Wrestling with the Fundamental Theorem of Calculus

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

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

Any piecewise continuous function is integrable: If ( )" ( )= lim f x lim f x J, " + x! c x! c then take an interval of length where the variation is less than $ V ( x " x )< ( J + 1) 2 1 $ 2( J + 1) + 1: # # i i i" 1 i i i" 1 J + other intervals $ $ < + = $ . 2 2 < J ( ) ( ) + V x " x
Bernhard Riemann (1852, 1867) On the representation of a function as a trigonometric series Riemann gave an example of a function that has a jump discontinuity in every subinterval of [0,1], but which can be integrated over the interval [0,1].
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: Any continuous function is integrab
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 and 48: Base three numbers: 10211.2012 1/3
Page 49 and 50: Cantor’s set eliminates all terci
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
Page 75 and 76:
( )= ( )! ( ) f x 2x sin 1 1 x cos
Page 77 and 78:
( )= ( )! ( ) f x 2x sin 1 1 x cos
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Wrestling with the Fundamental Theorem of Calculus

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