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Unformatted text preview: Taylor’s Formula: The Beast, Two-Variable Version! Shubhodeep Mukherji March 11, 2011 1 Taylor’s Formula in Single-Variate Calculus 1.1 Taylor Polynomial of Order n Suppose f ( x ) has derivatives of all orders throughout some interval containing the point x = a . If so, then the n th order taylor polynomial generated by f ( x ) at x = a is the polynomial P n ( x ) = f ( a ) + f ( a )( x- a ) + f 00 ( a ) 2! ( x- a ) 2 + ··· + f ( k ) ( a ) k ! ( x- a ) k + ··· + f ( n ) ( a ) n ! ( x- a ) n . 1.2 The Error Term R n ( x ) = f ( n +1) ( c ) ( n + 1)! ( x- a ) n +1 for some c between a and x . 1.3 Taylor’s Formula f ( x ) = P n ( x ) + R n ( x ) = f ( a ) + f ( a )( x- a ) + f 00 ( a ) 2! ( x- a ) 2 + ··· + f ( k ) ( a ) k ! ( x- a ) k + ··· + f ( n ) ( a ) n ! ( x- a ) n + f ( n +1) ( c ) ( n + 1)! ( x- a ) n +1 . 2 Second Derivative Test 2.1 Local Max, Local Min, Critical Point, Saddle Point Local Maximum occurs if f ( a,b ) ≥ f ( x,y ) for an open disk centered at ( a,b ). Local Minimum occurs if f ( a,b ) ≤ f ( x,y ) for an open disk centered at ( a,b ). A Critical Point occurs at the point( a,b ) if f x and f y are both zero or if one or both of f x and f y do not exist....
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This note was uploaded on 08/03/2011 for the course ECON 101 taught by Professor Profeessor during the Spring '11 term at Aachen University of Applied Sciences.
- Spring '11