Wrestling with the Fundamental Theorem of Calculus

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

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

If lim f' ( x) and lim f' ( x) exist, x! c x! c then they must be equal and they must equal " + f' () c . Mean Value Theorem: ( )! () f x f c f'()= c lim ! x" c x! c = lim ! x" c f' ( k) , x < k < c f ( x)! f () c f'()= c lim = lim f' ( k) , + + x" c x! c x" c c < k < x The derivative of a function cannot have any jump discontinuities!
Riemann’s function: f( x)= ! n= 1 { } nx n " 2 At x = a 2b , gcd ( a, 2b)= 1, the function jumps by ! 2 2 8b The key to the integrability is that given any positive number, no matter how small, there are only a finite number of places where is jump is larger than that number. Conclusion: F x f t dt x ( )= () ! 0 exists and is well - defined for all x, but F is not differentiable at any rational number with an even denominator.
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: If lim f' ( x) and lim f' ( x) exis
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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