pdfsharp c# : Change link in pdf control SDK system azure wpf web page console TRENCH_REAL_ANALYSIS56-part277

552
AnswerstoSelectedExercises
2:2:16
(p.70)
No 2:2:21
(p.71)(a)
Œ1;1,Œ0;1/
(b)
S
1
nD1
.2n;.2nC1//,
.0;1/
(c)
S
1
nD1
.n;.nC1//,.1;1/[.1;1/[.1;1/
(d)
S
1
nD1
Œn;.nC
1
2
/,Œ0;1/
2:2:23
(p.71)(a)
.1;1/
(b)
.1;1/
(c)
x
0
¤.2kC
3
2
/;kDinteger
(d)
x ¤
1
2
(e)
x ¤1
(f)
x ¤.kC
1
2
/;k Dinteger
(g)
x ¤.kC
1
2
/;k D
integer
(h)
x¤0
(i)
x¤0
Section2.3 pp. 8488
2:3:4
(p. 85)(b)
p.c/Dq.c/andp
0
.c/Dq
0
C
.c/
2:3:5
(p.85)
f.k/.x/Dn.n1/.nk1/xnk1jxjif1kn1;f.n/.x/DnŠ
ifx>0; f.n/.x/DnŠifx<0;f.k/.x/D0ifk>nandx¤0;f.k/.0/doesnot
existifkn.
2:3:7
(p. 85)(a)
c0 D D acbs, sD bcCas
(b)
c.x/ D eaxcosbx, s.x/ D
eaxsinbx
2:3:15
(p. 86)(b)
f.x/D1ifx 0,f.x/D1ifx >0;thenf
0
.0C/D0,but
f
0
C
.0/doesnotexist.
(c)
continuousfromtheright
2:3:22
(p. 87)
Thereisnosuchfunction(Theorem2.3.9).
2:3:24
(p. 87)
Counterexample: Letx
0
D0,f.x/D jxj
3=2
sin.1=x/ifx ¤ ¤ 0,and
f.0/ D0.
2:3:27
(p. 88)
Counterexample:Letx
0
D0,f.x/Dx=jxjifx¤0,f.0/D0.
Section2.4 pp. 9698
2:4:2
(p.96)
1 2:4:3
(p.96)
1
2
2:4:4
(p.96)
1
2:4:5
(p.96)
.1/
n1
n
2:4:6
(p. 96)
1
2:4:7
(p.96)
0
2:4:8
(p.96)
1
2:4:9
(p.96)
0
2:4:10
(p.96)
0
2:4:11
(p. 96)
0 2:4:12
(p. 96)
1
2:4:13
(p.96)
0
2:4:14
(p.96)
1
2
2:4:15
(p. 96)
0 2:4:16
(p. 96)
0
2:4:17
(p.96)
1
2:4:18
(p.96)
1
2:4:19
(p. 96)
1 2:4:20
(p. 96)
e
2:4:21
(p.96)
1
2:4:24
(p.96)
1=e 2:4:22
(p. 96)
0
2:4:23
(p. 96)
1if˛0,0if˛>0
2:4:25
(p. 96)
e
2
2:4:26
(p.96)
1
2:4:27
(p.96)
0
2:4:28
(p.96)
0
2:4:29
(p. 96)
1if˛>0,1if˛0
2:4:30
(p.96)
1
2:4:31
(p. 97)
1 2:4:32
(p.97)
1=1202:4:33
(p.97)
1
2:4:34
(p. 97)
1
2:4:35
(p.97)
1if˛0,0if˛>0
2:4:36
(p.97)
1
2:4:37
(p. 97)
1 2:4:38
(p.97)
0 2:4:39
(p.97)
0
2:4:40
(p.97)
0 2:4:41
(p.97)(b)
Supposethatg
0
iscontinuousatx
0
andf.x/D
g.x/ifxx
0
,f.x/D1Cg.x/ifx>x
0
.
