devexpress asp.net pdf viewer : Copy pages from pdf to new pdf control Library platform web page asp.net windows web browser Calculus29-part760

11.11 Taylor’sTheorem
291
EXAMPLE11.11.2
Findapolynomialapproximationforsinxaccurateto0:005.
FromTaylor’stheorem:
sinx=
XN
n=0
f
(n)
(a)
n!
(x a)
n
+
f
(N+1)
(z)
(N+1)!
(x a)
N+1
:
Whatcanwesayaboutthesizeoftheterm
f
(N+1)
(z)
(N+1)!
(x a)
N+1
?
Everyderivativeofsinxissinxorcosx,sojf
(N+1)
(z)j1.Thefactor(x a)
N+1
is
abitmoredicult,sincex acouldbequitelarge. . Let’spicka=0andjxj=2;ifwe
cancomputesinxforx2[ =2;=2],wecanofcoursecomputesinxforallx.
WeneedtopickN sothat
x
N+1
(N+1)!
<0:005:
Sincewehavelimitedxto[ =2;=2],
x
N+1
(N+1)!
<
2
N+1
(N+1)!
:
ThequantityontherightdecreaseswithincreasingN,soallweneedtodoisndanN
sothat
2
N+1
(N+1)!
<0:005:
AlittletrialanderrorshowsthatN=8works,andinfact2
9
=9!<0:0015,so
sinx=
X8
n=0
f(n)(0)
n!
x
n
0:0015
=x 
x
3
6
+
x
5
120
x
7
5040
0:0015:
Figure 11.11.1showsthe graphsofsinx andand theapproximationon[0;3=2]. Asx
getslarger,theapproximationheadstonegativeinnityveryquickly,sinceitisessentially
actinglike x
7
.
Copy pages from pdf to new pdf - copy, paste, cut PDF pages in C#.net, ASP.NET, MVC, Ajax, WinForms, WPF
Easy to Use C# Code to Extract PDF Pages, Copy Pages from One PDF File and Paste into Others
deleting pages from pdf file; extract pdf pages for
Copy pages from pdf to new pdf - VB.NET PDF Page Extract Library: copy, paste, cut PDF pages in vb.net, ASP.NET, MVC, Ajax, WinForms, WPF
Detailed VB.NET Guide for Extracting Pages from Microsoft PDF Doc
delete pages of pdf online; deleting pages from pdf document
292
Chapter 11 1 SequencesandSeries
5
4
3
2
1
0
1
1
2
3
4
5
.
.....
....
.....
.....
.....
.....
....
.....
.....
.....
.....
......
.....
.....
......
......
......
......
.......
......
........
........
........
..........
............
...................
.........................................
..................
............
..........
........
........
.......
.......
......
.......
.....
......
.....
......
.....
.....
....
.....
.....
.....
....
.....
....
....
....
.....
....
....
....
....
....
....
....
...
....
....
....
...
....
....
...
....
...
....
...
...
....
...
...
...
....
...
...
...
...
...
...
...
...
..
...
...
...
...
..
...
...
..
...
..
...
..
...
..
...
..
..
...
..
..
...
..
..
..
...
..
..
..
..
..
...
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
..
.
..
..
..
..
..
..
..
.
..
..
..
..
.
.
.....
....
.....
.....
.....
.....
....
.....
.....
.....
.....
......
.....
.....
......
......
......
......
.......
......
........
........
........
..........
............
...................
.........................................
..................
............
..........
.........
........
.......
.......
.......
......
......
......
.....
......
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
....
.....
.....
....
.....
.....
.....
.....
....
.....
.....
.....
.....
.....
......
.....
.....
......
......
......
......
......
.......
.......
........
.........
..........
............
................
........................
Figure 11.11.1
sinxandapolynomialapproximation. (AP)
Wecanextractabitmoreinformationfromthisexample.Ifwedonotlimitthevalue
ofx,westillhave
f
(N+1)
(z)
(N+1)!
x
N+1
x
N+1
(N+1)!
sothatsinxisrepresentedby
XN
n=0
f(n)(0)
n!
x
n
xN+1
(N+1)!
:
Ifwecanshowthat
lim
N!1
x
N+1
(N+1)!
=0
foreachxthen
sinx=
X1
n=0
f
(n)
(0)
n!
x
n
=
X1
n=0
( 1)
n
x
2n+1
(2n+1)!
;
thatis,thesinefunctionisactuallyequaltoitsMaclaurinseriesforallx. Howcanwe
provethatthelimitiszero? SupposethatN islargerthanjxj,andletM M bethelargest
integerlessthanjxj(ifM=0thefollowingiseveneasier). Then
jx
N+1
j
(N+1)!
