57
Lesson 12 Differentiation and Integration 1

Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

  • Upload
    others

  • View
    1

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

Lesson 12Differentiation and Integration

1

Page 2: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

2

• We have seen two applications:

– signal smoothing

– root finding

• Today we look

– differentation

– integration

• These will form the basis for solving ODEs

Page 3: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

3

Differentiation of Fourier series

Page 4: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

4

� *SV�JYRGXMSRW�SR�XLI�TIVMSHMG�MRXIVZEP� [I�LEZI�XLI�*SYVMIV�VITVIWIRXEXMSR

f(�) =��

k=��fk

k�

� -XW HIVMZEXMZI MW��JSVQEPP] �SFZMSYW�

f �(�) ���

k=��kfk

k�

� ;LIR�HSIW�XLMW�GSRZIVKI# ;LIRIZIV fk HIGE]W�JEWX�IRSYKL

� 2YQIVMGEPP]� [I�SFXEMR�XLI�ETTVS\MQEXMSR�

f �(�) ��

��, . . . , ����

���

� � ��

�� Ff

[LIVI f = (f(�1), . . . , f(�m))� ERH F MW�XLI�(*8

Page 5: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

4

� *SV�JYRGXMSRW�SR�XLI�TIVMSHMG�MRXIVZEP� [I�LEZI�XLI�*SYVMIV�VITVIWIRXEXMSR

f(�) =��

k=��fk

k�

� -XW HIVMZEXMZI MW��JSVQEPP] �SFZMSYW�

f �(�) ���

k=��kfk

k�

� ;LIR�HSIW�XLMW�GSRZIVKI# ;LIRIZIV fk HIGE]W�JEWX�IRSYKL

� 2YQIVMGEPP]� [I�SFXEMR�XLI�ETTVS\MQEXMSR�

f �(�) ��

��, . . . , ����

���

� � ��

�� Ff

[LIVI f = (f(�1), . . . , f(�m))� ERH F MW�XLI�(*8

Page 6: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

5

Pointwise convergence of derivative of at zero

500 1000 1500 2000

10-12

10-9

10-6

0.001

1

Numerical derivativeDirect interpolation

number of pointsnumber of points

f(�) = ecos(10��1)

0 500 1000 1500 2000

10-13

10-10

10-7

10-4

0.1

Page 7: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

6

� 8LI N XL�SVHIV�HIVMZEXMZI MW�

f (N)(�) ���

k=��( k)N fk

k�

� 2YQIVMGEPP]� [I�SFXEMR�XLI�ETTVS\MQEXMSR�

f (N)(�) ��

��, . . . , ����

���

� � ��

��

N

Ff

[LIVI f = (f(�1), . . . , f(�m))� ERH F MW�XLI�(*8

Page 8: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

7

Pointwise convergence of 10th derivative of at zero

Numerical 10th derivativeDirect interpolation

number of pointsnumber of points

f(�) = ecos(10��1)

