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1 II ( : )

2 紀元前 3000 年 紀元前 300 年 17 世紀 18 世紀 19 世紀 積分 古代エジプト 古代ギリシャ積分法の起源 微分 フェルマー デカルト 微分積分学の黎明期 ニュートンライプニッツ コーシー 微分積分学の誕 厳密化と発展 リーマン

3 : : ( 287? 212 )

4 紀元前 3000 年 紀元前 300 年 17 世紀 18 世紀 19 世紀 積分 古代エジプト 古代ギリシャ積分法の起源 微分 フェルマー デカルト 微分積分学の黎明期 ニュートンライプニッツ コーシー 微分積分学の誕 厳密化と発展 リーマン

5 : : ( ) ( )

6 紀元前 3000 年 紀元前 300 年 17 世紀 18 世紀 19 世紀 積分 古代エジプト 古代ギリシャ積分法の起源 微分 フェルマー デカルト 微分積分学の黎明期 ニュートンライプニッツ コーシー 微分積分学の誕 厳密化と発展 リーマン

7 , : ( ) ( )

8 紀元前 3000 年 紀元前 300 年 17 世紀 18 世紀 19 世紀 積分 古代エジプト 古代ギリシャ積分法の起源 微分 フェルマー デカルト 微分積分学の黎明期 ニュートンライプニッツ コーシー 微分積分学の誕 厳密化と発展 リーマン

9 : ϵ-δ : ( ) ( )

10 (? 1708 ) 12

11 : ( - ) ( ) ( ) ( )

12 紀元前 3000 年 紀元前 300 年 17 世紀 18 世紀 19 世紀 積分 古代エジプト 古代ギリシャ積分法の起源 微分 フェルマー デカルト 微分積分学の黎明期 ニュートンライプニッツ コーシー 微分積分学の誕 厳密化と発展 リーマン

13 ( )

14 ( ) ( ) ナイル川 土地

15 ( )

16 9 9 9

17 ( 9 ) 5 ( 3 ) + 4 ( ) = 7 ( 3 ) = ( ) π πr 2 = π(9/2) 2 64 π =

18 ( 1650 ) 87 48

19 ( 1650 ) : ( ) ( )

20 9 1/ / Géométrie dans l Égypte antique antique

21 ( 287? 212 ) π = ( ) ( ) (= ) : ( =) < π < 31 (= ) 7 : 2 2 ( )

22 ( 287? 212 ) π = ( ) ( ) (= ) : ( =) < π < 31 (= ) 7 : 2 2 ( )

23 ( 287? 212 ) π = ( ) ( ) (= ) : ( =) < π < 31 (= ) 7 : 2 2 ( )

24 ( 287? 212 ) ( =) < π < 31 7 : (= ) l n : 1 n L n : 1 n l n < π < L n : 6 n l n L n

25 ( 287? 212 ) ( =) < π < 31 7 : (= ) l n : 1 n L n : 1 n l n < π < L n : 6 n l n L n

26 ( 287? 212 ) ( =) < π < 31 7 : (= ) l n : 1 n L n : 1 n l n < π < L n : 12 n l n L n

27 ( 287? 212 ) ( =) < π < 31 7 : (= ) l n : 1 n L n : 1 n l n < π < L n : 24 n l n L n

28 ( 287? 212 ) ( =) < π < 31 7 : (= ) l n : 1 n L n : 1 n l n < π < L n : 48 n l n L n

29 ( 287? 212 ) ( =) < π < 31 7 : (= ) l n : 1 n L n : 1 n l n < π < L n : 96 n l n L n

30

31 ( ) ナイル川 土地

32 f(x) x x = a, x = b

33 f(x) x x = a, x = b

34 f(x) x x = a, x = b

35 f(x) x x = a, x = b

36 f(x) x x = a, x = b

37 n [a, b] x 0 = a, x n = b a b n 1 x 1 < x 2 < < x n 1 [a, b] n : a = x 0 < x 1 < < x n 1 < x n = b = max 1 i n x i ( x i = x i x i 1 )

