This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: 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 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 √ 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 = y 2 = 0. The same thing happens on the y-axis, but if we look along the line y = x we have x...
View Full Document