mth121
of 124
Transcript of mth121
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W
WK
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WK
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K K
K
WKKKKK
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KK
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K
K
K
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]
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--
WK
WWoKoKoKoK
WK
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--
K
WKK
Wyx2333K
WK
FE212x29xyx32237 yxK
xxx 234 2 +xx +24K)22(
0x=K
W
012
21
1 .......... axaxaxaxan
nn
n +++++
0nanKna
0aKWW
9 5,1,9 2 5,,9
2xx 59 2 + xx
2 11,2 1 11,2x x211
1 3,5,1 3 3,5,3 xx 353 + xx
K
WW
)232()75()2)354()463()1 2222 +++++ xxxxxxxx
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--
W
98))2(7()]3()5[()2()232()75()2
37)1)4(()56()43()354()463()1
22222
22222
+=+++=++
+=++++=+++
xxxxxxxxxx
xxxxxxxxxx
WK
WW)13)(32( 2 + xxxW
35116339226
)1)(3())(3()3)(3()1)(2())(2()3)(2()13)(32(
23223
222
+=++=
+++++=+
xxxxxxxx
xxxxxxxxxx
W
32233
32233
222
222
22
33)())()((
33)())()((
2)())((
2)())((
))((
yxyyxxyxyxyxyx
yxyyxxyxyxyxyx
yxyxyxyxyx
yxyxyxyxyx
yxyxyx
+==
+++=+=+++
+==
++=+=++
=+
WW
3222)32()3)112()2)107)(107()1 yxzyxx ++
W
322332233
4224222222222
222
2754368)3()3)(2(3)3()2(3)2()32()3
121444)11()11)(2(2)2()112()2
10049)10()7()107)(107()1
yxyyxxyyxyxxyx
zzyyzzyyzy
xxxx
+=+=
++=++=+
==+
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--
W76223 +
xxW2)24)1==
xxW
xW
524712247)2(6)22(27)2(6)2(2762)2
2177961287)16(6)64(27)4(6)4(2762)1
2323
2323
=+=+=+=+
=+=+=+=+
xx
xx
FE
WK
W1692 + xx3xW 12+x
3x 1692 + xx
xx 32
1612 x
3612 x
20
1692 + xx3x12+x20
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--
WW
22453234
323222
11109)61)55432)4
1754)32)272)1
yxxyyxxxxx
xyyxyxxx
++++
++
WW
)43)(12)(54()8)42()843()4
)25)(14)(23()7)9112()1147()3
)252)(523()6)185()724()2
)583)(75()5)272()543()1
223
22
22232
222
++
+++
+++++
++++
uuuuuuuu
cccssss
xxxxwwww
xxxxxxx
WW
] ]
[ ][ ]
[ ] [ ]
[ ] [ ]32
32
32
322
22
)21(4)12)3()6
2)1()11)2()5
)2()10)4()4
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7)2(7)2()8)34)(34()2
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yyx
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yxyxyxyx
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+
++
++
++
++++
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2;1)4
1;355)3
5;45)2
3;17)1
53
23
2
2
=
=+
=+
=+
xxx
xxxx
xxx
xxx
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--
WW
56,15162024)4
52,459232)3
32,931136)2
3,201765)1
2235
2234
224
3
++
++
++
+++
xxxx
xxxxxx
xxxxx
xxxx
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8/2/2019 mth121
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--
K
W
K
FKKE
K
WW)2)(4()2)(4()330612)2610)1
223baxbaxxyxyyxxx +++
WEFKKE310xx6x2W
)35(2)3(2)5(2610223 +=+=+ xxxxxxx
EFKKExy6W
)152(6)512(6)5(6)1(6)2(630612
22 ===yxxyyxxyyxyxyxxyxyxyyx
EFKKEba 2W
]
)2(2)2)(2(
)44)(2()4()4()2()2)(4()2)(4(
baxxba
xxbaxxbabaxbax
==
++=++=++
K
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--
WW144216 23 + yyyW
W
)144()216(144216 2323 =+ yyyyyy
FKKEW
)72(2)72(3)144()216(144216 22323 ==+ yyyyyyyyy
72 yW
)23)(72(
)72(2)72(3)144()216(144216
2
22323
=
==+
yy
yyyyyyyyy
cbxax ++2
W1=a
2xcbKK
WW1872 + xxW
7=b18=c187K29W79)2( =+
189)2( =KKW
)9)(2(1872 +=+ xxxx
KK
W1a
qpnm ,,,W