2:4:44
(p. 97)(a)
1
(b)
e
(c)
1 2:4:45
(p.98)
e
L
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AnswerstoSelectedExercises
553
Section2.5 pp. 107112
2:5:2
(p.107)
f
.nC1/
.x
0
/=.nC1/Š.2:5:4
(p.107)(b)
Counterexample:Letx
0
D0
andf.x/Dxjxj.
2:5:5
(p. 108)(b)
Letg.x/D1Cjxx
0
j,sof.x/D.xx
0
/.1Cjxx
0
j/.
2:5:6
(p. 108)(b)
Letg.x/D1Cjxx
0
j,sof.x/D.xx
0
/2.1Cjxx
0
j/.
2:5:10
(p. 109)(b)(i)
1,2,2,0
(ii)
0,,3=2,4C
3
=2
(iii)

2
=4,2,6C
2
=4,4
(iv)
2,5,16,65
2:5:11
(p. 109)(b)
0,1,0,5
2:5:12
(p. 110)(b)(i)
0,1,0,5
(ii)
1,0,6,24
(iii)
p
2,3
p
2,11
p
2,
57
p
2
(iv)
1,3,14,88
(a)
min
(b)
neither
(c)
min
(d)
max
(e)
min
(f)
neither
(g)
min
(h)
min
2:5:14
(p. 110)
f.x/De1=x
2
ifx¤0,f.0/D0(Exercise2:5:1
(p.107)
)
2:5:15
(p. 111)
Noneifb
2
4c<0;localminatx
1
D.bC
p
b24c/=2andlocal
maxatx
1
D.b
p
b24c/=2ifb24c>0;ifbD4cthenxDb=2isacritical
point,butnotalocalextremepoint.
2:5:16
(p. 111)(a)
1
6
20
3
(b)
1
83
(c)
2
512
p
2
(d)
1
4.63/4
2:5:20
(p. 112)(a)
M
3
h=3,whereM
3
Dsup
jxcjh
jf
.3/
.c/j
(b)
M
4
h
2
=12whereM
4
Dsup
jxcjh
jf
.4/
.c/j
2:5:21
(p. 112)
kDh=2
Section3.1 pp. 125128
3:1:8
(p. 126)(b)
monotonicfunctions
(c)
LetŒa;bDŒ0;1andP Df0;1g. . Let
f.0/Df.1/D
1
2
andf.x/Dxif0<x<1.Thens.P/D0andS.P/D1,butneither
isaRiemannsumoffoverP.
3:1:9
(p. 127)(a)
1
2
,
1
2
(b)
1
2
,1
3:1:10
(p. 127)
e
b
e
a
3:1:11
(p. 127)
1cosb
3:1:12
(p. 127)
sinb
3:1:14
(p. 127)
f.a/Œg
1
g.a/Cf.d/.g
2
g
1
/Cf.b/Œg.b/g
2
3:1:15
(p. 127)
f.a/Œg
1
g.a/Cf.b/Œg.b/g
p
C
P
p1
mD1
f.a
m
/.g
mC1
g
m
/
3:1:16
(p. 127)(a)
Ifg1andf isarbitrary,then
R
b
a
f.x/dg.x/D0.
