Engineering Calculus Notes 237

Engineering Calculus Notes 237 - : h ( x 1 ,x 2 ,x 3 ) = (...

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3.2. LINEAR AND AFFINE FUNCTIONS 225 Linearity In both of the cases reviewed above, the tangent approximation to a function (real- or vector-valued) is given by polynomials of degree one in the variable. Analogously, in trying to approximate a function f ( x 1 ,x 2 ,x 3 ) of 3 variables, we would expect to look for a polynomial of degree one in these variables: p ( x 1 ,x 2 ,x 3 ) = a 1 x 1 + a 2 x 2 + a 3 x 3 + c where the coeFcients a i , i = 1 , 2 , 3 and c are real constants. To formulate this in vector terms, we begin by ignoring the constant term (which in the case of our earlier approximations is just the value of the function being approximated, at the time of approximation). A degree one polynomial with zero constant term (also called a homogeneous polynomial of degree one) h ( x 1 ,x 2 ,x 3 ) = a 1 x 1 + a 2 x 2 + a 3 x 3 has two important properties: Scaling: If we multiply each variable by some common real number α , the value of the function is multiplied by
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Unformatted text preview: : h ( x 1 ,x 2 ,x 3 ) = ( a 1 )( x 1 ) + ( a 2 )( x 2 ) + ( a 3 )( x 3 ) = ( )( a 1 x 1 + a 2 x 2 + a 3 x 3 ) = h ( x 1 ,x 2 ,x 3 ); in vector terms, this can be written h ( x ) = h ( x ) . This property is often referred to as homogeneity of degree one . Additivity: If the value of each variable is a sum of two values, the value of the function is the same as its value over the rst summands plus its value over the second ones: h ( x 1 + y 1 ,x 2 + y 2 ,x 3 + y 3 ) = ( a 1 )( x 1 + y 1 ) + ( a 2 )( x 2 + y 2 ) + ( a 3 )( x 3 + y 3 ) = ( a 1 x 1 + a 2 x 2 + a 3 x 3 ) + ( a 1 y 1 + a 2 y 2 + a 3 y 3 ) = h ( x 1 ,x 2 ,x 3 ) + h ( y 1 ,y 2 ,y 3 ) or in vector terms h ( x + y ) = h ( x ) + h ( y ) ....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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