Newton-Leipniz. 2 Newton-Leibniz b a

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1 Newto-epz Newto-ez d. d A A A A 9. d A R A R R R 9.. F g g g --

2 g grge 9. d ξ d ω! d l l d A R ω ξ! d l d R 9. ω Newto-Cotes uss grge uss Roerg Newto-Cotes 9.. d A 9. grge 9.. grge 9.

3 A A d d. d [ ] T Newto-ez d Newto-ez d Newto-ez [] R η 9.6

4 R ξ d ξ η [ ] [ ] R η 9.. d η [ ] d d [ ] ξ ξ [ ] ξ ξ η [ ] ξ ξ d ξ [ ] 9.7 T E 9.7

5 T [ ] 9.8 E ξ T C -/; T T [ ] t.5*; []; or;<; { [][-]; } tt*; tt[]; [ ] [ ] 9..6 d 5

6 t [.5.5 ] d rctg rctg & Spso Spso 9.. Spso Spso Spso grge 9. T T T 6

7 7 6 d A d A 6 d A Spso [ ] 6 S d 9. grge Spso 9.. Spso Spso d ] [ 6 S ] [ 6 d Spso c c c c Spso Spso 9.. Spso ] [ Spso ] [ 6 d 5 88 ζ

8 8 Newto N N N N N! N ξ ξ grge N grge d N d S d d d N d d N ] [ d! ξ d! ξ

9 ζ d! ζ Spso Spso Spso [ ] / Spso d d [ ] S 9. Spso s E d S η 8 η [ ] Spso S / S S / s; 9

10 or;<; ss***; or;<; ss**-*; ss*/; 9..6 Spso Spso I d S / Spso.5 S Spso S Spso S / ε 88 5 η η ε

5.78 9-5 Spso.5.5.75 S.5.98.8.6.5.7859 9-6 Spso 6 5 6 S /6.

11 9. Newto--Cotes Spso Newto--Cotes grge Newto--Cotes 9.. Newto--Cotes / t 9. A t t t t t dt!!!! t t t t t dt c t t t t t dt!! K 9. A c 9.5 c d c K 9.6 Cotes 9. Cotes Cotes 9. c t dt c tdt d [ ] Spso

12 c t t dt c 6 t t dt d & [ ] 6 d [ ] Cotes Cotes c c t t dt Newto-Cotes Newto-Cotes k k 9.8 /

577 78 5888 85 6 8 5 5 9 8 8 6 5 5 5 989 78 78 98 96 85 85 7 9 5 96 9 8 989

13 c d c d c c d d c d d d d d d c d Newto-Cotes [] Newto-Cotes

14 ξ ω! d ξ R[ ] ω! d ω Newto-Cotes Newto-Cotes Newto-Cotes c Newto-Cotes Cotes d c c 9.6 % ε % c c c ε ε ε c > ε c c ε ε 9. - c 9. Cotes >7 c 7 Newto-Cotes

15 9.5 Roerg Roerg Roerg 9.5. [ ] 9. T.. T T T T T T T T T 9. k k... k k k k / T k T T k k 9. k k k k / k T k T k > ε k / 5

16 ε t t [ ] ; ; -; t.5*; /* t */ s*ep; t k s T k k s T T k T k k k / Wles>ep {.5*; k k > ε [ ] k k } t; or;<; tt*-*; t.5*tt*; sst-t; tt; ;

17 I e d k T k / T / / k / / T T T / k /8 /8 5/8 7/8 T / k T / T / T T.6 < ε I Roerg 9. { T k } Roerg T T 7

6879.556.686.677.69 9-6 k 5 7 9 5 T / T / 6 6 6 6 6 6 6 6 6.99.