Section3.3 pp. 149151
3:3:7
(p.150)(a)
uDcD
2
3
(b)
uDcD0
(c)
uD.e2/=.e1/;cD
p
u
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554
AnswerstoSelectedExercises
Section3.4 pp. 165171
3:4:4
(p. 166)
(a) (i)
p2
(ii)
p>0
(iii)
0
(b) (i)
p2
(ii)
p>0
(iii)
0
(c) (i)
none
(ii)
p>0
(iii)
1=p
(d) (i)
p0
(ii)
0<p<1
(iii)
1=.1p/
(e) (i)
none
(ii)
none
3:4:5
(p. 166)(a)
(b)
1
2
(c)
divergent
(d)
1
(e)
1
(f)
0
3:4:8
(p.166)(a)
divergent
(b)
convergent
(c)
divergent
(d)
convergent
(e)
convergent
(f)
divergent
3:4:9
(p. 166)(a)
p<2
(b)
p <1
(c)
p>1
(d)
1<p<2
(e)
none
(f)
none
(g)
p<1
3:4:11
(p. 167)(a)
pq<1
(b)
p;q<1
(c)
1<p<2q1
(d)
q>1,
pCq>1
(e)
pCq>1
(f)
qC1<p<3qC1
3:4:12
(p. 167)
deggdegf 2
3:4:18
(p. 168)
(a) (i)
p>1
(ii)
0<p1
(b) (i)
p>1
(ii)
p1
(c) (i)
p>1
(ii)
0p1
(d) (i)
p>0
(ii)
none
(e) (i)
1<p<4
(ii)
0<p1
(f) (i)
p>
1
2
(ii)
0<p
1
2
3:4:25
(p. 169)
(a) (i)
p>1
(ii)
2<p1
(b) (i)
p>1
(ii)
none
(c) (i)
p<1
(ii)
none
(d) (i)
none
(ii)
none
(e) (i)
p<1
(ii)
p>1
Section4.1 pp. 192195
4:1:3
(p.192) (a)
2
(b)
1
(c)
0 4:1:4
(p.192)(a)
1=2
(b)
1=2
(c)
1=2
(d)
1=2
4:1:11
(p. 192)(d)
p
A
4:1:14
(p. 193)(a)
1
(b)
1
(c)
1
(d)
1
(e)
0
4:1:22
(p.193)
Ifs
n
D1andt
n
D1=n,then.lim
n!1
s
n
/=.lim
n!1
t
n
/D1=0D1,
butlim
n!1
s
n
=t
n
D1.
4:1:24
(p. 193)(a)
1,0
(b)
1,1ifjrj>1;2,2ifrD1;0,0ifrD1;1,
1ifjrj<1
(c)
1,1ifr<1;0,0ifjrj<1;
1
2
,
1
2
ifrD1;1, 1ifr>1
(d)
1,1
(e)
jtj,jtj
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AnswerstoSelectedExercises
555
4:1:25
(p. 194)(a)
1,1
(b)
2,2
(c)
3,1
(c)
p
3=2,
p
3=2
4:1:34
(p. 194)(b)
Iffs
n
gDf1;0;1;0;:::g,thenlim
n!1
t
n
D
1
2
Section4.2 pp. 199200
4:2:2
(p. 199)(a)
lim
m!1
s
2m
D1,lim
m!1
s
2mC1
D1
(b)
lim
m!1
s
4m
D1,lim
m!1
s
4mC2
D1,lim
m!1
s
2mC1
D0
(c)
lim
m!1
s
2m
D0,lim
m!1
s
4mC1
D1,lim
m!1
s
4mC3
D1
(d)
lim
n!1
s
n
D0
(e)
lim
m!1
s
2m
D1,lim
m!1
s
2mC1
D0
(f)
lim
m!1
s
8m
Dlim
m!1
s
8mC2
D1,lim
m!1
s
8mC1
D
p
2,
lim
m!1
s
8mC3
Dlim
m!1
s
8mC7
D0,lim
m!1
s
8mC5
D
p
2,
lim
m!1
s
8mC4
Dlim
m!1
s
8mC6
D1
4:2:3
(p. 199)
f1;2;1;2;3;1;2;3;4;1;2;3;4;5;:::g
4:2:8
(p.200)
Letft
n
gbeanyconvergentsequenceandfs
n
gDft
1
;1;t
2
;2;:::;t
n
;n;:::g.