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--
bnpmqcpqamn =+== )3,)2,)1
W
))((2 qnxpmxcbxax ++=++
pqb0>c0c0>bW1,3,3,2 ==== qpnmW
)13)(32(3116 2 ++=++ xxxx
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qp =nm =cbxax ++22)( pmn +
W
))((22 yxyxyx +=
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14449 2 x222 )12()7(14449 = xxW)127)(127()12()7(14449 222 +== xxxx
W
))(()))(() 22332233 yxyxyxyxiiyxyxyxyxi ++=+++=
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--
WW64)28)1 333 + abaW
E338 ba +3333 )()2(8 baba +=+)iiW
)24)(2(
])())(2()2)[(2()()2(8
22
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)1)((
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)()()2222222
++=
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+=+=+
qpqp
qpqpqp
qpqpqpqpqqpp
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--
W
22
23
23
22
22
22
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22
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WW
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--
Q
P
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65
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+
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+ xxx
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+
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xxx
x
xxxxx
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xxx
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x
xx
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8/2/2019 mth121
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--
WW
103
12
4)3
12
3
1)2
2)1
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+
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nm
mn
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mmnmmn
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K
K
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--
WWyx
yx
xx
xxx
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1
1
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5
2
5
3
1
2
2
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y
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1
1
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5
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23
5
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23
5
23
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23
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22
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23
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5
2
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1
1
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3
2
x
yx +
111
+
z
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3
2
3
22
2y
x
yx +=
+yx
z
z
yx
+=
+
1
11
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--
WW
( )( )
1)8
9
15562)7
352
1310)6
103
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8
8)4
6
9)3
153
20)2
32
4)1
3
23
2
23
2
2
2
2
2
3
23
322
+
+
+
+
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+
+
+
x
xxx
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a
a
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2
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2
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1
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1
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12