500 1000 1500 200010-8

10-6

10-4

0.01

1

200 400 600 800 1000

10-14

10-11

10-8

10-5

0.01

10

Page 9: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

8

Integration of Fourier series

Page 10: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

9

� % JYRGXMSR�HI½RI�HSR�XLI�TIVMSHMG�MRXIVZEP�LEW�XLI MRHI½RMXI�MRXIKVEP

�f � =

��

k=��,k �=0

fk

kk� + f0� + C

� 8LMW�[MPP GSRZIVKI [LIRIZIV�XLI�*SYVMIV�WIVMIW�HSIW�

� 2YQIVMGEPP]� [I�SFXEMR�XLI�ETTVS\MQEXMSR�

�f � �

���, . . . , ��

�����������

1�

� � �1�

01

� � �1�

�����������

Ff +�e�0 Ff +C

[LIVI f = (f(�1), . . . , f(�m))� ERH F MW�XLI�(*8

� 8LMW�MW WXEFPI FIGEYWI�XLI�IVVSV�MR�IEGL�GSQTYXIH fk MW�QYPXMTPMIH�F]�E FSYRHIHRYQFIV

Page 11: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

9

� % JYRGXMSR�HI½RI�HSR�XLI�TIVMSHMG�MRXIVZEP�LEW�XLI MRHI½RMXI�MRXIKVEP

�f � =

��

k=��,k �=0

fk

kk� + f0� + C

� 8LMW�[MPP GSRZIVKI [LIRIZIV�XLI�*SYVMIV�WIVMIW�HSIW�

� 2YQIVMGEPP]� [I�SFXEMR�XLI�ETTVS\MQEXMSR�

�f � �

���, . . . , ��

�����������

1�

� � �1�

01

� � �1�

�����������

Ff +�e�0 Ff +C

[LIVI f = (f(�1), . . . , f(�m))� ERH F MW�XLI�(*8

� 8LMW�MW WXEFPI FIGEYWI�XLI�IVVSV�MR�IEGL�GSQTYXIH fk MW�QYPXMTPMIH�F]�E FSYRHIHRYQFIV

Page 12: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

9

� % JYRGXMSR�HI½RI�HSR�XLI�TIVMSHMG�MRXIVZEP�LEW�XLI MRHI½RMXI�MRXIKVEP

�f � =

��

k=��,k �=0

fk

kk� + f0� + C

� 8LMW�[MPP GSRZIVKI [LIRIZIV�XLI�*SYVMIV�WIVMIW�HSIW�

� 2YQIVMGEPP]� [I�SFXEMR�XLI�ETTVS\MQEXMSR�

�f � �

���, . . . , ��

�����������

1�

� � �1�

01

� � �1�

�����������

Ff +�e�0 Ff +C

[LIVI f = (f(�1), . . . , f(�m))� ERH F MW�XLI�(*8

� 8LMW�MW WXEFPI FIGEYWI�XLI�IVVSV�MR�IEGL�GSQTYXIH fk MW�QYPXMTPMIH�F]�E FSYRHIHRYQFIV

Page 13: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

10

Pointwise convergence of integral of at zero

number of points

f(�) = ecos(10��1)

200 400 600 800 1000

10-15

10-11

10-7

0.001

Page 14: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

11

� 0IXXMRK C = 0 EX�IEGL�WXEKI� [I�GER MXIVEXI XLMW N XMQIW�

�· · ·

�f �N �

���, . . . , ��

�����������

1�

� � �1�

01

� � �1�

�����������

N

Ff+f0

N !�N

� 8LI WXEFMPMX] SJ�XLMW�ETTVS\MQEXMSR�MW�QEMRXEMRIH

Page 15: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

12

Differentiation of Taylor series

Page 16: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

13

� *SV�JYRGXMSRW�MR�XLI�HMWO� [I�LEZI�XLI�8E]PSV�WIVMIW

f(z) =��

k=0

fkzk

� (IVMZEXMZI �

f �(z) =��

k=0

kfkzk�1

� 2YQIVMGEPP]� [I�SFXEMR�XLI (m � 1) � m QEXVM\�ETTVS\MQEXMSR�

f �(z) ��1 | · · · | zm�2

����

0 12

� � �m � 1

����T f

[LIVI f = (f(z1), . . . , f(zm))� ERH T MW�XLI�HMWGVIXI�8E]PSV�XVERWJSVQ

Page 17: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

13

� *SV�JYRGXMSRW�MR�XLI�HMWO� [I�LEZI�XLI�8E]PSV�WIVMIW

f(z) =��

k=0

fkzk

� (IVMZEXMZI �

f �(z) =��

k=0

kfkzk�1

� 2YQIVMGEPP]� [I�SFXEMR�XLI (m � 1) � m QEXVM\�ETTVS\MQEXMSR�

f �(z) ��1 | · · · | zm�2

����

0 12

� � �m � 1

����T f

[LIVI f = (f(z1), . . . , f(zm))� ERH T MW�XLI�HMWGVIXI�8E]PSV�XVERWJSVQ

Page 18: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

14

50 100 150 200

10-14

10-11

10-8

10-5

0.01

First derivative

Error approximating exp(z) for z = {.1,.5,1.}exp(.1i)

number of points

Page 19: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

14

50 100 150 200

10-14

10-11

10-8

10-5

0.01

First derivative

50 100 150 200

10-8

10-5

0.01

10

104

107

10th derivative

Error approximating exp(z) for z = {.1,.5,1.}exp(.1i)

