# a6 - ≤ x ≤ 1 and 0 ≤ y ≤ 1 4 Consider f R 2 → R...

This preview shows page 1. Sign up to view the full content.

Math 237 Assignment 6 Due: Friday, Oct 30th 1. Find the 2nd degree Taylor polynomial for the following functions at the given point. a) f ( x, y ) = x 2 y 3 - xy at (1 , 1). b) f ( x, y ) = (1 + x ) y at (0 , 0). 2. Let f ( x, y ) = ln( x + 2 y ). a) Show that for ( x, y ) suﬃciently close to (3 , - 1) we have f ( x, y ) ( x - 3) + 2( y + 1). b) Prove that if x 3 and y ≥ - 1, then the error in the approximation in a) satisﬁes | f ( x, y ) - [( x - 3) + 2( y + 1)] | ≤ 3[( x - 3) 2 + ( y + 1) 2 ] . 3. Let f ( x, y ) = 1 + 2 x + y . a) Find the linear approximation L (0 , 0) ( x, y ) of f . b) Use Taylor’s theorem to show that the error in the linear approximation L (0 , 0) ( x, y ) is at most 3 4 ( x 2 + y 2 ) if 0
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ≤ x ≤ 1 and 0 ≤ y ≤ 1. 4. Consider f : R 2 → R deﬁned by f ( x, y ) = 2 x 2 + 3 y 2 , and let a ∈ R 2 . Prove that f ( x, y ) ≥ L a ( x, y ) for all ( x, y ) ∈ R 2 . 5. Prove that if f : R 2 → R has continuous partial derivatives in some neighborhood N ( a, b ) of ( a, b ), then for all ( x, y ) ∈ N ( a, b ) there exists a point ( c, d ) on the line segment from ( a, b ) to ( x, y ) such that f ( x, y ) = f ( a, b ) + f x ( c, d )( x-a ) + f y ( c, d )( y-b ) ....
View Full Document

## This note was uploaded on 10/12/2010 for the course MATH 237 taught by Professor Wolczuk during the Fall '08 term at Waterloo.

Ask a homework question - tutors are online