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22222
222
++
+
+
+
+
+
++++
+
+
x
x
x
x
xxxxxx
xxxxxxx
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8/2/2019 mth121
21/124
--
WW
14233
10116
21415112
)4
18152
30
6103
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1
1
12
11
12
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3
3
1
12
31
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2
2
2
2
2
2
2
2
2
2
+
+
++
+
++
+
+++
++
+
yy
yy
yyyy
zz
zz
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xxx
xxx
x
xx
xxx
WW
( )
x
x
p
p
ba
abba
aa
a
h
xhx
h
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11
14
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3
11
2
11
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11
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22
11
21
122
+
+
+
+
+
++
K
WFEKK
W712 =+x3=xxK3=xx371)3(2 =+K
K
W3=x2=x0652 =+ xxK
xWx =K
W
W11532 =++ xx1137 =+xK
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--
W273 =x93 =xKW
1065 =x12=x
W
WFExW
0=+ baxab0a
WW)1(5)15)(2()3064
3)2952)1 +=++==+ xxxxxx
WEW
44259552 ===+ xxx
E3
44
3W
8)6(3
4
4
3
3
46
4
36066
4
306
4
3=
=
=+=+= xxxxx
E2,5,5 2 xxW
3
1
3
1
6
226026
5211552115)1(5)15)(2( 22
=====+
=++=+++=++
xxxx
xxxxxxxxxx
WK
KK
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8/2/2019 mth121
23/124
--
WW5
5
51)2
3
524
3)1
=
+
=
xx
x
x
x
x
x
W E3xK)3( xK
46
24
6
6246
552455243
524)3(
3)3(
===
+=+=
=
xx
x
xxxxxxx
xx
x
xx
3K
E5xK)5( xK
51025555255255
5
5)5(
5)5(1)5(
5
5)5(
51)5(
==+=+==+
=
+
=
+
xxxxxx
xx
x
xxx
xx
x
xx
5=x
K
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8/2/2019 mth121
24/124
--
W
8
76
5
340)8
)1(5)3)(3(5)7
2
19
4
17
2
1)6
3
2
2
1
4
3)5
0)83(5)172(4)4
207114)3
2203)2
40102)1
+=
=+
=+
=+
=+
+=
=+
=+
x
x
x
xxxx
xx
x
xx
xx
y
x
23
75
4
12)14
2
42
2
34)13
33
392)12
4
3
5
3)11
2
4
3)10
72
5
2
3)9
+
=
+
=
=
+
=
+
+
+
+=
=
+
xx
x
x
x
x
mm
m
x
x
x
x
x
x
x
x
xx
( )[ ]
[ ]
( )
9)2()7()12
1)4(3)19
)2(])1(2[4)19
072)1(236)18
23545)17
3
4
2
3
3
5)15
4
122
4
3)14
22
22
22
+=
++=+
+=++
=+
=
=
+=
+
zz
yy
yy
yyy
xxx
xxx
xx
x
WWW
002 =++ acbxax
K
cbxax ++2
W000 === BAABWW062)202510)1 22 =+=++ xxxxW
E)5)(3(1582 ++=++ xxxxW
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25/124
--
}505303{0)5)(3(1582 ==+==+=++=++ xxxxxxxx
01582 =++ xx3=x5=xK
K
Eqpnm ,,,1aW
==+
====+=+
2022
332032
0)2)(32(62 2
xx
xxxxxxx
062 2 =+ xx2=x2
3=x
ABWBA =20>BBA =K
WW49)1()205)1 22 =+= xxW
EW
55505522 ==+=+ xxx
W5=x5=x
EW
817617
177149149)1( 2
====
==+=+=+
xx
xxxx
W6=x8=xK
W
a1a
2
2
bK
2
2
bxKK
-
8/2/2019 mth121
26/124
--
27WW01582)2062)1 22 =+=+ xxxxW
E1=a12
22
=
W
9)1(181282082 2222 =+=+== xxxxxxx
W
}231431{
31919)1( 2
====
===
xxxx
xxx
E2=a1aW
2
154)15(
2
1)82(
2
1158201582 2222 =+=+=+=+ xxxxxxxx
42
4
2
22
=
=
bW
2
446
2
446
2
446
2223
2232
223)2(4
21544 22
=
=
=
==+=++=++
xxx
xxxxx
W
W
W0,02 =++ acbxaxa
acbbx
2
42 =K
acb 42 W
acb 42 Wa
acbbx
2
42 =
acb 42 Wa
bx
2
=
acb 42 K
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27/124
--
WW0763)3096)20252)1 222 =++=++=+ xxxxxxW
E252 === cbaW
091625)2)(2(4)5(4 22 >=== acb
W
2
1
4
2
4
352
4
8
4
35
4
35
)2(2
9)5(
2
42
==
===+
=
=
=
= xxa
acbbx
E961 === cbaW
03636)9)(1(4)6(422 === acb
W3)1(2
)6(
2=
=
=
a
bx
E763 === cbaW
0488436)7)(3(4)6(4 22
-
8/2/2019 mth121
28/124
--
W
0)114()52()609)5()5015148)4
073)30721898)20152)1
2222
222
===+
==+=
xxxtt
xxyyxx
W
222222
222
)32(4)6)1()5)1()8()4
36)5()30482)281)1