number of points

Page 20: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

15

Integration of Taylor series

Page 21: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

16

� -RXIKVEP � �f(z) z =

��

k=0

fk

k + 1zk+1 + C

� 2YQIVMGEPP]� [I�SFXEMR�XLI (m + 1) � m QEXVM\�ETTVS\MQEXMSR�

�f(z) z �

�1 | · · · | zm

������

01

12

. . .1

m+1

������T f + C

[LIVI f = (f(z1), . . . , f(zm))� ERH T MW�XLI�HMWGVIXI�8E]PSV�XVERWJSVQ

� 2S[�XLMW�MW WXEFPI FSXL�SR�ERH�MRWMHI�XLI�YRMX�GMVGPI�

Page 22: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

17

� -RXIKVEP � �f(z) z =

��

k=0

fk

k + 1zk+1 + C

� 2YQIVMGEPP]� [I�SFXEMR�XLI (m + 1) � m QEXVM\�ETTVS\MQEXMSR�

�f(z) z �

�1 | · · · | zm

������

01

12

. . .1

m+1

������T f + C

[LIVI f = (f(z1), . . . , f(zm))� ERH T MW�XLI�HMWGVIXI�8E]PSV�XVERWJSVQ

� 2S[�XLMW�MW WXEFPI FSXL�SR�ERH�MRWMHI�XLI�YRMX�GMVGPI�

Page 23: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

18

First integral 10th integral

Error approximating exp(z) for z = {.1,.5,1.}exp(.1i)

number of points

50 100 150 200

10-14

10-11

10-8

10-5

0.01

50 100 150 200

10-16

10-14

10-12

10-10

10-8

Page 24: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

19

Differentiation of Laurent series

Page 25: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

20

� *SV�JYRGXMSRW�SR�XLI�GMVGPI� [I�LEZI�XLI�0EYVIRX�WIVMIW

f(z) =��

k=��fkzk

� (IVMZEXMZI �

f �(z) =��

k=��kfkzk�1 =

��

k=��,k �=�1

(k + 1)fk+1zk

� 2YQIVMGEPP]� [I�SFXEMR�XLI��WUYEVI �ETTVS\MQEXMSR� [LIVI�[I GLERKIH�SYV�FEWMW�

f �(z) ��z��1 | · · · | z��1

��

���

. . .�

�� Ff

[LIVI f = (f(z1), . . . , f(zm))�

� 'PIEVP]� MX�[MPP�SRP]�FI�WSQI[LEX�EGGYVEXI�SR�XLI�YRMX�GMVGPI� ERH�XLI IVVSV�[MPP�KVS[

Page 26: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

21

� *SV�JYRGXMSRW�SR�XLI�GMVGPI� [I�LEZI�XLI�0EYVIRX�WIVMIW

f(z) =��

k=��fkzk

� (IVMZEXMZI �

f �(z) =��

k=��kfkzk�1 =

��

k=��,k �=�1

(k + 1)fk+1zk

� 2YQIVMGEPP]� [I�SFXEMR�XLI��WUYEVI �ETTVS\MQEXMSR� [LIVI�[I GLERKIH�SYV�FEWMW�

f �(z) ��z��1 | · · · | z��1

��

���

. . .�

�� Ff

[LIVI f = (f(z1), . . . , f(zm))�

� 'PIEVP]� MX�[MPP�SRP]�FI�WSQI[LEX�EGGYVEXI�SR�XLI�YRMX�GMVGPI� ERH�XLI IVVSV�[MPP�KVS[

Page 27: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

22

Integration of Taylor series

Page 28: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

23

� *SV�JYRGXMSRW�SR�XLI�GMVGPI� [I�LEZI�XLI�0EYVIRX�WIVMIW

f(z) =��

k=��fkzk

� -RXIKVEP��

f(z) z =��

k=��,k �=�1

1

k + 1fkzk+1 + f�1 z + C

� 2YQIVMGEPP]� [I�SFXEMR�XLI�ETTVS\MQEXMSR�[LIVI�[I GLERKIH�FEWIW�

�f(z) z �

�z�+1 | · · · | z�+1

�����������

1�+1

. . .1

�10

1. . .

