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Unformatted text preview: (3/23/08) Section 14.3 Partial derivatives with two variables Overview: In this section we begin our study of the calculus of functions with two variables. Their derivatives are called partial derivatives and are obtained by differentiating with respect to one variable while holding the other variable constant. We describe the geometric interpretations of partial derivatives, show how formulas for them can be found with differentiation formulas with one variable, and demonstrate how they can be estimated from tables and level curves. Topics: • Limits of functions with two variables • Continuity of functions with two variables • Partial derivatives • A geometric interpretation of partial derivatives • Estimating partial derivatives from tables • Estimating partial derivatives from level curves Limits of functions with two variables In studying functions of one variable we used one and twosided limits. We cannot talk of twosided or onesided limits of functions of two variables. Instead we find limits by studying the values of functions along paths, as in the next definition. † Definition 1 Suppose that the function z = f ( x,y ) is defined in a circle with its center at the point ( x ,y ) , except possibly at the point ( x ,y ) itself. Then the limit of f ( x,y ) as ( x,y ) approaches ( x ,y ) is L and we write lim ( x,y ) → ( x ,y ) f ( x,y ) = L (1) if the number f ( x,y ) approaches L as ( x,y ) approaches ( x ,y ) along all paths that lie in the circle and do not contain the point ( x ,y ) (Figure 1). Here L can be a number or ±∞ . x y ( x ,y ) Three paths to ( x ,y ) FIGURE 1 † The formal definition of this limit for numbers L reads as follows: The limit of f ( x,y ) is L as ( x,y ) → ( x ,y ) if for each epsilon1 > 0 there is a δ > 0 such that  f ( x,y ) L  < epsilon1 for all points ( x,y ) negationslash = ( x ,y ) within a distance δ of ( x ,y ). The definitions for L = ±∞ are similar. 301 p. 302 (3/23/08) Section 14.3, Partial derivatives with two variables Example 1 What is lim ( x,y ) → (3 , 2) ( x 2 + y 2 )? Solution As ( x,y ) → (3 , 2), the number x tends to 3 and the number y tends to 2. Then, because A ( x ) = x 2 is continuous for all x and B ( y ) = y 2 is continuous for all y , x 2 → 3 2 and y 2 → 2 2 , so that lim ( x,y ) → (3 , 2) ( x 2 + y 2 ) = 3 2 + 2 2 = 9 + 4 = 13 . square Example 2 What is the limit of z = 1 radicalbig x 2 + y 2 as ( x,y ) → (0 , 0)? Solution Because radicalbig x 2 + y 2 is positive for ( x,y ) negationslash = (0 , 0) and tends to 0 as ( x,y ) → (0 , 0), lim ( x,y ) → (0 , 0) 1 radicalbig x 2 + y 2 = ∞ . square The result of Example 2 is illustrated in Figure 2, which shows the graph of z = 1 radicalbig x 2 + y 2 . The zcoordinates of points on the surface tend to ∞ as their x and ycoordinates tend to zero....
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This note was uploaded on 09/13/2010 for the course MATH Math 20A taught by Professor Eggers during the Summer '08 term at UCSD.
 Summer '08
 Eggers
 Math, Derivative

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