Engineering Calculus Notes 207

Engineering Calculus Notes 207 - x [0 , 1], f ( x ) := lim...

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2.4. REGULAR CURVES 195 and in particular f ( a ) = c f ( a 1 ) = 2 a + b 3 = 2 c + d 3 f ( a 2 ) = a + 2 b 3 = c + 2 d 3 f ( b ) = d. Now, ¯ f is defned by ¯ f ( a ) = c ¯ f ( a ) = c ¯ f ( a 1 ) = c 1 = c + 2 d 3 ¯ f ( a 2 ) = c 2 = 2 c + d 3 ¯ f ( b ) = d and ¯ f is aFne on each o± the intervals I 1 = [ a,a 1 ], I 2 = [ a 1 ,a 2 ], and I 3 = [ a 2 ,b ]. Show that the slopes m j o± the graph o± ¯ f on I j satis±y m 1 = m 3 = 2 m m 2 = m. (b) Now, we construct a sequence o± ±unctions f k on [0 , 1] via the recursive defnition f 0 = id f k +1 = ¯ f k . Show that | f k ( x ) f k +1 ( x ) | ≤ p 2 3 P k +1 (2.26) ±or all x [0 , 1]. This implies that ±or each
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Unformatted text preview: x [0 , 1], f ( x ) := lim f k ( x ) is well-defned or each x [0 , 1]. We shall accept without proo the act that Equation ( 2.26 ) (which implies a property called uniform convergence ) also guarantees that f is continuous on [0 , 1]. Thus its graph is a continuous curvein act, it is an arc....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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