midterm_w08_post

midterm_w08_post - Let f R 2 → R and g R → R such that...

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Math 237 - Midterm Winter 2008 NOTE: The questions on this exam does not exactly reﬂect which questions will be on this semesters midterm. That is, some questions asked on this midterm may not be asked on our midterm and there may be some questions on our midterm not asked here. 1. Short Answer Problems a) Let f : R 2 R . Write the precise deﬁnition of f being diﬀerentiable at ( a,b ). b) If f x and f y are both continuous at ( a,b ) what two things can we conclude about f ( x,y ) at ( a,b )? c) What is the equation for the tangent plane to the surface 0 = f ( x,y,z ) at ( a,b,c )? d) State the deﬁnition of the directional derivative D u f ( a,b ) at a point ( a,b ) in the direction of a unit vector u . 2. Let f ( x,y ) = p 1 + x 2 - y 2 . a) Sketch the domain of f . What is the range of f ? b) Sketch the level curve 1 = f ( x,y ), the cross-section x = 1 and the cross section y = 1 of the surface z = f ( x,y ). 3. Determine if each of the following limits exist. Evaluate the limits that exist. a) lim ( x,y ) (1 , 0) ( x 2 + xy + y 2 + 1) sin( x 2 + y 2 ) x 2 + y 2 . b) lim ( x,y ) (0 , 0) 3 xy 2 x + y 3 . 4. State and prove the Squeeze Theorem for f : R 2 R . 5.
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Unformatted text preview: Let f : R 2 → R and g : R → R such that f ( x,y ) = ± x + 1 y 2 ² g ( xy 2 ) , y 6 = 0 . Use the Chain Rule to show that 2 xf x-yf y = 2 f . 6. Let f ( x,y ) = ln(1 + x + 2 y ). a) Find the linear approximation L (0 , 0) ( x,y ) of f . b) Use Taylor’s Theorem to show that | R 1 , (0 , 0) ( x,y ) | ≤ 7 2 ( x 2 + y 2 ) for x ≥ 0, y ≥ 0. 7. Let f ( x,y,z ) = e xy + z . a) Find the rate of change of f at the point (1 ,-1 , 1) in the direction of the vector u = (4 ,-2 , 3). b) In what direction from (1 ,-1 , 1) does f change most rapidly and what is the maximum rate of change. 8. Consider the function f ( x,y ) = ( x 3 + y 3 x 2 + y 2 , ( x,y ) 6 = (0 , 0) , ( x,y ) = (0 , 0) . a) What is the domain of f ? b) Where is f continuous on its domain? c) Where is f diﬀerentiable on its domain? d) Based on your answer in part c), what can you conclude about the continuity of f x and f y ?...
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This note was uploaded on 04/18/2010 for the course MATH 237 taught by Professor Wolczuk during the Spring '08 term at Waterloo.

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