertical cross sections of graphs in Section 14.1, horizontal cross sections and level curves in Section 14.2, partial derivatives in Section 14.3, Chain Rules in Section 14.4, directional derivatives and gradient vectors in Section 14.5, and tangent planes in Section 14.6. Functions with three variables are covered in Section 14.7 and functions with more than three variables in Section 14.8. Section 14.1 Functions of two variables Overview: In this section we discuss domains, ranges and graphs of functions with two variables. Topics: The domain, range, and graph of z = f ( x , y ) Fixing x or y : vertical cross sections of graphs Drawing graphs of functions The domain, range, and graph of z = f ( x , y ) The definitions and notation used for functions with two variables are similar to those for one variable. Definition 1 A function f of the two variables x and y is a rule that assigns a number f ( x,y ) to each point ( x,y ) in a portion or all of the xy-plane. f ( x,y ) is the value of the function at ( x,y ) , and the set of points where the function is defined is called its domain . The range of the function is the set of its values f ( x,y ) for all ( x,y ) in its domain. If a function z = f ( x,y ) is given by a formula, we assume that its domain consists of all points ( x,y ) for which the formula makes sense, unless a different domain is specified. Example 1 (a) What is the domain of f ( x,y ) = x 2 + y 2 ? (b) What are the values f (2 , 3) and f (- 2 ,- 3) of this function at (2 , 3) and (- 2 ,- 3)? (c) What is its range? Solution (a) Because the expression x 2 + y 2 is defined for all x and y , the domain of f is the entire xy-plane. (b) f (2 , 3) = 2 2 + 3 2 = 13 and f (- 2 ,- 3) = (- 2) 2 + (- 3) 2 = 13. (c) The values x 2 + y 2 of the function are all nonnegative and for every z 0 it has the value z at all points ( x,y ) on the circle x 2 + y 2 = z . Consequently, the range of f is the closed infinite interval [0 , ). square Recall that the graph of a function f of one variable is the curve y = f ( x ) in an xy-plane consisting of the points ( x,y ) with x in the domain of the function and y = f ( x ). The graph of a function of two variables is a surface in three-dimensional space....
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