Section4.3 pp. 228234
4:3:4
(p. 229)(b)
No;consider
P
1=n
4:3:8
(p. 229)(a)
convergent
(b)
convergent
(c)
divergent
(d)
divergent
(e)
convergent
(f)
convergent
(g)
divergent
(h)
convergent
4:3:10
(p. 229)(a)
p>1
(b)
p>1
(c)
p>1
4:3:15
(p. 230)(a)
convergent
(b)
convergentif0<r<1,divergentifr1
(c)
divergent
(d)
convergent
(e)
divergent
(f)
convergent
4:3:17
(p. 231)(a)
convergent
(b)
convergent
(c)
convergent
(d)
convergent
4:3:18
(p. 231)(a)
divergent
(b)
convergentifandonlyif0<r<1orrD1and
p<1
(c)
convergent
(d)
convergent
(e)
convergent
4:3:19
(p. 231)(a)
divergent
(b)
convergent
(c)
convergent
(d)
convergentif
˛<ˇ1,divergentif˛ˇ1
4:3:20
(p. 231)(a)
divergent
(b)
convergent
(c)
convergent
(d)
convergent
4:3:21
(p. 231)(a)
P
.1/
n
(b)
P
.1/
n
=n,
P
.1/
n
n
C
1
nlogn
(c)
P
.1/
n
2
n
(d)
P
.1/
n
4:3:27
(p.232)(a)
conditionallyconvergent
(b)
conditionallyconvergent
(c)
abso-
lutelyconvergent
(d)
absolutelyconvergent
4:3:28
(p. 232)
Letkandsbethedegreesofthenumeratoranddenominator,respec-
tively. IfjrjD D 1,theseriesconvergesabsolutelyifandonlyifs s   kC2. Theseries
convergesconditionallyifsDkC1andrD1,anddivergesinallothercases,where
skC1andjrjD1.
4:3:30
(p. 232)(b)
P
.1/
n
=
p
n
4:3:41
(p. 233)(a)
0
(b)
2Aa
0
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556
AnswerstoSelectedExercises
Section4.4 pp. 253256
4:4:1
(p. 253)(a)
F.x/D0;jxj1
(b)
F.x/D0;jxj1
(c)
F.x/D0;1<x1
(d)
F.x/Dsinx;1<x<1
(e)
F.x/D1; 1<x1;F.x/D0;jxj>1
(f)
F.x/Dx;1<x<1
(g)
F.x/Dx2=2;1<x<1
(h)
F.x/D0;1<x<1
(i)
F.x/D1;1<x<1
4:4:5
(p. 254)(a)
F.x/D0
(b)
F.x/D1;jxj<1;F.x/D0;jxj>1
(c)
F.x/Dsinx=x
4:4:6
(p. 254)(c)
F
n
.x/Dxn;S
k
DŒk=.kC1/;k=.kC1/
4:4:7
(p. 254)(a)
Œ1;1
(b)
Œr;r[f1g[f1g;0<r<1
(c)
Œr;r[f1g;0<
r<1
(d)
Œr;r;r>0
(e)
.1;1=r[Œr;r[Œ1=r;1/[f1g;0<r<1
(f)
Œr;r;r>0
(g)
Œr;r;r>0
(h)
.1;r[Œr;1/[f0g;r>0
(i)
Œr;r;r>0
4:4:12
(p. 254)(b)
LetS D.0;1,F
n
.x/Dsin.x=n/,G
n
.x/D1=x
2
;thenF D0,
GD1=x
2
,andtheconvergenceisuniform,butkF
n
G
n
k
S
D1.
4:4:14
(p. 255)(a)
3
(b)
1
(c)
1
2
(d)
e1
4:4:17
(p. 255)(a)
compact subsetsof.
1
2
;1/
(b)
Œ
1
2
;1/
(c)
closedsub-
setsof
1
p
5
2
;
1C
p
5
2
!
(d)
.1;1/
(e)
Œr;1/; r > > 1
(f)
compactsubsets
of.1;0/[.0;1/
4:4:19
(p. 255)(a)
LetS D.1;1/,f
n
D a
n
(constant),where
P
a
n
converges
conditionally,andg
n
Dja
n
j.
(b)
“absolutely"
4:4:20
(p. 255)(a)(i)
meansthat
P
jf
n
.x/jconvergespointwiseand
P
f
n
.x/con-
vergesuniformlyonS,while
(ii)
meansthat
P
jf
n
.x/jconvergesuniformlyonS.
