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Unformatted text preview: Lecture 4 : Monday April 6th [email protected] Key concepts : Tangent hyperplane, Gradient, Directional derivative, Level curve Know how to find equation of tangent hyperplane, gradient, directional derivatives, direction of fastest increase. 4.1 Differentiability Recall that f : R n R is differentiable at a point a R n if all partial derivatives f j ( a ) exist and lim x a f ( x ) f ( a ) n j =1 f j ( a )( x j a j ) d ( x,a ) = 0 . It is not enough that all the partial derivatives f j ( a ) exist for f to be differentiable at a . Remember here that x a means ( x 1 ,x 2 ,...,x n ) ( a 1 ,a 2 ,...,a n ). Lets do some examples. In all the examples p = ( x,y ). Example 1. The function f ( x,y ) = xy is differentiable at a = (0 , 0), since f x ( a ) = 0 = f y ( a ) as we checked in a previous example, and lim p a xy p x 2 + y 2 = 0 . Lets use the  definition of limits to see that this limit is zero. Let g ( x,y ) be the function in the limit. For every &gt; 0 we must find a &gt; 0 such that d ( p,a ) &lt; ensures  g ( x,y )  &lt; . We claim that  xy  x 2 + y 2 . To see this, note that it is equivalent to  x  2 +  y  2  xy  0. But this is true since (  x    y  ) 2 =  x  2 +  y  2 2  xy   x  2 +  y  2  xy  . Now we use this to show that f is differentiable:  g ( x,y )  &lt; fl fl fl xy p x 2 + y 2 fl fl fl &lt; x 2 + y 2 p x 2 + y 2 &lt; p x 2 + y 2 &lt; d ( p,a ) &lt; . 1 So putting = in the definition of the limit we get lim g ( x,y ) = 0 as required for f to be differentiable at (0 , 0). Theres an easier way to check the limit is zero, using squeezing: note that  xy  p x 2 + y 2  xy  p y 2 =  x  so the function in the limit is between 0 and  x  . Since both of these have a limit of zero as ( x,y ) (0 , 0), so must xy/ p x 2 + y 2 . Example 2. This example is almost a follow up to the last one, where we now put f ( x,y ) = p  xy  . This function is not differentiable at a = (0 , 0), since f x (0 , 0) = f y (0 , 0) = 0 as we saw last lecture, but lim ( x,y ) a p  xy  p x 2 + y 2 does not exist. To see that the limit fails, along the linedoes not exist....
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This note was uploaded on 02/12/2010 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.
 Spring '07
 Enright
 Math, Derivative

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