notes10 - CM221A ANALYSIS I NOTES ON WEEK 10 n times...

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Unformatted text preview: CM221A ANALYSIS I NOTES ON WEEK 10 n times differentiable functions. We say that f is n times differentiable on ( a,b ) if each derivative of order up to n exists at every point of the interval (the derivative of order two is the derivative of the first derivative, the derivative of order three is obtained by differentiation the derivative of order three and so on). We say that it is n times continuously differentiable if the final derivative is continuous (the function and its first ( n- 1) derivatives are automatically continuous). If the interval is [ a,b ] then we require the one-sided derivatives of all orders up to n to exist at the end-points if the interval. The usual notation for the derivative of order n is f ( n ) , so that f ( n ) ( x ) = d d x f ( n- 1) ( x ). TAYLORS THEOREM Theorem (Taylors formula). If f is n times continuously differentiable on the interval ( a- ,a + ) then, for each h (- , ), there exists a point c lying between a and a + h (that is, c [ a,a + h ] if h is positive and c [ a + h,a ] if h is negative), such that f ( a + h ) = f ( a )+ f ( a ) h + f (2) ( a ) h 2 2! + f (3) ( a ) h 3 3! + + f ( n- 1) ( a ) h m- 1 ( m- 1)! + R m ( a,h ) , where R m ( a,h ) = f ( m ) ( c ) h m m ! . Proof. Let g ( x ) = ( x- a ) m and f ( x ) = f ( x )- m- 1 X n =0 f ( n ) ( a ) ( x- a ) n n ! . Note that the functions f and g and their first ( n- 1) derivatives vanish at x = a . Therefore, applying Cauchys Mean Value Theorem, we obtain f ( b ) g ( b ) = f ( x )- f ( a ) g ( x )- g ( a ) = f ( c 1 ) g ( c 1 ) = f ( c 1 )- f ( a ) g ( c 1 )- g ( a ) =...
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notes10 - CM221A ANALYSIS I NOTES ON WEEK 10 n times...

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