2220hw1sol - Math 2220 Problem Set 1 Solutions Spring 2010...

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Math 2220 Problem Set 1 Solutions Spring 2010 Section 2.1 : In the following two problems you are asked to do two things: (a) determine several level curves f ( x, y ) = c , being sure to label each curve with the appropriate value of c , and (b) sketch the graph, using the information from (a). 12. f ( x, y ) = x 2 + y 2 - 9. Solution. The level sets are the circles x 2 + y 2 = 9 + c , as shown at the left below. The graph is on the right. The graph intersects the xz -plane in the parabola z = x 2 - 9 and it intersects the yz -plane in the parabola z = y 2 - 9. 18. f ( x, y ) = 3 - 2 x - y . Solution. The level curves are the lines 3 - 2 x - y = c , or y = - 2 x + (3 - c ). These are shown on the left below. The graph z = 3 - 2 x - y is a plane shown at the right. 1
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Math 2220 Problem Set 1 Solutions Spring 2010 Section 2.2 : In the next four problems, evaluate the limit if it exists, or explain why the limit does not exist. 8. lim ( x,y ) (0 , 0) | y | radicalbig x 2 + y 2 Solution. Along the x -axis we have y = 0 and the value of | y | radicalbig x 2 + y 2 is 0 x 2 = 0 and along the y -axis we have x = 0 so | y | radicalbig x 2 + y 2 = | y | radicalbig y 2 = 1. The limits along the two axes are therefore 0 and 1. Since these are different, the limit as ( x, y ) (0 , 0) does not exist. 14. lim ( x,y ) (0 , 0) xy x 2 + y 2 Solution. Along the x -axis we have y = 0 so xy x 2 + y 2 = 0 y 2 = 0. The same thing happens on the y -axis, but if we look along the line y = x we have x 2 x 2 + x 2 = 1 2 . Thus we have
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