=
jxj
N+1
jxj
N
jxj
N 1

jxj
M+1
jxj
M
jxj
M 1

jxj
2
jxj
1
jxj
N+1
111
jxj
M
jxj
M 1

jxj
2
jxj
1
=
jxj
N+1
jxj
M
M!
:
ThequantityjxjM=M!isaconstant,so
lim
N!1
jxj
N+1
jxj
M
M!
=0
VB.NET PDF Page Insert Library: insert pages into PDF file in vb.
As String = Program.RootPath + "\\" Output.pdf" Dim doc1 Dim doc2 As PDFDocument = New PDFDocument(inputFilePath2 GetPage(2) Dim pages = New PDFPage() {page0
extract pages from pdf reader; delete pages of pdf preview
VB.NET PDF Page Delete Library: remove PDF pages in vb.net, ASP.
to delete a range of pages from a PDF document. Dim filepath As String = "" Dim outPutFilePath As String = "" Dim doc As PDFDocument = New PDFDocument(filepath
cutting pdf pages; delete pages from pdf
11.11 Taylor’sTheorem
293
andbytheSqueezeTheorem(11.1.3)
lim
N!1
xN+1
(N+1)!
=0
asdesired.Essentiallythesameargumentworksforcosxande
x
;unfortunately,itismore
diculttoshowthatmostfunctionsareequaltotheirMaclaurinseries.
EXAMPLE 11.11.3
Findapolynomialapproximationfore
x
nearx=2accurateto
0:005.
FromTaylor’stheorem:
e
x
=
XN
n=0
e
2
n!
(x 2)
n
+
e
z
(N+1)!
(x 2)
N+1
;
sincef
(n)
(x)=e
x
foralln.Weareinterestedinxnear2,andweneedtokeepj(x 2)
N+1
j
incheck,sowemayaswellspecifythatjx 2j1,sox2[1;3]. . Also
ez
(N+1)!
e3
(N+1)!
;
so we need d to nd an n N N that t makes e
3
=(N+1)!   0:005. This s time N N = = 5 5 makes
e
3
=(N+1)!<0:0015,sotheapproximatingpolynomialis
e
x
=e
2
+e
2
(x 2)+
e
2
2
(x 2)
2
+
e
2
6
(x 2)
3
+
e
2
24
(x 2)
4
+
e
2
120
(x 2)
5
0:0015:
Thispresentsanadditionalproblemforapproximation,sincewealsoneedtoapproximate
e
2
, and any approximation n we e use e will increase e the e error, , but t we e will l not pursue this
complication.
Notewellthatintheseexampleswefoundpolynomialsofacertainaccuracyonlyon
asmallinterval,eventhoughtheseriesforsinxande
x
convergeforallx;thisistypical.
Togetthesameaccuracyonalargerintervalwouldrequiremoreterms.
Exercises11.11.
1. Findapolynomialapproximationforcosxon[0;],accurateto10
3
)
2. Howmanytermsoftheseriesforlnxcenteredat1arerequiredsothattheguaranteederror
on[1=2;3=2]isatmost10
3
? Whatiftheintervalisinstead[1;3=2]? )
3. FindtherstthreenonzerotermsintheTaylorseriesfortanxon[ =4;=4],andcompute
theguaranteederror termasgivenby Taylor’s theorem. (Youmaywanttouse e Sage or a
similaraid.) )
C# PDF File & Page Process Library SDK for C#.net, ASP.NET, MVC
RasterEdge XDoc.PDF allows you to easily move PDF document pages position Copying and Pasting Pages. You can use specific APIs to copy and get a specific page of
extract page from pdf acrobat; cut pdf pages
C# PDF Page Rotate Library: rotate PDF page permanently in C#.net
Copy this demo code to your C# application to rotate the first page of One is used for rotating all PDF pages to 180 in clockwise and output a new PDF file
delete pages from pdf in reader; copying a pdf page into word
294
Chapter11 Sequences s andSeries
4. Showthatcosxis s equaltoitsTaylorseriesforallxbyshowingthatthelimitoftheerror
termiszeroasN approachesinnity.
5. Showthate
x
isequaltoitsTaylorseriesforallxbyshowingthatthelimitoftheerrorterm
iszeroasN approachesinnity.
Theseproblemsrequirethetechniquesofthischapter,andareinnoparticularorder.Some
problemsmaybedoneinmorethanoneway.
Determinewhethertheseriesconverges.
1.
X1
n=0
n
n+4
)
2.
