exam2-v1-solutions

# exam2-v1-solutions - Math233 Exam 2 - Solutions x2 + y 2...

This preview shows pages 1–2. Sign up to view the full content.

Math233 Exam 2 - Solutions Fall 2006 Problem 1 (15pts). Determine the set of points at which the function is continuous. f ( x,y ) = ( p x 2 + y 2 ln( x 2 + y 2 ) if ( x,y ) 6 = (0 , 0) 1 if ( x,y ) = (0 , 0) f is continuous on R 2 except maybe at (0 , 0) (as a product of continuous functions). To check continu- ity at (0 , 0) , we examine the limit of f ( x,y ) as ( x,y ) tends to (0 , 0) . For that we use polar coordinates (see very similar example done in class). Let x = r cos θ and y = r sin θ , then x 2 + y 2 = r 2 and ( x,y ) (0 , 0) is equivalent to r 0 . lim ( x,y ) (0 , 0) p x 2 + y 2 ln( x 2 + y 2 ) = lim r 0 r 2 ln( r 2 ) = lim r 0 2 r ln( r ) = lim r 0 2 ln( r ) 1 /r = lim r 0 2 1 /r - 1 /r 2 (L’Hopital’s Rule) = lim r 0 - 2 r = 0 The limit exists but it different from f (0 , 0) , therefore f is not continuous at (0 , 0) . Problem 2 (10pts). Give the deﬁnition of the partial derivatives by completing the following: f x ( a,b ) = lim h 0 f ( a + h,b

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## exam2-v1-solutions - Math233 Exam 2 - Solutions x2 + y 2...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online