Wrestling with the Fundamental Theorem of Calculus

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$Math 237 Worksheet: line integrals and gradient vector fields ...$

$Math 237 Worksheet: line integrals and gradient vector fields ...$

Cantor’s set eliminates all tercimals from 0.1 to 0.2. From 0.01 to 0.02 and 0.21 to 0.22 From 0.001 to 0.002, 0.021 to 0.022, 0.201 to 0.202, and 0.221 to 0.222 Cantor’s set eliminates all tercimals that terminate or contain a 1.
Cantor’s set eliminates all tercimals from 0.1 to 0.2. From 0.01 to 0.02 and 0.21 to 0.22 From 0.001 to 0.002, 0.021 to 0.022, 0.201 to 0.202, and 0.221 to 0.222 Cantor’s set eliminates all tercimals that terminate or contain a 1. What’s left? 2 2 2 2 2 1 0 02020202 9 81 9 9 9 1 1 . = + + 3 + 4 + ! = ! " 9 = 1 4
Page 1 and 2: David M. Bressoud Macalester Colleg
Page 3 and 4: S. F. LaCroix (1802):“Integral ca
Page 9 and 10: Bernhard Riemann (1852, 1867) On th
Page 11 and 12: Any continuous function is integrab
Page 13 and 14: Bernhard Riemann (1852, 1867) On th
Page 15 and 16: At x = a 2b , gcd ( a, 2b)= 1, the
Page 17 and 18: If lim f' ( x) and lim f' ( x) exis
Page 19 and 20: Riemann’s function: f( x)= ! n= 1
Page 21 and 22: If F is differentiable at x = a, ca
Page 23 and 24: " 2 x sin ( 1 ) Fx x , x ! 0, ( )=
Page 25 and 26: " 2 x sin ( 1 ) Fx x , x ! 0, ( )=
Page 27 and 28: # 2x sin( 1 f x x x x ( )= )! cos (
Page 29 and 30: # 2x sin( 1 f x x x x ( )= )! cos (
Page 31 and 32: Consider: " ( ) 2 $ x sin 1 x ! Fx
Page 33 and 34: What if the derivative, F ' , is bo
Page 35 and 36: “Cantor’s Set” First describe
Page 37 and 38: “Cantor’s Set” First describe
Page 39 and 40: Cantor’s Set 0 1/9 2/9 1/3 2/3 7/
Page 41 and 42: Cantor’s Set 0 1/9 2/9 1/3 2/3 7/
Page 43 and 44: 1 1 1 1 1 1 1 1 [ 0, 0. 1)( , . 99,
Page 45 and 46: Base three numbers: 10211 1! 1+ 1!
Page 47: Base three numbers: 10211.2012 1/3
Page 51 and 52: The Fundamental Theorem of Calculus
Page 53 and 54: Create a new set like the Cantor se
Page 55 and 56: We’ll call this set SVC (for Smit
Page 57 and 58: Volterra’s construction: " 2 x si
Page 59 and 60: Volterra’s construction: f x 1( )
Page 61 and 62: We follow the same procedure to cre
Page 63 and 64: Volterra’s function, V(x), is wha
Page 65 and 66: We’ll call this set SVC (for Smit
Page 67 and 68: No matter how we partition [0,1], t
Page 69 and 70: Conclusion: Volterra’s function V
Page 71 and 72: The Lebesgue Integral Henri Lebesgu
Page 73 and 74: ( )= ( )! ( ) f x 2x sin 1 1 x cos

Wrestling with the Fundamental Theorem of Calculus

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