{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

A1_SOLN

A1_SOLN - Math 237 Assignment 1 Due Friday Sept 19th 1 For...

This preview shows pages 1–4. Sign up to view the full content.

Math 237 Assignment 1 Due: Friday, Sept 19th 1. For each of the following function f : R 2 R i) Sketch the domain of f and state the range of f . ii) Sketch level curves and cross sections. iii) Sketch the surface z = f ( x,y ). a) f ( x,y ) = x 2 - y 2 + 1. Solution: i) The domain is R 2 the range is z R . ii) Level Curves: k = x 2 - y 2 + 1 x 2 - y 2 = k - 1 Cross sections: z = c 2 - y 2 + 1 z = x 2 - d 2 + 1 iii) 1

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

View Full Document
2 b) f ( x,y ) = radicalbig x 2 + y 2 Solution: i) The domain is R 2 the range is z [0 , ). ii) Level Curves: k = radicalbig x 2 + y 2 x 2 + y 2 = k 2 , k 0 Cross sections: z = radicalbig c 2 + y 2 z 2 - y 2 = c 2 , z 0 z = x 2 + d 2 z 2 - x 2 = d 2 , z 0 iii)
3 2. Find the limit, if it exists, or show that the limit does not exist. a) lim ( x,y ) (0 , 0) x 4 + y 4 x 2 + y 2 Solution: Try approaching the limit along lines y = mx . This gives lim ( x,y ) (0 , 0) x 4 + y 4 x 2 + y 2 = lim ( x,y ) (0 , 0) x 4 + m 4 x 4 x 2 + m 2 x 2 = lim x 0 x 2 + m 4 x 2 1 + m 2 = 0 Since all these limits give the same value, the limit may be equal to 0. We will try to prove it using the squeeze theorem. Observe that x 4

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.

{[ snackBarMessage ]}