1. Let f (50,31) 2 sin(%my) and 9(23, y) = 2" be functions in the domain 1:, y 2 0. .___._"_I__-- (a) Sketch the level curves g(m,y) : c for c : 116, %, i, %,1. Use the same set of coordinate axes for the level curves, with the :1: and y ranging from O to 4. Label your level curves with the value of 0, (preferably) with a different color. (b) In a new picture, sketch the level curves f (m, y) : c for c : 1,0, 1. Use the same set of coordinate axes for the level curves, with the a: and y ranging from O to 4. Label your level curves with the value of 6, each with a different color. (0) Suppose that [($41) is the WWW (SO that the altitude at the coordinate (a, b) is f (a, b)). Suppose that three people are standing at points A = (Vi), B = (2,2,0) and C = (1%,0) on the surface 2 = f (31:, y) Can Person A see Person B? can Person A see Person C? can Person B see Person C"? Justify your answer. 2. Solve the following problems. You must justify your answer carefully. (a) Compute the following limit or show it does not exist. 1 4 3 ' cos ((321%) a: 3; 11m . (m,y)+(0,0) m2 + 3/2 (b) Compute the following limit or show it does not exist. 3 lim my . (m,y)->(0,0) 532 + 3/6 Hint: "The right path and easy path are usually two different things, and when it's time to choose, always choose the right path over the easy."1 (c) Consider the function F (m, y) = x2y3 | y. Directly compute the limit lim F($ay+h) _ F($:y). h>O h You answer should be a function of a: and y. 3. Let 2)1/2 m _ ( ($,y)%(0,0) "0 {C (m,y)=(0,0) Determine Whether there is a choice of C that makes f (3:, 3;) continuous everywhere. Hint: In order to nd the limit of f at (0, 0), you might nd it useful to write f using polar coordinates and then use the squeeze theorem