4:4:27
(p. 256)(a)
X1
nD0
.1/
n
x
2nC1
nŠ.2nC1/
(b)
X1
nD0
.1/
n
x
2nC1
.2nC1/.2nC1/Š
Section4.5 pp. 275280
4:5:2
(p. 276)(a)
1=3e
(b)
1
(c)
1
3
(d)
1
(e)
1
4:5:8
(p. 276)(a)
1
(b)
1
2
(c)
1
4
(d)
4
(e)
1=e
(f)
1
4:5:10
(p. 277)
x.1Cx/=.1x/
3
4:5:12
(p.277)
e
x
2
4:5:16
(p. 277)
X1
nD1
.1/
n1
n2
.x1/
n
IRD1
4:5:17
(p.277)
Tan
1
xD
X1
nD0
.1/
n
x2nC1
.2nC1/
If
.2n/
.0/D0If
.2nC1/
.0/D.1/
2
.2n/Š;
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AnswerstoSelectedExercises
557
6
DTan
1
1
p
3
D
X1
nD0
.1/
n
.2nC1/3nC1=2
4:5:22
(p. 278)
coshxD
X1
nD0
x
2n
.2n/Š
,sinhxD
X1
nD0
x
2nC1
.2nC1/Š
4:5:23
(p. 278)
.1x/
P
1
nD0
x
n
D1convergesforallx
4:5:24
(p. 278)(a)
xCx
2
C
x
3
3
3x
5
40
C
(b)
1x
x
2
2
C
5x
3
6
C
(c)
1
x
2
2
C
x
4
24
721x
6
720
C
(d)
x
2
x
3
2
C
x
4
6
x
5
6
C
4:5:27
(p.279)(a)
1CxC
2x
2
3
C
x
3
3
C
(b)
1x
x
2
2
C
3x
3
2
C
(c)
1C
x
2
2
C
5x
4
24
C
61x
6
720
C
(d)
1C
x
2
6
C
7x
4
360
C
31x
6
15120
C
(e)
2x
2
C
x
4
12
x
6
360
C
4:5:28
(p.279)
F.x/D
5
.13x/.1C2x/
D
3
13x
C
2
1C2x
D
1
X
nD0
Œ3
nC1
.2/
nC1
x
n
4:5:29
(p. 279)
1
Section5.1 pp. 299302
5:1:1
(p. 299)(a)
.3;0;3;3/
(b)
.1;1;4/
(c)
.
1
6
;
11
12
;
23
24
;
5
36
/
5:1:3
(p. 299)(a)
p
15
(b)
p
65=12
(c)
p
31
(d)
p
3
5:1:4
(p. 299)(a)
p
89
(b)
p
166=12
(c)
3
(d)
p
31
5:1:5
(p. 299)(a)
12
(b)
1
32
(c)
27
5:1:7
(p. 299)
XDX
0
CtU.1<t<1/inallcases.
5:1:8
(p. 299)
:::UandX
1
X
0
arescalarmultiplesofV.
5:1:9
(p. 299)(a)
XD.1;3;4;2/Ct.1;3;5;3/
(b)
XD.3;1;2;1;4;/Ct.1;1;1;3;7/
(c)
XD.1;2;1/Ct.1;3;0/
5:1:10
(p. 300)(a)
5
(b)
2
(c)
1=2
p
5
5:1:11
(p.300)(a)(i)
˚
.x
1
;x
2
;x
3
;x
4
/
ˇ
ˇ
jx
i
j3.iD1;2;3/withatleastoneequality
(ii)
˚
.x
1
;x
2
;x
3
;x
4
/
ˇ
ˇ
jx
i
j3.i D1;2;3/
(iii)
S
(iv)
˚
.x
1
;x
2
;x
3
;x
4
/
ˇ
ˇ
jx
i
j>3foratleastoneofiD1;2;3
(b)(i)
S
(ii)
S
(iii)
;
(iv)
˚
.x;y;´/
ˇ
ˇ
´¤1orx2Cy>1
5:1:12
(p. 300)(a)
open
(b)
neither
(c)
closed
5:1:18
(p. 300)(a)
.;1;0/
(b)
.1;0;e/
5:1:19
(p. 300)(a)
6
(b)
6
(c)
2
p
5
(d)
2L
p
n
(e)
1
5:1:29
(p. 302)
˚
.x;y/
ˇ
ˇ
x
2
Cy
2
D1
558
AnswerstoSelectedExercises
5:1:33
(p. 302)
:::ifforAthereisanintegerRsuchthatjX
r
j>AifrR.
