Tutorial 2

# Tutorial 2 - At each of the following points(i determine...

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MATH1211/10-11(1)/Tu2 THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 1st Semester: Tutorial 2 1. For each of the following limits, either evaluate it or explain why it fails to exist. (a) lim ( x,y ) ( - 1 , 2) 2 x 2 + y 2 x 2 + y 2 (b) lim ( x,y ) (0 , 0) x 2 x 2 + y 2 2. Find the gradient f ( a ) where f ( x,y ) = e xy + ln( x - y ) , a = (2 , 1) . (Recall that if f : X R n R then its gradient is the vector f ( x ) = ± ∂f ∂x 1 , ∂f ∂x 2 , ··· , ∂f ∂x n ² . ) 3. Find the matrix D f ( a ) of partial derivatives where f : R 2 R 3 and a are given by f ( x,y ) = (2 x - y, y 3 , x sin xy ) , a = (1 , - 1) . 4. Let f : R 2 R be the function deﬁned by f ( x,y ) = ( 1 if x = 0 or y = 0 , 0 if both x,y 6 = 0
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Unformatted text preview: . At each of the following points, (i) determine whether f is continuous, (ii) ﬁnd f x and f y (or show that they do not exist), and (iii) determine whether f is diﬀerentiable : (a) ( a,b ) where both a,b 6 = 0 (b) ( a, 0) where a 6 = 0 (c) (0 ,b ) where b 6 = 0 (d) (0 , 0) 5. Explain why the function f ( x,y ) = x sin y x 2 + y 2 + 1 is diﬀerentiable at every point in its domain. 6. Find an equation for the plane tangent to the graph of z = 4 cos xy at the point ( π/ 3 , 1 , 2). 1...
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