1�+1

�����������

Ff+e�1Ff z+C

[LIVI f = (f(z1), . . . , f(zm))�

� 8LMW�[MPP�FI WXEFPI

� ,S[IZIV� [I�GERRSX��IEWMP] �MXIVEXI�MRXIKVEPW�

Page 29: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

24

� *SV�JYRGXMSRW�SR�XLI�GMVGPI� [I�LEZI�XLI�0EYVIRX�WIVMIW

f(z) =��

k=��fkzk

� -RXIKVEP��

f(z) z =��

k=��,k �=�1

1

k + 1fkzk+1 + f�1 z + C

� 2YQIVMGEPP]� [I�SFXEMR�XLI�ETTVS\MQEXMSR�[LIVI�[I GLERKIH�FEWIW�

�f(z) z �

�z�+1 | · · · | z�+1

�����������

1�+1

. . .1

�10

1. . .

1�+1

�����������

Ff+e�1Ff z+C

[LIVI f = (f(z1), . . . , f(zm))�

� 8LMW�[MPP�FI WXEFPI

� ,S[IZIV� [I�GERRSX��IEWMP] �MXIVEXI�MRXIKVEPW�

Page 30: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

25

Integration of Chebyshev series

Page 31: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

26

� ;I�[ERX�XS�GSQTYXI

�f(x) x =

��

k=0

fk

�Tk(x) x

� 8LI�½VWX�X[S QSQIRXW EVI�

T0(x) x = x+C = T1(x)+C ERH�

T1(x) x =x2

2+C =

T2(x) � T0(x)

4+C

� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�

� x

aTk(x) x =

� J�1� (x)

J�1� (a)

Tk(J(z))J �(z) z =1

4

� J�1� (x)

J�1� (a)

�zk + z�k

� �1 � 1

z2

�z

=1

4

� J�1� (x)

J�1� (a)

�zk + z�k � zk�2 � z�k�2

�z

=1

4

�zk+1

k + 1+

z1�k

1 � k� zk�1

k � 1� z�k�1

�k � 1

�+ C �2S[ z = J�1

� (x)�

=1

2(k + 1)

zk+1 + z�k�1

2� 1

2(k � 1)

zk�1 + z1�k

2+ C

=Tk+1(x)

2(k + 1)� Tk�1(x)

2(k � 1)+ C

Page 32: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

26

� ;I�[ERX�XS�GSQTYXI

�f(x) x =

��

k=0

fk

�Tk(x) x

� 8LI�½VWX�X[S QSQIRXW EVI�

T0(x) x = x+C = T1(x)+C ERH�

T1(x) x =x2

2+C =

T2(x) � T0(x)

4+C

� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�

� x

aTk(x) x =

� J�1� (x)

J�1� (a)

Tk(J(z))J �(z) z =1

4

� J�1� (x)

J�1� (a)

�zk + z�k

� �1 � 1

z2

�z

=1

4

� J�1� (x)

J�1� (a)

�zk + z�k � zk�2 � z�k�2

�z

=1

4

�zk+1

k + 1+

z1�k

1 � k� zk�1

k � 1� z�k�1

�k � 1

�+ C �2S[ z = J�1

� (x)�

=1

2(k + 1)

zk+1 + z�k�1

2� 1

2(k � 1)

zk�1 + z1�k

2+ C

=Tk+1(x)

2(k + 1)� Tk�1(x)

2(k � 1)+ C

Page 33: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

26

� ;I�[ERX�XS�GSQTYXI

�f(x) x =

��

k=0

fk

�Tk(x) x

� 8LI�½VWX�X[S QSQIRXW EVI�

T0(x) x = x+C = T1(x)+C ERH�

T1(x) x =x2

2+C =

T2(x) � T0(x)

4+C

� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�

� x

aTk(x) x =

� J�1� (x)

J�1� (a)

Tk(J(z))J �(z) z =1

4

� J�1� (x)

J�1� (a)

�zk + z�k

� �1 � 1

z2

�z

=1

4

� J�1� (x)

J�1� (a)

�zk + z�k � zk�2 � z�k�2

�z

=1

4

�zk+1

k + 1+

z1�k

1 � k� zk�1

k � 1� z�k�1

�k � 1

�+ C �2S[ z = J�1

� (x)�

=1

2(k + 1)

zk+1 + z�k�1

2� 1

2(k � 1)

zk�1 + z1�k

2+ C

=Tk+1(x)

2(k + 1)� Tk�1(x)