Section5.2 pp. 314316
5:2:1
(p. 314)(a)
10
(b)
3
(c)
1
(d)
0
(e)
0
(f)
0
5:2:3
(p. 315)(b)
a=.1Ca
2
/
5:2:4
(p. 315)(a)
1
(b)
1
(c)
no
(d)
1
(e)
no
5:2:5
(p. 315)(a)
0
(b)
0
(c)
none
(d)
0
(e)
none
5:2:6
(p. 316)(a)
...ifD
f
isunboundedandforeachM thereisanR suchthat
f.X/>MifX2D
f
andjXj>R.
(b)
Replace“>M”by“<M”in
(a)
.
5:2:7
(p.316)
lim
X!0
f.X/D0ifa
1
Ca
2
CCa
n
>b;nolimitifa
1
Ca
2
CCa
n
banda
2
1
Ca
2
2
CCa
2
n
¤0;lim
X!0
f.X/D1ifa
1
Da
2
DDa
n
D0andb>0.
5:2:8
(p. 316)
No;forexample,lim
x!1
g.x;
p
x/D0.
5:2:9
(p. 316)(a)
R
3
(b)
R
2
(c)
R
3
(d)
R
2
(e)
˚
.x;y/
ˇ
ˇ
xy
(f)
R
n
5:2:10
(p. 316)(a)
R
3
f.0;0;0/g
(b)
R
2
(c)
R
2
(d)
R
2
(e)
R
2
5:2:11
(p. 316)
f.x;y/Dxy=.x
2
Cy
2
/if.x;y/¤.0;0/andf.0;0/D0
Section5.3 pp. 335339
5:3:1
(p.335)(a)
2
p
3
.xCycosxxysinx/2
r
2
3
.xcosx/
(b)
12y
p
3
e
xCy
2
C2´
(c)
2
p
n
.x
1
Cx
2
CCx
n
/
(d)
1=.1CxCyC´/
5:3:2
(p. 335)
2
1
2
5:3:3
(p. 335)(a)
5=
p
6
(b)
2e
(c)
0
(d)
0
5:3:5
(p. 335)(a)
f
x
Df
y
D1=.xCyC2´/,f
´
D2=.xCyC2´/
(b)
f
x
D 2xC3y´C2y,f
y
D3x´C2x,f
´
D3xy
(c)
f
x
De
,f
y
Dx´e
,
f
´
Dxye
(d)
f
x
D2xycosx
2
y,f
y
Dx
2
cosx
2
y,f
´
D1
5:3:6
(p. 335)(a)
f
xx
Df
yy
Df
xy
D f
yx
D1=.xCyC2´/
2
,f
Df
´x
D
f
Df
´y
D2=.xCyC2´/
2
,f
´´
D4=.xCyC2´/
2
(b)
f
xx
D2,f
yy
Df
´´
D0,f
xy
Df
yx
D3´C2,f
Df
´x
D3y,f
Df
´y
D3x
(c)
f
xx
D0,f
yy
Dx´
2
e
,f
´´
Dxy
2
e
,f
xy
Df
yx
D´e
,f
Df
´x
Dye
,
f
Df
´y
Dxe
(d)
f
xx
D 2ycosx2y4x2y2sinx2y,f
yy
Dx4sinx2y,f
´´
D 0,f
xy
Df
yx
D
2xcosx
2
y2x
3
ysinx
2
y,f
Df
´x
Df
Df
´y
D0
5:3:7
(p. 336)(a)
f
xx
.0;0/Df
yy
.0;0/D0,f
xy
.0;0/D1,f
yx
.0;0/D1
(b)
f
xx
.0;0/Df
yy
.0;0/D0,f
xy
.0;0/D1,f
yx
.0;0/D1
5:3:8
(p.336)
f.x;y/Dg.x;y/Ch.y/,whereg
xy
existseverywhereandhisnowhere
differentiable.