18 I T I T I-T T T T T T T T T ~ T T T T T / T -/ T T 9.5 Spso T [ [ T ] ] T [ ] [ T T [ 6 ] ] ] Spso T S T 9.5 { T k } 9.5 Spso { S k }. Spso { { C k } S k } 9.5 Cotes 6 C S S 5 5 S S 9.6 8

19 Cotes Roerg { R } k C R C 9.7 Roerg Roerg {T k } {T k } Spso Roerg T T T [ k T T T Roerg 9- T T T k ] / T T T [ k ] T T T T T T T T T 9- Roerg 9.8 Roerg T Roerg T - 9

20 T. T [ ] T T [ k k t[][].5**; or;<; { /; } T t[][]t[-][]/; ork;k<;k T T t[][] t[][]* *k-*}; T * ; or;<; { or; <; } *; ] t[][]* t[][-]- t[-][-]/-; Roerg Roerg I e d ε < 7. T Roerg

21 Roerg 9.6 uss d A 9-9 grge d A A A 9- A A A A A A A A A A 6 6 d

22 A A.... A A uss 9.6. uss d g uss. uss g. uss uss uss 9. uss [ ] ρ ρ d ρ [ ] g ρ d... ρ d ρ g g g d > { g } [ ] ρ g [ ] ρ egedre

23 P d [ ] P! d [-] ρ Cesev T cos rccos [-] ρ uss q p p.. p q c p c ρ [ ] { p } p { p } p ρ [ ] [ ] ρ 9.6. uss uss uss uss [ ] grge d l d l d g 9. uss 9. g uss uss uss-egedre egedre

24 [ ] t - uss d d g 9. uss d - d egedre grge d d [ ] d... g g ω ω g egedre d ω g d g uss uss egedre uss uss ~7 uss egedre uss uss uss uss g q ± ± ± ± ± ± ± ± ± ± ± ± ±

25 ± ± ± > g 9.6. uss g s s s g or ;<; ssg[]*-/*[]/; ss*-/; uss uss-egedre I d s g g 9-8 uss g g s uss g g g s

26 9.6.6 uss P P P ' ' ξ P... ξ! uss uss R ξ! uss Newto-Cotes uss uss 5 uss uss l l < uss l l d g l l σ < l d g l g uss d ~ ~ ε s g 6

27 7 g s ~ ~ g s s ~ ~ g ε d g g ε uss uss uss Newto 9.7 grge Newto-Cotes uss Newto-Cotes uss grge 9.7. grge ] [ ] [ ] [ ] [ d c d c z l l 9. grge q p q p q p q p l l q p q p q q p p l l grge

28 9.7. Newto-Cotes Newto-Cotes grge Newto- Cotes grge I dd 9.5 dd l 9.6 c l dd l d l c c 9.6 c d c Newto-Cotes Newto-Cotes Cotes c Cotes c c c 9.7. Newto-Cotes Newto-Cotes c 8

29 k k Newto-Cotes gerge Newto-Cotes P q p - /p l r p q I llc r s l / p l r l / q l r l r l s l s rs r s p q C #dee p #dee #dee {t rsq[p]; lot c[][]l[p]r[p]ll; 9 9.8

30 lot sors; sc %% & & ; or;<p; sc %% &q[]&l[]&r[]; l /p; so; or;<p; or;<q[]; {lr[]-l[]/q[]; orro;r<;r {r-*lr*l/l[]; ors;s<;s {s-*ls*l/ sosoc[r][s]*rs; } } soso*l*l; } prt % so; } uss uss grge ; uss Newto-Cotes uss S uss uss 9.9 d d dd

31 [ p 9 q ] p q g p q g [ p 9 q ] p q g egedre egedre dd uss 9. uss Newto-Cotes uss S p q ll u v C uv l / p l / q p l u l / q u l v l / u Newto-Cotes uss I u v 9. dd l.895 Q 9- Newto-Cotes Q Q Q

32 9- uss Q Q Newto-Cotes uss I d d π.677 Q 9- Newto-Cotes Q Q Q Q Q uss Q Q Q Q Q uss uss l l l 9-9-

33 O O O 9.8. Roerg Roerg ; 9-6

34 / g[][]--/* or;<; {.5*; g[][]--/*; } / - or;<; {or;<-; g[][]*g[][-]-g[-][-]/-; * }

35 9.8. e Z * 9- * Z 9.7 *

36 6 6 8 / / / / * * * z 9.9 z * O * * 6 z 9.5 z * O ] [..5 * /.6 /.685 /.9 / * * * z

37 z * *

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