38 Riemann ( ) f(x) : [a, b] ( ) : a = x 0 < x 1 < < x n 1 < x n = b : [a, b]

39 Riemann ( ) f(x) : [a, b] ( ) : a = x 0 < x 1 < < x n 1 < x n = b : [a, b] 1 i n [x i 1, x i ] ξ i ( )

40 Riemann ( ) f(x) : [a, b] ( ) : a = x 0 < x 1 < < x n 1 < x n = b : [a, b] 1 i n [x i 1, x i ] ξ i ( ) {ξ i } n S(, {ξ i }) = f(ξ i ) x }{{} i i=1 ( x i = x i x i 1 ) Riemann ( ξ i ) Riemann f(x)

41 : a = x 0 < x 1 < < x 5 < x 6 = b [a, b] 6 S(, {ξ i }) = f(ξ i ) x i i=1 ( x i = x i x i 1 )

42 : a = x 0 < x 1 < < x 5 < x 6 = b [a, b] 6 S(, {ξ i }) = f(ξ i ) x i i=1 ( x i = x i x i 1 ) 12 S(, {ξ i}) = f(ξ i) x i i=1 ( x i = x i x i 1)

43 : a = x 0 < x 1 < < x 5 < x 6 = b [a, b] 6 S(, {ξ i }) = f(ξ i ) x i i=1 ( x i = x i x i 1 ) S(, {ξ i }) = 25 i=1 f(ξ i ) x i ( x i = x i x i 1)

44 : a = x 0 < x 1 < < x 5 < x 6 = b [a, b] 6 S(, {ξ i }) = f(ξ i ) x i i=1 ( x i = x i x i 1 ) S(, {ξ i }) = 49 i=1 f(ξ i ) x i ( x i = x i x i 1)

45 : a = x 0 < x 1 < < x 5 < x 6 = b [a, b] 6 S(, {ξ i }) = f(ξ i ) x i i=1 ( x i = x i x i 1 ) lim S(, {ξ i}) 0

46 f(x) : [a, b] ( ) [a, b] Riemann S(, {ξ i }) σ f(x) [a, b] (Riemann) b ( ) f(x) dx = σ = lim S(, {ξ i}) 0 a [a, b] f(x) 0 0

47 f(x) : [a, b] ( ) [a, b] Riemann S(, {ξ i }) σ f(x) [a, b] (Riemann) b ( ) f(x) dx = σ = lim S(, {ξ i}) 0 a [a, b] f(x) 1 2 (Riemann )

48 f(x) (Riemann) f(x) [a, b] ξ i [x i 1, x i ] Riemann S(, {ξ i }) f(x) = { x (x 0) 1 (x = 0) [ 1, 1]

49 Riemann f(x) = x (0 x 1) Riemann : 0 = x 0 < x 1 < < x n 1 < x n = 1 x 0 = 0, x 1 = 1 n, x 2 = 2 n,..., x n 1 = n 1 n, x n = 1 ξ i = x i [x i 1, x i ], x i = x i x i 1 = 1 n Riemann S(, {ξ i }) n n i S(, {ξ i }) = f(ξ i ) x i = n 1 n i=1 i=1 = 1 n(n + 1) 1 n2 2 2 (n ) Riemann f(x) = sin x ξ i

50 ( ) f(x) I F(x) f(x) ( ) a, b I b a f(x) = F(b) F(a) ξ i π 0 sin x dx = ( cos(π)) ( cos(0)) = 2

51 紀元前 3000 年 紀元前 300 年 17 世紀 18 世紀 19 世紀 積分 古代エジプト 古代ギリシャ積分法の起源 微分 フェルマー デカルト 微分積分学の黎明期 ニュートンライプニッツ コーシー 微分積分学の誕 厳密化と発展 リーマン

52 ( ) ( ) ϵ-δ ( ) ϵ-δ Wikimedia Commons GFDL

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