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taylorhandout - Taylors Formula The Beast Two-Variable...

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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 0 ( 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 0 ( 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. A Saddle Point occurs at a critical point if in every open disk centered at (a,b) there are points where f ( x, y ) > f ( a, b ) and other points where f ( x, y ) < f ( a, b ).
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