# 10.07 - Section 11.4: Tangent planes and linear...

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: Section 11.4: Tangent planes and linear approximation (concluded) Theorem 11.4.8: If f ( x , y ) is a continuous function that is differentiable w.r.t. x and w.r.t. y at ( x , y ) = ( a , b ) (i.e., if f x ( a , b ) and f y ( a , b ) are both defined), and if f 1 ( x , y ) and f 2 ( x , y ) both exist in a neighborhood of ( a , b ) and are both continuous at ( a , b ), then f is differentiable at ( a , b ), and the graph z = f ( x , y ) has a tangent plane at the point ( a , b , c ), where c = f ( a , b ), namely the plane z – c = f x ( a , b ) ( x – a ) + f y ( a , b ) ( y – b ). We say that the function L ( x , y ) = f ( a , b ) + f x ( a , b ) ( x – a ) + f y ( a , b ) ( y – b ) is the linearization of f at ( a , b ). The approximation f ( x , y ) ≈ L ( x , y ) (valid for ( x , y ) ≈ ( a , b )) is called the linear approximation to f at ( a , b ). We can phrase this in terms of the notion of differentials discussed in Honors Calc I; writing x – a as dx and y – b as dy , where dx and dy are independent variables, we can introduce a dependent variable dz defined as dz = f x ( a , b ) dx + f y ( a , b ) dy . In the notation of differentials, our remarks about linear approximation can be written compactly as ∆ z ≈ dz where ∆ z is defined as f ( x + dx , y + dy ) – f ( x , y )....
View Full Document

## This note was uploaded on 02/13/2012 for the course MATH 241 taught by Professor Staff during the Fall '11 term at UMass Lowell.

### Page1 / 7

10.07 - Section 11.4: Tangent planes and linear...

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

View Full Document
Ask a homework question - tutors are online