Intro to Analysis in-class_Part_36

Intro to Analysis in-class_Part_36 - < g x i.e that h:=...

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5.3 Calculus of derivatives 77 Note: s y as t x , by continuity of f . Theorem 5.3.5 ((Baby) Inverse Fn Thm) . f : ( a,b ) ( c,d ) is C 1 and f 0 ( x ) > 0 on ( a,b ) . Then f is invertible and is C 1 with ( f - 1 ) 0 ( y ) = 1 /f 0 ( x ) , if y = f ( x ) . Proof. Use the chain rule to differentiate both sides of the identity f - 1 ( f ( x )) = x .
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78 Math 413 Differentiation 5.4 Higher derivatives and Taylor’s Thm 5.4.1 Interpretations of f 00 Definition 5.4.1. If f 0 is defined in a neighbourhood of x and differentiable at x , then f is twice differentiable at x , with f 00 ( x ) = ( f 0 ) 0 ( x ). If f 00 is continuous, then f C 2 . Theorem 5.4.2. Suppose f 00 exists in some neighbourhood of x . (i) f 0 ( x ) = 0 ,f 00 ( x ) > 0 = x is a strict local min. (ii) If x is a local max, then f 00 ( x ) 0 . (iii) f 00 ( x ) > 0 = the graph of f ( x ) lies below any secant line. Proof. (i) f 00 ( x ) > 0 means f 0 is increasing on some open interval ( a,b ) around x , with f 0 ( a ) < 0 and f 0 ( b ) > 0 (since f 0 ( x ) = 0). Then f is decreasing on ( a,x ) and increasing on ( x,b ). (ii) This is implied by the contrapositive of (1). (iii) Let g be an affine function whose graph intersects f ’s at x = s and x = t . We need to show f ( x
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Unformatted text preview: ) < g ( x ), i.e. that h := f-g is negative on ( s,t ) (note h ( s ) = h ( t ) = 0). If f 00 ( x ) > 0 for a < x < b , then h is also, since g 00 = 0. Suppose h were not negative on ( s,t ); then it would have a local max on this interval at some point c . By Part (1), this would imply h 00 ( c ) ≤ 0. < . Part (i) holds, in particular, for g ( t ) = f ( x ) + f ( x )( x-t ). 5.4.2 Taylor’s Thm Definition 5.4.3. If f ∈ C n , the Taylor expansion of f at a is T n ( a,x ) = f ( a ) + f ( a )( x-a ) + 1 2 f 00 ( a )( x-a ) 2 + ··· + 1 n ! f ( n ) ( a )( x-a ) n , where f ( n ) = ( f ( n-1) ) is defined by induction. Theorem 5.4.4. If f ( n ) exists at a , then T n ( a,x ) is the unique polynomial of degree n in powers of ( x-a ) having n th-order agreement with f ( x ) at a ....
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis in-class_Part_36 - < g x i.e that h:=...

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