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Unformatted text preview: EE 103 Lecture Notes, Spring 2010 (SEJ) Appendix A-5 Taylor's 1 st and 2 nd Order Approximations for Functions of Several Variables Taylors Theorem for a function of one variable, 1 1 : f R R : Let 1 1 : f R R have an ( 1) st n continuous derivative. Let x x . Then, there exists a number , 0,1 x x so that 1 1 1 ! ( 1 ) ! k n n k n k x x x x f x f x f x f k n where , x x x x x , i.e., is a point between x and x . The term 1 1 ( 1 ) ! n n x x f n is often called the "error" or "remainder" term and note that we generally do not know the value of the number ; rather, we only know that it exists. Consider the two special cases, 1 n and 2 n . 1 n : Then, 2 ( ) ( ) '( )( ) ''( )( ) / 2 f x f x f x x x f x x If we drop the "remainder" term, we have what is often called Taylor's first order approximation , or linear approximation , ( )...
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- Spring '09