+=+=+=
===
xxxxxx
xxx
W
05102)601544)503102)4
027)306)20108)1
222
222
=+=+=+
=+==+
xxxxxx
xxxxxx
W
0352)90232)817)7
02
15
3
2)601
4
3
2
1)50353)4
0142)301)20152)1
222
222
222
=+=++=
=+=+=+
=++=+=
xxxxxx
xxxxxx
xxxxxx
Wk015)403)3081)20916)1 2222 =++=+=++=++ kxxkyykxxkxx
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]
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8/2/2019 mth121
30/124
--
WKK
WWoKoKoK
WK
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8/2/2019 mth121
31/124
--
??
waoKSekiKKK
WK
KW
Wnn
n K11KK
WK223344W
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478
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42)1
22
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bcaddc
ba=
WW
0511)2
3142)1
W
246)14()32(31
42)1 ===
5)5(0)15()01(05
11)2 ===
-
8/2/2019 mth121
32/124
--
33W
W33FE
FEW
122133112332132231
322113312312332211
3231
2221
1211
333231
232221
131211
333231
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aaaaaaaaa
aaaaaaaaa
aa
aa
aa
aaa
aaa
aaa
aaa
aaa
aaa
++==
WW
420
315
201
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478
362
931
)1
W
25524214321267224)324()137()968()729()833()461(
78
62
31
478
362
931
478
362
931
)1
=++=
++=
=
180)6(02004
)054()132()210()252()030()411(
20
15
01
420
315
201
420
315
201
)2
=+=
++=
=
KW
WK
WW
432
210
051
)201
51)1
-
8/2/2019 mth121
33/124
--
WEW
50511
0151 ==
EW
18
420
315
201
432
210
051
=
=
W
FEKWW
4218
3182
991
)231
21)1
WEW
125.031425.0
3145.025.0
3121 ====
EW
7652553
478
362
931
3
4738
3632
9331
4218
3182
991
===
=
WK
WW
3182
4218
991
)213
12)1
-
8/2/2019 mth121
34/124
--
WEW
13121
1312 ==
EW
765
4218
3182
991
3182
4218
991
=
=
WK
WW
511
300
211
)221
21)1
WEW
021
21=
EW
0
511
300
211
=
KW
Wnm
mnK11K
-
8/2/2019 mth121
35/124
--
W3223145133W
( )
==
=
=
=
042
7.001
325
,011726.0,
2
5.00
1
,
1210
57
35
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57
35 EDCBA
W
WFEKK
WW
A
GF
=
+
=0427.021
325
,06227.001
3x1223
BA GEFEFE=K
KW
WWFE
FEK
WW
=
=
7
1
4
0
3
2
136
742BA
W
ABBA + )2)1
W
=
+++
++=+
673
840
714336
170422)1 BA
=
=
819
644
)1(734)6(3
714022)2 AB
-
8/2/2019 mth121
36/124
--
WK
WKW
ABBBABAB +++++ )3)()2)1
WEW
=+=+
673
840BAAB
EW
=
+
=++=++13110
942
7
1
4
0
3
2
673
840)()( BABBAB
EW
=++=++=++
13110
942)( BABBABABB
W
WFEFE
FEFEFEK
WW
=
=
4.0
2
1
3
0
5.0
5
2
0
6
1
3
BA
W
2)32)23)1
+ BBAA
W
=
=
=
15
6
0
18
3
9
53
23
03
63
13
33
5
2
0
6
1
3
33)1 A
-
8/2/2019 mth121
37/124
--
=
+
=
+
=+
2.4
6
2
0
1
2
8.0
4
2
6
0
1
5
2
0
6
1
3
4.0
2
1
3
0
5.0
2
5
2
0
6
1
3
12)2 BA
=
=
=
2.0
1
5.0
5.1
0
25.0
2
4.02
22
1
2
32
02
5.0
2
4.0
2
1
3
0
5.0
2)3
B
WFKE
WKW)2(5.1)23)1 AA
WEW
==
15
6
0
18
3
9
33 AA
EW
===
15
6
0
18
3
9
3)25.1()2(5.1 AAA
W
WK
WW
( ) ( )
=
=
4
1
3
3.0021)2
10
4
1
3
3.0021)1 ba
-
8/2/2019 mth121
38/124
--
W
23023)10x3.0()4x0()1x2()3x1()1 =++=+++=a
EbKK
W
WkmnkFEnm
K
WW
=
=
=
6323
12015
3021
110120
9070
1510
100
90
80
40
25
30CBA
W
BCBAAB )3)2)1
W
( ) ( )
( ) ( )
=
=34256150
67508300
110
90
15
1008025
120
70
10
1008025
110
9015
904030
120
7010
904030
)1 AB
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
=
=2180013600850
1530010000150
2400160075
100
90110120
80
40110120
25
30110120
100909070
80409070
25309070
100
901510
80
401510
25
301510
)2 BA
EK
-
8/2/2019 mth121
39/124
--
WK
WW
BAAB
WW
( )3.0021
10
4
1
3
=A
W
=
=
302010
2.1084
3.0021
9.0063
3.010010210110
3.04042414
3.01012111
3.03032313
A
W
( ) ( )3.0021
