Fall 2007 - Mengi's Class - Exam 2 (Version A)

# Fall 2007 - Mengi's Class - Exam 2 (Version A) - Midterm 2...

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

Unformatted text preview: Midterm 2, Math 20C (Lecture C) November 28th, 2007 1. Consider the function f ( x, y ) = x 2 +4 y 2 4 xy ( x, y ) 6 = (0 , 0) ( x, y ) = (0 , 0) a) (3 points) Plot the contour diagrams of z = f ( x, y ) for z =- 1 , , 1. Setting z =- 1, we have 0 = x 2 + 4 xy + 4 y 2 = ( x + 2 y ) 2 . Setting z = 1, we have 0 = x 2- 4 xy + 4 y 2 = ( x- 2 y ) 2 . The function takes the value z = 0, only at ( x, y ) = (0 , 0). Therefore the level sets when z =- 1 and z = 1 are the lines x =- 2 y and x = 2 y , respectively. The level set z = 0 is just the origin.-4.8-4-3.2-2.4-1.6-0.8 0.8 1.6 2.4 3.2 4 4.8-2.4-1.6-0.8 0.8 1.6 2.4 z=-1 z=1 z=0 Answer for the other version: The level sets when z =- 2 and z = 2 are the lines y =- 2 x and y = 2 x , respectively. The level set z = 0 is just the origin.-4.8-4-3.2-2.4-1.6-0.8 0.8 1.6 2.4 3.2 4 4.8-2.4-1.6-0.8 0.8 1.6 2.4 z=-2 z=2 z=0 b) (3 points) Is f ( x, y ) continuous at ( x, y ) = (0 , 0)? Justify your answer. No, it is not continuous at (0 , 0), because lim ( x,y ) → (0 , 0) f ( x, y ) does not exist. Suppose we approach the origin along the line y = kx . The function along this line is f ( x, kx ) = x 2 +4 k 2 x 2 4 kx 2 = 1+4 k 2 4 k . For instance along the line y = x ( k = 1), the function is constant and equal to 5 4 . But along the line y =- x ( k =- 1), the function takes the value- 5 4 . Since along different paths leading to (0 , 0) the function f ( x, y ) approaches different values, the limit as ( x, y ) → (0 , 0) does not exist....
View Full Document

## This note was uploaded on 04/29/2008 for the course MATH 20C taught by Professor Helton during the Fall '08 term at UCSD.

### Page1 / 6

Fall 2007 - Mengi's Class - Exam 2 (Version A) - Midterm 2...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online