2(k � 1)+ C

Page 34: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

26

� ;I�[ERX�XS�GSQTYXI

�f(x) x =

��

k=0

fk

�Tk(x) x

� 8LI�½VWX�X[S QSQIRXW EVI�

T0(x) x = x+C = T1(x)+C ERH�

T1(x) x =x2

2+C =

T2(x) � T0(x)

4+C

� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�

� x

aTk(x) x =

� J�1� (x)

J�1� (a)

Tk(J(z))J �(z) z =1

4

� J�1� (x)

J�1� (a)

�zk + z�k

� �1 � 1

z2

�z

=1

4

� J�1� (x)

J�1� (a)

�zk + z�k � zk�2 � z�k�2

�z

=1

4

�zk+1

k + 1+

z1�k

1 � k� zk�1

k � 1� z�k�1

�k � 1

�+ C �2S[ z = J�1

� (x)�

=1

2(k + 1)

zk+1 + z�k�1

2� 1

2(k � 1)

zk�1 + z1�k

2+ C

=Tk+1(x)

2(k + 1)� Tk�1(x)

2(k � 1)+ C

Page 35: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

26

� ;I�[ERX�XS�GSQTYXI

�f(x) x =

��

k=0

fk

�Tk(x) x

� 8LI�½VWX�X[S QSQIRXW EVI�

T0(x) x = x+C = T1(x)+C ERH�

T1(x) x =x2

2+C =

T2(x) � T0(x)

4+C

� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�

� x

aTk(x) x =

� J�1� (x)

J�1� (a)

Tk(J(z))J �(z) z =1

4

� J�1� (x)

J�1� (a)

�zk + z�k

� �1 � 1

z2

�z

=1

4

� J�1� (x)

J�1� (a)

�zk + z�k � zk�2 � z�k�2

�z

=1

4

�zk+1

k + 1+

z1�k

1 � k� zk�1

k � 1� z�k�1

�k � 1

�+ C �2S[ z = J�1

� (x)�

=1

2(k + 1)

zk+1 + z�k�1

2� 1

2(k � 1)

zk�1 + z1�k

2+ C

=Tk+1(x)

2(k + 1)� Tk�1(x)

2(k � 1)+ C

Page 36: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

26

� ;I�[ERX�XS�GSQTYXI

�f(x) x =

��

k=0

fk

�Tk(x) x

� 8LI�½VWX�X[S QSQIRXW EVI�

T0(x) x = x+C = T1(x)+C ERH�

T1(x) x =x2

2+C =

T2(x) � T0(x)

4+C

� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�

� x

aTk(x) x =

� J�1� (x)

J�1� (a)

Tk(J(z))J �(z) z =1

4

� J�1� (x)

J�1� (a)

�zk + z�k

� �1 � 1

z2

�z

=1

4

� J�1� (x)

J�1� (a)

�zk + z�k � zk�2 � z�k�2

�z

=1

4

�zk+1

k + 1+

z1�k

1 � k� zk�1

k � 1� z�k�1

�k � 1

�+ C �2S[ z = J�1

� (x)�

=1

2(k + 1)

zk+1 + z�k�1

2� 1

2(k � 1)

zk�1 + z1�k

2+ C

=Tk+1(x)

2(k + 1)� Tk�1(x)

2(k � 1)+ C

Page 37: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

26

� ;I�[ERX�XS�GSQTYXI

�f(x) x =

��

k=0

fk

�Tk(x) x

� 8LI�½VWX�X[S QSQIRXW EVI�

T0(x) x = x+C = T1(x)+C ERH�

T1(x) x =x2

2+C =

T2(x) � T0(x)

4+C

� *SV k > 1� [I�HS�XLI GLERKI�SJ�ZEVMEFPIW x = J(z) XS�QET�XS�XLI�YRMX�GMVGPI�

� x

aTk(x) x =

� J�1� (x)

J�1� (a)

Tk(J(z))J �(z) z =1

4

� J�1� (x)

J�1� (a)

�zk + z�k

� �1 � 1

z2

�z

=1

4

� J�1� (x)

J�1� (a)

�zk + z�k � zk�2 � z�k�2

�z

=1

4

�zk+1

k + 1+

z1�k

1 � k� zk�1

k � 1� z�k�1

�k � 1

�+ C �2S[ z = J�1

� (x)�

=1

2(k + 1)

zk+1 + z�k�1

2� 1

2(k � 1)

zk�1 + z1�k

2+ C

=Tk+1(x)