1
12
+
1
34
+
1
56
+
1
78
+)
3.
X1
n=0
n
(n+4)2
)
4.
X1
n=0
n!
8n
)
5. 1 
3
4
+
5
8
7
12
+
9
16
+)
6.
X1
n=0
1
p
n+4
)
7.
X1
n=0
sin
3
(n)
n2
)
8.
X1
n=0
n
en
)
9.
X1
n=0
n!
135(2n 1)
)
10.
X1
n=1
1
n
p
n
)
11.
1
234
+
2
345
+
3
456
+
4
567
+)
12.
X1
n=1
135(2n 1)
(2n)!
)
13.
X1
n=0
6
n
n!
)
14.
X1
n=1
( 1)
n 1
p
n
)
C# PDF Page Insert Library: insert pages into PDF file in C#.net
Apart from the ability to inserting a new PDF page into existing PDF to delete PDF page using C# .NET, how to reorganize PDF document pages and how to
cut pages from pdf preview; delete pages out of a pdf
C# PDF Page Delete Library: remove PDF pages in C#.net, ASP.NET
How to delete a range of pages from a PDF document. String filepath = @""; String outPutFilePath = @""; PDFDocument doc = new PDFDocument(filepath); // Detele
pdf extract pages; acrobat extract pages from pdf
11.12 Additionalexercises
295
15.
X1
n=1
2
n
3
n 1
n!
)
16. 1+
5
2
22
+
5
4
(24)2
+
5
6
(246)2
+
5
8
(2468)2
+)
17.
X1
n=1
sin(1=n))
Findtheintervalandradiusofconvergence;youneednotchecktheendpointsoftheintervals.
18.
X1
n=0
2
n
n!
x
n
)
19.
X1
n=0
x
n
1+3n
)
20.
X1
n=1
x
n
n3n
)
21. x+
1
2
x
3
3
+
13
24
x
5
5
+
135
246
x
7
7
+)
22.
X1
n=1
n!
n2
x
n
)
23.
X1
n=1
( 1)
n
n23n
x
2n
)
24.
X1
n=0
(x 1)
n
n!
)
Findaseriesforeachfunction,usingtheformulaforMaclaurinseriesandalgebraicmanip-
ulationasappropriate.
25. 2
x
)
26. ln(1+x))
27. ln
1+x
1 x
)
28.
p
1+x)
29.
1
1+x2
)
30. arctan(x))
31. Usetheanswertothepreviousproblemtodiscoveraseriesforawell-knownmathematical
constant.)
C# PDF Image Extract Library: Select, copy, paste PDF images in C#
Open a document. PDFDocument doc = new PDFDocument(inputFilePath); PDFPage page = (PDFPage)pdf.GetPage(0); // Extract all images on one pdf page.
copy web page to pdf; extract page from pdf preview
C# PDF Annotate Library: Draw, edit PDF annotation, markups in C#.
RootPath + "\\" 2.pdf"; String outputFilePath = Program.RootPath + "\\" Annot_6.pdf"; // open a PDF file PDFDocument doc = new PDFDocument(inputFilePath
extract pages from pdf document; cut pages from pdf file
A
Selected Answers
1.1.1. (2=3)x+(1=3)
1.1.2. y= 2x
1.1.3. ( 2=3)x+(1=3)
1.1.4. y=2x+2,2, 1
1.1.5. y= x+6,6,6
1.1.6. y=x=2+1=2,1=2, 1
1.1.7. y = 3=2, y-intercept: 3=2, , no
x-intercept
1.1.8. y=( 2=3)x 2, 2, 3
1.1.9. yes
1.1.10. y=0,y= 2x+2,y=2x+2
1.1.11. y=75t(tinhours);164minutes
1.1.12. y=(9=5)x+32,( 40; 40)
1.1.13. y=0:15x+10
1.1.14. 0:03x+1:2
1.1.15. (a) y
=
(
0
0x<100
(x=10) 10 100x1000
x 910
1000<x
1.1.16. y
=
(
0:15x
0x19450
0:28x 2528:50 19450<x47050
0:33x 4881
47050<x97620
1.1.17. (a)P P = 0:0001x+2
(b)x= 10000P+20000
1.1.18. (2=25)x (16=5)
1.2.1. (a)x
2
+y
2
=9
(b)(x 5)2+(y 6)=9
(c)(x+5)
2
+(y+6)
2
=9
1.2.2. (a) ) x = = 2,y y =3,m m = = 3=2,
y=(3=2)x 3,
p
13
(b) x= 1,y=3,m= 3,
y= 3x+2,
p
10
(c) x = 2,y y = 2,m=1,
y=x,
p
8
1.2.6. (x+2=7)
2
+(y 41=7)
2
=1300=49
1.3.1. fxjx3=2g
1.3.2. fxjx6= 1g
1.3.3. fxjx6=1andx6= 1g
1.3.4. fxjx<0g
1.3.5. fxjx2Rg,i.e.,allx
297
298
AppendixA SelectedAnswers
1.3.6. fxjx0g
1.3.7. fxjh rxh+rg
1.3.8. fxjx1orx<0g
1.3.9. fxj 1=3<x<1=3g
1.3.10. fxjx0andx6=1g
1.3.11. fxjx0andx6=1g
1.3.12. R
1.3.13. fxjx3g,fxjx0g
1.3.14. A=x(500 2x),fxj0x250g
1.3.15. V = = r(50 r
2
), fr r j j 0 0 <r r 
p
50=g
1.3.16. A=2r
2
+2000=r,frj0<r<1g
2.1.1.  5,  2:47106145,  2:4067927,
2:400676, 2:4
2.1.2.  4=3, 24=7,7=24,3=4
2.1.3.  0:107526881,  0:11074197,
0:1110741,
1
3(3+x)
!