AnswerstoSelectedExercises
559
5:3:18
(p.337)(a)
df D.3x
2
C4y
2
C2ysinxC2xycosx/dxC.8xyC2xsinx/dy,
d
X
0
f D16dx,.d
X
0
f/.XX
0
/D16x
(b)
df De
xy´
.dxCdyCd´/,d
X
0
f Ddxdyd´,
.d
X
0
f/.XX
0
/Dxy´
(c)
df D.1Cx
1
C2x
2
CCnx
n
/
1
P
n
jD1
jdx
j
,d
X
0
f D
P
n
jD1
jdx
j
,
.d
X
0
f/.XX
0
/D
P
n
jD1
jx
j
,
(d)
df D2rjXj
2r2
P
n
jD1
x
j
dx
j
,d
X
0
f D2rn
r1
P
n
jD1
dx
j
,
.d
X
0
f/.XX
0
/D2rn
r1
P
n
jD1
.x
j
1/,
5:3:19
(p.337)(b)
Theunitvectorinthedirectionof.f
x
1
.X
0
/;f
x
2
.X
0
/;:::;f
x
n
.X
0
//
providedthatthisisnot0;ifitis0,then@f.X
0
/=@ˆD0foreveryˆ.
5:3:24
(p.338)(a)
´D2xC4y6
(b)
´D2xC3yC1
(c)
´D.x/=2Cy=2
(d)
´DxC10yC4
Section5.4 pp. 356360
5:4:2
(p. 357)(a)
5duC34dv
(b)
0
(c)
6du18dv
(d)
8du
5:4:3
(p. 357)
h
r
Df
x
cosCf
y
sin,h
Dr.f
x
sinCf
y
cos/,h
´
Df
´
5:4:4
(p.357)
h
r
Df
x
sincosCf
y
sinsinCf
´
cos,h
Drsin.f
x
sinC
f
y
cos/,h
Dr.f
x
coscosCf
y
cossinf
´
sin/
5:4:6
(p. 357)
h
y
Dg
x
x
y
Cg
y
Cg
w
w
y
,h
´
Dg
x
x
´
Cg
´
Cg
w
w
´
5:4:13
(p.358)
h
rr
Df
xx
sin
2
cos
2
Cf
yy
sin
2
sin
2
Cf
´´
cos
2
Cf
xy
sin
2
sin2C
f
sin2sinCf
sin2cos,
h
r
D.f
x
sinCf
y
cos/sinC
r
2
.f
yy
f
xx
/sin
2
sin2Crf
xy
sin
2
cos2C
r
2
.f
´y
cosf
´x
sin/sin2
5:4:16
(p. 358)(a)
1CxC
x
2
2
y
2
2
C
x
3
6
xy
2
2
(b)
1xyC
x
2
2
CxyC
y
2
2
x
3
6
x
2
y
2
xy
2
2
y
3
6
(c)
0
(d)
xy´
5:4:21
(p. 359)(a)
.d
2
.0;0/
p/.x;y/D.d
2
.0;0/
q/.x;y/D2.xy/
2
Section6.1 pp. 376378
6:1:3
(p. 376)(a)
2
4
3
4 6
2 4 2
7
2 3
3
5
(b)
2
6
6
4
2
4
3 2
7 4
6
1
3
7
7
5
6:1:4
(p. 376)(a)
2
4
8
8 16 24
0
0
4 12
12 16 28 44
3
5
(b)
2
4
2 6
0
0 2 4
2
2 6
3
5
560
AnswerstoSelectedExercises
6:1:5
(p. 376)(a)
2
4
2
2
6
6
7 3
0 2
6
3
5
(b)
2
4
1
7
3
5
5 14
3
5
6:1:6
(p. 376)(a)
2
4
13 25
16 31
16 25
3
5
(b)
29
50
6:1:10
(p. 377)
AandBaresquareofthesameorder.
