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Unformatted text preview: (8/30/08) Math 20C. Lecture Examples. Section 14.1, Part 1. Functions of two variables Definition 1 A function f of the two variables x and y is a rule z = f ( x, y ) that assigns a number denoted 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 f 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? Answer: (a) The domain of f is the entire xy-plane. (b) f (2 , 3) = 13 f (- 2 ,- 3) = 13. (c) The range of f is the closed infinite interval [0 , ). Definition 2 The graph of z = f ( x, y ) is the surface z = f ( x, y ) formed by the points ( x, y, z ) in xyz-space with ( x, y ) in the domain of the function and z = f ( x, y ) (Figure 1). z = f ( x, y ) FIGURE 1 Lecture notes to accompany Section 14.1, Part 1 of Calculus, Early Transcendentals by Rogawski. 1 Math 20C. Lecture Examples. (8/30/08) Section 14.1, Part 1, p. 2 Fixing x or y : vertical cross sections of graphs One way to study the graph z = f ( x, y ) of a function of two variables is to study the graphs of the functions of one variable that are obtained by holding x or y constant. Example 2 Determine the shape of the surface z = x 2 + y 2 by studying its cross sections in the planes x = c perpendicular to the x-axis. Answer: The intersection of the surface z = x 2 + y 2 with the plane x = c is a parabola that opens upward and whose vertex is at the origin if c = 0 and is c 2 units above the xy-plane if c negationslash = 0 Figure A2a The surface has the bowl-like shape in Figure A2b y x z x = c c 2 Figure A2a...
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