1
9
2.1.4.
3+3x+x
2
1+x
!3
2.1.5. 3:31,3:003001,3:0000,
3+3x+x
2
!3
2.1.6. m
2.2.1. 10,25=2,20,15,25,35.
2.2.2. 5,4:1,4:01,4:001,4+t!4
2.2.3.  10:29, 9:849, 9:8049,
9:8 4:9t! 9:8
2.3.1. 7
2.3.2. 5
2.3.3. 0
2.3.4. undened
2.3.5. 1=6
2.3.6. 0
2.3.7. 3
2.3.8. 172
2.3.9. 0
2.3.10. 2
2.3.11. doesnotexist
2.3.12.
p
2
2.3.13. 3a
2
2.3.14. 512
2.3.15.  4
2.3.16. 0
2.3.18. (a)8,(b)6,(c)dne,(d) 2,(e) 1,
(f)8,(g)7,(h)6,(i) 3,(j) 3=2,
(k)6,(l)2
2.4.1.  x=
p
169 x2
2.4.2.  9:8t
2.4.3. 2x+1=x
2
2.4.4. 2ax+b
2.4.5. 3x
2
2.4.8.  2=(2x+1)
3=2
2.4.9. 5=(t+2)
2
2.4.10. y= 13x+17
2.4.11.  8
3.1.1. 100x
99
3.1.2.  100x
101
3.1.3.  5x
6
3.1.4. x
 1
3.1.5. (3=4)x
1=4
3.1.6.  (9=7)x
16=7
3.2.1. 15x
2
+24x
3.2.2.  20x
4
+6x+10=x
3
3.2.3.  30x+25
3.2.4. 6x
2
+2x 8
AppendixA SelectedAnswers
299
3.2.5. 3x
2
+6x 1
3.2.6. 9x
2
x=
p
625 x2
3.2.7. y=13x=4+5
3.2.8. y=24x 48 
3
3.2.9.  49t=5+5, 49=5
3.2.11.
Xn
k=1
ka
k
x
k 1
3.2.12. x
3
=16 3x=4+4
3.3.1. 3x
2
(x
3
5x+10)+x
3
(3x
2
5)
3.3.2. (x
2
+5x 3)(5x
4
18x
2
+6x 7)+
(2x+5)(x
5
6x
3
+3x
2
7x+1)
3.3.3.
p
625 x2
2
p
x
x
p
x
p
625 x2
3.3.4.
1
x19
p
625 x2
20
p
625 x2
x21
3.3.5. f
0
=4(2x 3),y=4x 7
3.4.1.
3x
2
x3 5x+10
x
3
(3x
2
5)
(x3 5x+10)2
3.4.2.
2x+5
x5 6x3+3x2 7x+1
(x
2
+5x 3)(5x
4
18x
2
+6x 7)
(x5 6x3+3x2 7x+1)2
3.4.3.
1
2
p
x
p
625 x2
+
x
3=2
(625 x2)3=2
3.4.4.
1
x19
p
625 x2
20
p
625 x2
x21
3.4.5. y=17x=4 41=4
3.4.6. y=11x=16 15=16
3.4.8. y=19=169 5x=338
3.4.9. 13=18
3.5.1. 4x
3
9x
2
+x+7
3.5.2. 3x
2
4x+2=
p
x
3.5.3. 6(x
2
+1)
2
x
3.5.4.
p
169 x2 x
2
=
p
169 x2
3.5.5. (2x 4)
p
25 x2 
(x
2
4x+5)x=
p
25 x2
3.5.6.  x=
p
r2 x2
3.5.7. 2x
3
=
p
1+x4
3.5.8.