2(k + 1)� Tk�1(x)

2(k � 1)+ C

Page 38: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

27

� ;I�XLYW�LEZI�XLI�MRXIKVEXMSR�JSVQYPE

�f(x) x =

��

k=0

fk

�Tk(x) x

= C + f0T1(x) + f1T2(x) � T0(x)

4+

1

2

��

k=2

fk

�Tk+1(x)

k + 1� Tk�1(x)

k � 1

= C � f1

4+

�f0 � f2

2

�T1(x) +

1

2

��

k=2

�fk�1 � fk+1

k

�Tk(x)

� 2YQIVMGEPP]� [I�ETTVS\MQEXI

f(x) ��1 | · · · | Tn�1(x)

�Cf

ERH�LEZI�XLI (n + 1) � n QEXVM\�JSV�MRXIKVEXMSR�

�f(x) x �

�1 | · · · | Tn(x)

����������

� 14

1 � 12

14 � 1

4� � �

� � �1

2(n�2) � 12(n�2)

12(n�1)

12n

����������

Cf

[LIVI f = (f(x1), . . . , f(xn))� ERH C MW�XLI�HMWGVIXI�GSWMRI�XVERWJSVQ��('8

Page 39: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

27

� ;I�XLYW�LEZI�XLI�MRXIKVEXMSR�JSVQYPE

�f(x) x =

��

k=0

fk

�Tk(x) x

= C + f0T1(x) + f1T2(x) � T0(x)

4+

1

2

��

k=2

fk

�Tk+1(x)

k + 1� Tk�1(x)

k � 1

= C � f1

4+

�f0 � f2

2

�T1(x) +

1

2

��

k=2

�fk�1 � fk+1

k

�Tk(x)

� 2YQIVMGEPP]� [I�ETTVS\MQEXI

f(x) ��1 | · · · | Tn�1(x)

�Cf

ERH�LEZI�XLI (n + 1) � n QEXVM\�JSV�MRXIKVEXMSR�

�f(x) x �

�1 | · · · | Tn(x)

����������

� 14

1 � 12

14 � 1

4� � �

� � �1

2(n�2) � 12(n�2)

12(n�1)

12n

����������

Cf

[LIVI f = (f(x1), . . . , f(xn))� ERH C MW�XLI�HMWGVIXI�GSWMRI�XVERWJSVQ��('8

Page 40: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

27

� ;I�XLYW�LEZI�XLI�MRXIKVEXMSR�JSVQYPE

�f(x) x =

��

k=0

fk

�Tk(x) x

= C + f0T1(x) + f1T2(x) � T0(x)

4+

1

2

��

k=2

fk

�Tk+1(x)

k + 1� Tk�1(x)

k � 1

= C � f1

4+

�f0 � f2

2

�T1(x) +

1

2

��

k=2

�fk�1 � fk+1

k

�Tk(x)

� 2YQIVMGEPP]� [I�ETTVS\MQEXI

f(x) ��1 | · · · | Tn�1(x)

�Cf

ERH�LEZI�XLI (n + 1) � n QEXVM\�JSV�MRXIKVEXMSR�

�f(x) x �

�1 | · · · | Tn(x)

����������

� 14

1 � 12

14 � 1

4� � �

� � �1

2(n�2) � 12(n�2)

12(n�1)

12n

����������

Cf

[LIVI f = (f(x1), . . . , f(xn))� ERH C MW�XLI�HMWGVIXI�GSWMRI�XVERWJSVQ��('8

Page 41: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

27

� ;I�XLYW�LEZI�XLI�MRXIKVEXMSR�JSVQYPE

�f(x) x =

��

k=0

fk

�Tk(x) x

= C + f0T1(x) + f1T2(x) � T0(x)

4+

1

2

��

k=2

fk

�Tk+1(x)

k + 1� Tk�1(x)

k � 1

= C � f1

4+

�f0 � f2

2

�T1(x) +

1

2

��

k=2

�fk�1 � fk+1

k

�Tk(x)

� 2YQIVMGEPP]� [I�ETTVS\MQEXI

f(x) ��1 | · · · | Tn�1(x)

�Cf

ERH�LEZI�XLI (n + 1) � n QEXVM\�JSV�MRXIKVEXMSR�

�f(x) x �

�1 | · · · | Tn(x)

