MATH 411 Midterm: MATH411 BOYLE-M SPRING2012 0101 MID SOL

• School: University of Maryland
• Department: Mathematics
• Course: MATH 411
• Professor: All

Document Summary

Math 411 spring 2012 boyle exam 2 solutions: compute lim (x,y) (0,0) x2y + y2x x2 + y2 or show the limit doesn"t exist. 0 x2y + y2x x2 + y2 = (r cos( ))2(r sin( )) + (r sin( ))2(r cos( ))] r2. Because r 0 as (x, y) 0, it follows that lim (x,y) (0,0) x2y + y2x x2 + y2 = 0 : de ne f : r2 r by the rule. F (x1, x2) = 3x2 +(cid:0)(x1)2 + (x2)2(cid:1) sin(x1/x2) Determine whether f is di erentiable at the origin. We"ll show that at the origin, the derivative exists, with. Df (0) = ( f/ x, f/ y)(0) = (0, 3) . If h = (h1, h2) and h2 6= 0, then. ||h||2 and if h2 = 0 then ||f (0 + h) f (0) [0h1 + 3h2]|| = 0. ||f (0 + h) f (0) [0h1 + 3h2]||