lesson 12 differentiation and integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew...
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![Page 1: Lesson 12 Differentiation and Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/1.jpg)
Lesson 12Differentiation and Integration
1
![Page 2: Lesson 12 Differentiation and Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/2.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/3.jpg)
3
Differentiation of Fourier series
![Page 4: Lesson 12 Differentiation and Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/4.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/5.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/6.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/7.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/8.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/9.jpg)
8
Integration of Fourier series
![Page 10: Lesson 12 Differentiation and Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/10.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/11.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/12.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/13.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/14.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/15.jpg)
12
Differentiation of Taylor series
![Page 16: Lesson 12 Differentiation and Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/16.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/17.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/18.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/19.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/20.jpg)
15
Integration of Taylor series
![Page 21: Lesson 12 Differentiation and Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/21.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/22.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/23.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/24.jpg)
19
Differentiation of Laurent series
![Page 25: Lesson 12 Differentiation and Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/25.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/26.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/27.jpg)
22
Integration of Taylor series
![Page 28: Lesson 12 Differentiation and Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/28.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/29.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/30.jpg)
25
Integration of Chebyshev series
![Page 31: Lesson 12 Differentiation and Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/31.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/32.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/33.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/34.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/35.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/36.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/37.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/38.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/39.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/40.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/41.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/42.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/43.jpg)
29
Differentiation of Chebyshev series
![Page 44: Lesson 12 Differentiation and Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/44.jpg)
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� 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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/45.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/46.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/47.jpg)
31
�CT �
0(x) | · · · | CT �10(x)
�=
![Page 48: Lesson 12 Differentiation and Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/48.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/49.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/50.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/51.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/52.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/53.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/54.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/55.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/56.jpg)
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 Integration · 9 %jyrgxmsr hi½ri hsr xli tivmshmg mrxivzep lew xlimrhi½rmxi mrxikvep f / = k=,k=0 fˆ k bk 2bk + fˆ 0 + c 8lmw [mppgsrzivki[liriziv](https://reader033.vdocuments.net/reader033/viewer/2022060809/608e24e4efdc570df202ae45/html5/thumbnails/57.jpg)
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