����������

� 14

1 � 12

14 � 1

4� � �

� � �1

2(n�2) � 12(n�2)

12(n�1)

12n

����������

Cf

[LIVI f = (f(x1), . . . , f(xn))� ERH C MW�XLI�HMWGVIXI�GSWMRI�XVERWJSVQ��('8

Page 42: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

28

First integral 10th integral

Error approximating exp(x) for x = .1

number of points

10 20 30 40 50

10-13

10-10

10-7

10-4

0.1

10 20 30 40 50

10-14

10-12

10-10

10-8

Page 43: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

29

Differentiation of Chebyshev series

Page 44: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

30

� 0IX�W�XV]�HIGSQTSWMRK�XLI�XIVQW�

f �(x) =��

k=0

fkT �k(x)

� ;I�[ERX�XS�VI[VMXI T �k(x) MR�XIVQW�SJ T0(x), . . . , Tk�1(x)

� 0IX�W�XV]�ER�I\TIVMQIRX�

9WI Tk(x) = k x� WS T �k(x) = k k x�

1�x2

8LIVIJSVI� [I�GER RYQIVMGEPP] IZEPYEXI

CTk(x) = C

��T �

k(x1)���

T �k(xn)

��

XS�KIX�XLI�GSIJ½GMIRXW

Page 45: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

30

� 0IX�W�XV]�HIGSQTSWMRK�XLI�XIVQW�

f �(x) =��

k=0

fkT �k(x)

� ;I�[ERX�XS�VI[VMXI T �k(x) MR�XIVQW�SJ T0(x), . . . , Tk�1(x)

� 0IX�W�XV]�ER�I\TIVMQIRX�

9WI Tk(x) = k x� WS T �k(x) = k k x�

1�x2

8LIVIJSVI� [I�GER RYQIVMGEPP] IZEPYEXI

CTk(x) = C

��T �

k(x1)���

T �k(xn)

��

XS�KIX�XLI�GSIJ½GMIRXW

Page 46: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

30

� 0IX�W�XV]�HIGSQTSWMRK�XLI�XIVQW�

f �(x) =��

k=0

fkT �k(x)

� ;I�[ERX�XS�VI[VMXI T �k(x) MR�XIVQW�SJ T0(x), . . . , Tk�1(x)

� 0IX�W�XV]�ER�I\TIVMQIRX�

9WI Tk(x) = k x� WS T �k(x) = k k x�

1�x2

8LIVIJSVI� [I�GER RYQIVMGEPP] IZEPYEXI

CTk(x) = C

��T �

k(x1)���

T �k(xn)

��

XS�KIX�XLI�GSIJ½GMIRXW

Page 47: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

31

�CT �

0(x) | · · · | CT �10(x)

�=

Page 48: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

31

0 1. 0 3. 0 5. 0 7. 0 9. 00 0 4. 0 8. 0 12. 0 16. 0 20.0 0 0 6. 0 10. 0 14. 0 18. 00 0 0 0 8. 0 12. 0 16. 0 20.0 0 0 0 0 10. 0 14. 0 18. 00 0 0 0 0 0 12. 0 16. 0 20.0 0 0 0 0 0 0 14. 0 18. 00 0 0 0 0 0 0 0 16. 0 20.0 0 0 0 0 0 0 0 0 18. 00 0 0 0 0 0 0 0 0 0 20.0 0 0 0 0 0 0 0 0 0 0

�CT �

0(x) | · · · | CT �10(x)

�=

Page 49: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

31

0 1. 0 3. 0 5. 0 7. 0 9. 00 0 4. 0 8. 0 12. 0 16. 0 20.0 0 0 6. 0 10. 0 14. 0 18. 00 0 0 0 8. 0 12. 0 16. 0 20.0 0 0 0 0 10. 0 14. 0 18. 00 0 0 0 0 0 12. 0 16. 0 20.0 0 0 0 0 0 0 14. 0 18. 00 0 0 0 0 0 0 0 16. 0 20.0 0 0 0 0 0 0 0 0 18. 00 0 0 0 0 0 0 0 0 0 20.0 0 0 0 0 0 0 0 0 0 0

Problem: the operation is dense!

