272_c14

272_c14 - Mat 272 Calculus III Updated on Dr Firoz Chapter 14 Partial Derivatives Section 14.1 Functions of Several Variables Definition A function

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Mat 272 Calculus III Updated on 10/4/07 Dr. Firoz Chapter 14 Partial Derivatives Section 14.1 Functions of Several Variables Definition : A function f of two variables is a rule that assigns to each ordered pair of real numbers ( x , y ) in a set D a unique real number denoted by f ( x, y ) . The set D is the domain of f and its range is the set of values that f takes on, that is { ( , ) | ( , ) } f x y x y D . Definition : If f is a function of two variables with domain D, then the graph of f is the set of all points ( x , y , z ) in 3 such that ( , ) z f x y = and ( , ) x y D . Definition : The Level Curves (contour curve) of a function f of two variables are the curves with equation ( , ) f x y k = where k is a constant (in the range of f ) Functions of three or more variables : A function of three variables f , is a rule that assigns to each ordered triple ( , , ) x y z in a domain 3 D a unique real number denoted by ( , , ) f x y z Examples : 1. Find the domain of a) 1 ( , ) x y f x y x y + + = - and b) 1 ( , ) x y f x y x y + + = - Solution: a) The domain of f is {( , ) | } D x y x y = b) The domain of f is {( , ) | 1 0, } D x y x y x y = + + ≥ 2. Find the domain and range of 2 2 ( , ) 36 f x y x y = - - Solution: The domain of f is 2 2 {( , ) | 36} D x y x y = + that is all points inside and on the circle of radius 6. And the range of the function of f is 2 2 { | 36 ,( , ) } z z x y x y D = - - 3. Find the domain and range of f is 2 2 ( , ) 4 z h x y x y = = + Solution: We have seen in chapter 13 that the function h(x, y) is an elliptic paraboloid with vertex at (0, 0, 0), and opens upward. Horizontal traces are ellipses and vertical
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Mat 272 Calculus III Updated on 10/4/07 Dr. Firoz traces are parabolas. The domain is all the ordered pairs ( x, y ) in 2 , that is the xy- plane. The range is the set [0, ) of all nonnegative real numbers. 4. Sketch all the level curves of the function 2 2 ( , ) 36 f x y x y = - - for 0,1,2,3 k = 5. Find the level surfaces of the function 2 2 2 ( , , ) f x y z x y z = + + Solution: Choose different numerical values of ( , , ) f x y z and observe that 2 2 2 k x y z = + + represents spheres as level surfaces. See example 15, page # 897 at your text. Section 14.2 Limits and Continuity Definition : Let ( , ) f x y be a function of two variables whose domain D includes points arbitrarily close to ( , ) a b . Then the limit of ( , ) f x y as ( , ) x y approaches ( , ) a b is L and is written as s and continuity ( , ) ( , ) lim ( , ) x y a b f x y L = if for every number 0 ε > there is a corresponding number 0 δ > such that 2 2 ( , ) whenever ( , ) and 0 ( ) ( ) f x y L x y D x a y b - < < - + - < Definition: Continuous function The function ( , ) f x y is continuous on D if f is continuous at every point ( , ) a b in D. Examples:
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This note was uploaded on 05/09/2008 for the course MAT 272 taught by Professor Firoz during the Spring '08 term at ASU.

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272_c14 - Mat 272 Calculus III Updated on Dr Firoz Chapter 14 Partial Derivatives Section 14.1 Functions of Several Variables Definition A function

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