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Unformatted text preview: MA1104 Multivariable Calculus Lecture 5 Dr KU Cheng Yeaw Thursday Jan 29, 2008 Recap Last lecture ... • we introduced the TNB frame. • we began the study of (scalar) functions of several variables. We visualize them by their (3D) graphs. • We first extend the notion of limit and continuity to functions of several variables. Overview In this lecture, • we introduce continuity for functions of several variables • we introduce partial derivatives • we study tangent plane and linear approximation of functions of several variables Continuity Recall that evaluating limits of continuous functions of a single variable is easy. • It can be accomplished by direct substitution. • This is because the defining property of a continuous function is lim x → a f ( x ) = f ( a ) . Continuous functions of two variables are also defined by the direct substitution property. Definition of Continuity Suppose f ( x,y ) is defined in the interior of a circle centered at ( a,b ) . We say that f is continuous at ( a,b ) if lim ( x,y ) → ( a,b ) f ( x,y ) = f ( a,b ) . If f ( x,y ) is not continuous at ( a,b ) , then we call ( a,b ) a disconti nuity of f . f is said to be continuous on D ⊆ R 2 if f is continuous at each point in D . Notice that because we define continuity in terms of limits, we have the following immediate consequences: Continuity Theorems If f ( x,y ) and g ( x,y ) are continuous at ( a,b ) , then f ± g , f · g are all continuous at ( a,b ) . Further, f g is continuous at ( a,b ) , provided g ( a,b ) 6 = 0 . Subsequently, every polynomial in x and y is continuous on R 2 . Each rational function is continuous in its domain. For example, f ( x,y ) = x 2 + x 3 y x + y is continuous on D = { ( x,y ) : x + y 6 = 0 } . Example 1. Find all points where the given function is continuous: g ( x,y ) = ( x 4 x ( x 2 + y 2 ) if ( x,y ) 6 = (0 , 0) , if ( x,y ) = (0 , 0) . Solution. Notice g ( x,y ) is a rational function when ( x,y ) 6 = (0 , 0) , so it is continuous at ( x,y ) 6 = (0 , 0) . It remains to check continuity at (0 , 0) . We can check the limit at (0 , 0) as follows: for ( x,y ) 6 = (0 , 0) ,  g ( x,y )  = x 4 x ( x 2 + y 2 ) ≤ x 4 x ( x 2 ) =  x  . Since lim ( x,y ) → (0 , 0)  x  = 0 , we deduce that lim ( x,y ) → (0 , 0) g ( x,y ) = 0 = g (0 , 0) . Therefore, g ( x,y ) is continuous at (0 , 0) . Putting things together, g ( x,y ) is continuous on the entire plane R 2 . Just as for functions of one variable, composition is another way of combining two continuous functions to get a third. Continuity and Composition Suppose f ( x,y ) is continuous at ( a,b ) and g ( x ) is continuous at f ( a,b ) . Then h ( x,y ) = ( g ◦ f )( x,y ) = g ( f ( x,y )) is continuous at ( a,b ) . Example 2. Determine where f ( x,y ) = e x 2 y is continuous....
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This note was uploaded on 03/19/2012 for the course MATH 1104 taught by Professor Kuchengyeaw during the Spring '08 term at National University of Singapore.
 Spring '08
 Kuchengyeaw
 Multivariable Calculus, Scalar

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