�CT �

0(x) | · · · | CT �10(x)

�=

Page 50: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

32� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR

� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX

�����������

� 14

1 � 12

14 � 1

4� � �

� � �1

2(n�3) � 12(n�3)

12(n�2)

12(n�1)

�����������

u = Cf

� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�

un�2 = 2(n � 1)fn�1

un�3 = 2(n � 2)fn�2

un�4 = 2(n � 3)fn�3 + un�2

���

u0 = f1 +u2

2

� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#

8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]

Page 51: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

32

� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR

� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX

�����������

� 14

1 � 12

14 � 1

4� � �

� � �1

2(n�3) � 12(n�3)

12(n�2)

12(n�1)

�����������

u = Cf

� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�

un�2 = 2(n � 1)fn�1

un�3 = 2(n � 2)fn�2

un�4 = 2(n � 3)fn�3 + un�2

���

u0 = f1 +u2

2

� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#

8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]

Page 52: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

32

� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR

� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX

�����������

� 14

1 � 12

14 � 1

4� � �

� � �1

2(n�3) � 12(n�3)

12(n�2)

12(n�1)

�����������

u = Cf

� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�

un�2 = 2(n � 1)fn�1

un�3 = 2(n � 2)fn�2

un�4 = 2(n � 3)fn�3 + un�2

���

u0 = f1 +u2

2

� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#

8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]

Page 53: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

32

� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR

� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX

�����������

� 14

1 � 12

14 � 1

4� � �

� � �1

2(n�3) � 12(n�3)

12(n�2)

12(n�1)

�����������

u = Cf

� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�

un�2 = 2(n � 1)fn�1

un�3 = 2(n � 2)fn�2

un�4 = 2(n � 3)fn�3 + un�2

���

u0 = f1 +u2

2

� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#

8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]

Page 54: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

32

� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR

� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX

�����������

� 14

1 � 12

14 � 1

4� � �

� � �1

2(n�3) � 12(n�3)

12(n�2)

12(n�1)

�����������

u = Cf

� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�

un�2 = 2(n � 1)fn�1

un�3 = 2(n � 2)fn�2

un�4 = 2(n � 3)fn�3 + un�2

���

u0 = f1 +u2

2

� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#

8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]

Page 55: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

32

� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR

� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX

�����������

� 14

1 � 12

14 � 1

4� � �

� � �1

2(n�3) � 12(n�3)

12(n�2)

12(n�1)

�����������

u = Cf

� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�

un�2 = 2(n � 1)fn�1

un�3 = 2(n � 2)fn�2

un�4 = 2(n � 3)fn�3 + un�2

���

u0 = f1 +u2

2

� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#

8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]

Page 56: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

32

� -RWXIEH� [I�[MPP�YWI�XLI�JEGX�XLEX HMJJIVIRXMEXMSR�MW�XLI�STTSWMXI�SJ�MRXIKVEXMSR

� ;I�[ERX�XS�½RH�XLI�ZIGXSV�SJ�GSIJ½GMIRXW u = (u0, . . . , un�2)� WS�XLEX

�����������

� 14

1 � 12

14 � 1

4� � �

� � �1

2(n�3) � 12(n�3)

12(n�2)

12(n�1)

�����������

u = Cf

� ;I�GER�ETTP] FEGO[EVH�WYFWXMXYXMSR�

un�2 = 2(n � 1)fn�1

un�3 = 2(n � 2)fn�2

un�4 = 2(n � 3)fn�3 + un�2

���

u0 = f1 +u2

2

� ;LEX�EFSYX�XLI�PEWX�GSRHMXMSR � u14 = f0#

8LMW�GSRHMXMSR�MW RSX�RIGIWWEV] FIGEYWI�XLI�GSRWXERX�SJ�MRXIKVEXMSR�MW�EVFMXVEV]

Page 57: Lesson 12 Differentiation and Integration9 %JYRGXMSR HI½RI HSR XLI TIVMSHMG MRXIVZEP LEW XLIMRHI½RMXI MRXIKVEP f / = k=,k=0 fˆ k Bk 2Bk + fˆ 0 + C 8LMW [MPPGSRZIVKI[LIRIZIV XLI

33

First derivative 10th derivative

Error approximating exp(x) for x = .1

number of points

10 20 30 40 50

10-13

10-10

10-7

10-4

0.1

10 20 30 40 50

10-5

10-4

0.001

0.01

0.1

1

10