Rog_Sec_14_1_P_1

# Rog_Sec_14_1_P_1 - Math 20C Lecture Examples Section 14.1...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 10

Rog_Sec_14_1_P_1 - Math 20C Lecture Examples Section 14.1...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online