6:1:12
(p. 377)(a)
2
4
7
3 3
4
7 7
6 9 1
3
5
(b)
2
4
14 10
6 2
14
2
3
5
6:1:13
(p. 377)
2
4
7 6
4
9 7
13
5 0 14
3
5
,
2
4
5
6 0
4 12 3
4
0 3
3
5
6:1:15
(p. 377)(a)
6xy´ 3x´
2
3x
2
y
;
6 3 3
(b)
cos.xCy/
1 1
;
0 0
(c)
.1x´/ye
x´
xe
x´
x
2
ye
x´
;
2 1 2
(d)
sec
2
.xC2yC´/
1 2 1
;
2 4 2
(e)
jXj
1
x
1
x
2
 x
n
;
1
p
n
1 1  1
6:1:20
(p. 377)(a)
.2;3;2/
(b)
.2;3;0/
(c)
.2;0;1/
(d)
.3;1;3;2/
6:1:21
(p. 378)(a)
1
10
4 2
3 1
(b)
1
2
2
4
1
1
2
3
1 4
1 1
2
3
5
(c)
1
25
2
4
4
3 5
6 8
5
3
4 10
3
5
(d)
1
2
2
4
1 1
1
1
1
1
1
1 1
3
5
(e)
1
7
2
6
6
4
3 2 0
0
2
1 0
0
0
0 2 3
0
0 1
2
3
7
7
5
(f)
1
10
2
6
6
4
1
2
0
5
14 18
10
20
21
22 10 25
17
24 10 25
3
7
7
5
Section6.2 pp. 390394
6:2:12
(p.392)(a)
F
0
.X/D
2
6
4
2x
1
2
sin.xCyC´/ sin.xCyC´/ sin.xCyC´/
y´e
xy´
x´e
xy´
xye
xy´
3
7
5
;
JF.X/Dexy´sin.xCyC´/Œx.12x/.y´/´.xy/;
AnswerstoSelectedExercises
561
G.X/D
2
4
0
1
1
3
5
C
2
4
2 1
2
0 0
0
0 0 1
3
5
2
4
x1
yC1
´
3
5
(b)
F0.X/D
excosy exsiny
exsiny
excosy
; JF.X/De2x;
G.X/D
0
1
C
0 1
1
0

x
y=2
(c)
F0.X/D
2
4
2x 2y
0
0
2y 2´
2x
0
3
5
;JFD0;
G.X/D
2
4
2 2
0
0
2 2
2
0
2
3
5
2
4
x1
y1
´1
3
5
6:2:13
(p. 392)(a)
F0.X/D
.xCyC´C1/ex
ex
ex
.2xx2y2/ex 2yex
0
(b)
F
0
.X/D
2
6
6
6
4
g
0
1
.x/
g
0
2
.x/
:
:
:
g
0
n
.x/
3
7
7
7
5
(c)
F0.r;/D
2
4
exsiny´
´excosy´ yexcosy´
´eycosx´
eysinx´
xeycosx´
ye
´
cosxy xe
´
cosxy
e
´
sinxy
3
5
6:2:14
(p. 392)(a)
F
0
.r;/D
cos rsin
sin
rcos
;JF.r;/Dr
(b)
F
0
.r;;/D
2
4
coscos rsincos rcossin
sincos
rcoscos rsinsin
sin
0
rcos
3
5
;
JF.r;;/Dr
2
cos
(c)
F
0
.r;;´/D
2
4
cos rsin 0
sin
rcos 0
0
0
1
3
5
;JF.r;;´/Dr
6:2:20
(p. 393)(a)
0
0 4
0 
1
2
0
(b)
18 0
2 0
(c)
2
4
9 3
3 8
1
0
3
5
(d)
4 3 1
0
1 1
(e)
2 0
2 0
(f)
2
4
5 10
9 18
4 8
3
5
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