1
4
p
x(5 
p
x)3=2
3.5.9. 6+18x
3.5.10.
2x+1
1 x
+
x
2
+x+1
(1 x)2
3.5.11.  1=
p
25 x2 
p
25 x2=x
2
3.5.12.
1
2
169
x2
1
.
r
169
x
x
3.5.13.
3x
2
2x+1=x
2
2
p
x3 x2 (1=x)
3.5.14.
300x
(100 x2)5=2
3.5.15.
1+3x2
3(x+x3)2=3
3.5.16.
4x(x
2
+1)+
4x
3
+4x
2
p
1+(x2+1)2
!
.
2
q
(x2+1)2+
p
1+(x2+1)2
3.5.17. 5(x+8)
4
3.5.18.  3(4 x)
2
3.5.19. 6x(x
2
+5)
2
3.5.20.  12x(6 2x
2
)
2
3.5.21. 24x
2
(1 4x
3
)
3
3.5.22. 5+5=x
2
3.5.23.  8(4x 1)(2x
2
x+3)
3
3.5.24. 1=(x+1)
2
3.5.25. 3(8x 2)=(4x
2
2x+1)
2
3.5.26.  3x
2
+5x 1
3.5.27. 6x(2x 4)
3
+6(3x
2
+1)(2x 4)
2
300
AppendixA SelectedAnswers
3.5.28.  2=(x 1)
2
3.5.29. 4x=(x
2
+1)
2
3.5.30. (x
2
6x+7)=(x 3)
2
3.5.31.  5=(3x 4)
2
3.5.32. 60x
4
+72x
3
+18x
2
+18x 6
3.5.33. (5 4x)=((2x+1)
2
(x 3)
2
)
3.5.34. 1=(2(2+3x)
2
)
3.5.35. 56x
6
+72x
5
+110x
4
+100x
3
+
60x
2
+28x+6
3.5.36. y=23x=96 29=96
3.5.37. y=3 2x=3
3.5.38. y=13x=2 23=2
3.5.39. y=2x 11
3.5.40. y=
20+2
p
5
5
p
4+
p
5
x+
3
p
5
5
p
4+
p
5
4.1.1. 2n =2,anyintegern
4.1.2. n=6,anyintegern
4.1.3. (
p
2+
p
6)=4
4.1.4.  (1+
p
3)=(1 
p
3)=2+
p
3
4.1.11. t==2
4.3.1. 5
4.3.2. 7=2
4.3.3. 3=4
4.3.4. 1
4.3.5.  
p
2=2
4.3.6. 7
4.3.7. 2
4.4.1. sin(
p
x)cos(
p
x)=
p
x
4.4.2.
sinx
2
p
x
+
p
xcosx
4.4.3.  
cosx
sin
2
x
4.4.4.
(2x+1)sinx (x2+x)cosx
sin
2
x
4.4.5.
sinxcosx
p
1 sin
2
x
4.5.1. cos
2
x sin
2
x
4.5.2.  sinxcos(cosx)
4.5.3.
tanx+xsec
2
x
2
p
xtanx
4.5.4.
sec
2
x(1+sinx) tanxcosx
(1+sinx)2
4.5.5.  csc
2
x
4.5.6.  cscxcotx
4.5.7. 3x
2
sin(23x
2
)+46x
4
cos(23x
2
)
4.5.8. 0
4.5.9.  6cos(cos(6x))sin(6x)
4.5.10. sin=(cos+1)
2
4.5.11. 5t
4
cos(6t) 6t
5
sin(6t)
4.5.12. 3t
2
(sin(3t)+tcos(3t))=cos(2t)+
2t
3
sin(3t)sin(2t)=cos
2
(2t)
4.5.13. n=2,anyintegern
4.5.14. =2+n,anyintegern
4.5.15.
p
3x=2+3=4 
p
3=6
4.5.16. 8
p
3x+4 8
p
3=3
4.5.17. 3
p
3x=2 
p
3=4
4.5.18. =6+2n,5=6+2n,anyinteger
n
4.7.1. 2ln(3)x3
x
2
4.7.2.
cosx sinx
ex
4.7.3. 2e
2x
4.7.4. e
x
cos(e
x
)
4.7.5. cos(x)e
sinx
4.7.6. x
sinx
cosxlnx+
sinx
x
Documents you may be interested
Documents you may be interested