midterm_04_L1_

# midterm_04_L1_ - x ≥ 0 y ≥ 0 and z ≥ 0 that are...

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Math 100 (L1) Midterm Test 2004 Fall (1) (25 marks) Show that lim ( x,y ) (0 , 0) xy 2 x 2 + y 2 + x 2 y 2 = 0 , but lim ( x,y ) (0 , 0) xy 2 x 2 + y 4 + x 2 y 2 does not exist. (2) (25 marks) Assume that u ( x, t ) = t 1 / 2 y ( z ), with z = t - 1 / 2 x , satisﬁes the equation u t - Du xx = 0 , D = constant . Show that the function y = y ( z ) satisﬁes D d 2 y dz 2 + z 2 dy dz - 1 2 y = 0 . (3) (25 marks) Find the points on the surface xyz = 1, with
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Unformatted text preview: x ≥ 0, y ≥ 0 and z ≥ 0, that are closest to the origin. (4) (25 marks) Assume that ( x , y , z ) on the smooth surface z = f ( x, y ) is closes to the origin. Show that the normal line at ( x , y , z ) passes through the origin. 1...
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