10
4
1
3
10
4
1
3
3.0021
WW
( )
==
=
40
03
12
5.0032
1CBA
W
)()2)()1 BCACAB
W
=
=
1012
56
40
03
12
106
5.003)()1 CAB
EW
-
8/2/2019 mth121
40/124
--
==
1012
56)()( CABBCA
WW
ACABCBA
BCACCBA
+=+
+=+
)(
)(
ABCK
W
WAnmtAmnK
WKW
( )
=
==401
032
5.0
0
3
21 ttt CBA
KW
W
WAK(A)detKK
WW
=
=
=
=
=
510
604
316
035
410
102
124
301
11
32
01
11
E
DCBA
W
-
8/2/2019 mth121
41/124
--
107150722000
1
3
1
6
5
0
316
035
410
316
035
410
)det(
1000)12(002
0
2
0
2
4
1
102
124
301
102
124
301
)det(
5321132)det(
1)1(001
11)det(
=+=
=
=
=++=
=
=
==
=
==
=
D
C
B
A
)det(EEKWK
WKW
)det()4)det()3)det()2)det()1 DCCDBAAB
WKW
107010107)det()det()det()4
107010710)det()det()det()3
515)det()det()det()2551)det()det()det()1
===
===
===
===
CDDC
DCCD
ABBABAAB
W
WK
WK
W
=
36
12F
W
WFEK
-
8/2/2019 mth121
42/124
--
WW
=
=
=
=300
030
000
100020
001
10
02
00
01
DCBA
W
=
=
=
010
001
200
400
321
00
11GFE
WFKE
WKW
0330)det(
2121)det(
212)det(
001)det(
==
==
==
==
C
C
B
A
W
WKnInnIK
W
=
=
=
1000
0100
0010
0001
100
010
001
10
01432 III
WKWW
=
=
251
430
321
100
010
001
)2
100
010
001
251
430
321
)1 BA
-
8/2/2019 mth121
43/124
--
W
AB
A
=
=
=
251
430
321
)2
251430
321
)1
W
WaA
aAaFAE223344FWE
++
++
++
++
++
+
++
+
+
KcofAKKWW
=
=
232
241
231
)202
13)1 BA
W
=
+
+
+
+
+
=
=
+
+=
742
3612
11214
41
31
21
21
24
23
32
31
22
21
23
23
32
41
22
21
23
24
)2
31
20
3)1(
20)1
cofB
cofA
-
8/2/2019 mth121
44/124
--
W
WAK
adjAKWKW
==
==
7311
462
21214
)()2
32
10)()1
t
t
cofBadjB
cofAadjA
KW
WA1AJW
IAAAA == 11
WAW
)det(
1
A
adjAA =
WKW
EW
02)2(00213)det( ===A
W
=
==
5.11
5.00
2
32
10
)det(
1
A
adjAA
EW
-
8/2/2019 mth121
45/124
--
03066166128
3
4
3
2
1
1
232
241
231
232
241
231
)det( =+=
=
=B
W
=
==
30
7
10
1
30
1115
2
5
1
15
115
1
5
2
15
7
30
7311
462
21214
)det(
1
B
adjBB
WW
=
=
=
=31
62)4
43
21)3
000
520
1021
)2
140
152
301
1 DCBA)
WEWW
0150)4(02405
4
5
0
0
2
1
140
152
301
140
152
301
)det( =+===A
KWW
=
+
+
+
+
+
=
5515
4112
829
5201
1231
1530
40
01
10
31
14
30
40
52
10
12
14
15
cofA
WW
==
548
512
15129
)( tcofAadjA
-
8/2/2019 mth121
46/124
--
WW
=
==
3
1
15
4
15
83
1
15
1
15
215
4
5
3
15
548512
15129
)det(
1
A
adjAA
EWW
0000000
0
2
2
0
0
1
000
520
1021
000
520
1021
)det( =++=
=
=B
KEWW
026443
21)det( ===C
KWW
+
+= 12
34cofC
WW
==
13
24)( tcofCadjC
WW
=
==
5.05.1
12
2
13
24
)det(
1
C
adjC
C
EWW
0)6(631
62)det( ==
=D
K
-
8/2/2019 mth121
47/124
--
22W
9Wabcd0 bcadW
dc
ba
ac
bd
bcad
ac
bd
dc
ba
=
=
1
22K
W229KW
EW
=
=
=
5.11
5.00
2
32
10
02
13
32
)1(0
02
131
EW
=
=
=
5.05.1
12
213
24
43
2113
24
43
211
EK
0)6(631
62==
-
8/2/2019 mth121
48/124
--
WW
100
410
321
)6
522
113
021
)5
021
212
113
)4
63
70)3
13
52)2
53
21)1
WW
622
033
511
)4
143
012
001
)3
021
113
522
)2
35105
1412
713
)1
WW
=
=
=
3
2
5
12
0
5.0
1
6
7
2
3
10
5
4
6
0
2
1CBA
W
CABACCBA 3)332)22)1 +++WW
321
352
103
501
241
432
)4
7
1
0
1
3
4
2
3
1
4
7
1
3
2
0
1)3
23
34
12
241
532
121
)2
22
01
14
214
213)1
-
8/2/2019 mth121
49/124
--
WW
=
=
=
=
=
6
7
8
5
12
9
4
11
10
3
2
1
)5
034
120
351
)4
01430105
001
)321
105
)264
21
)1
ED
CBA
WW
=
612
126
12
1M
W
MMMMMMMM == 32 )3)2)det()1
WW
=
=
=
66
95
21
15
01
32CBA
PQRW
CRAIBQAQBAP ==++= )3)22)12
WW
=
12
32
k
kkM
k
WW
=211
322
112
A
KW
AAB
=
001
010
1001
-
8/2/2019 mth121
50/124
]
-
8/2/2019 mth121
51/124
--
K:
WWoooK
WK
-
8/2/2019 mth121
52/124
--
K??
FEKK
KK
KW
WK
WxyzW
zyx
zyxy
xyx
32)3
23)2
35.2)1
+
++
++
xyzW
zyx
xx
zxyx
++
++
+
12
WK
WxyzW
65.3232
32635.2
=+=+
+=++
yxzyx
yzxyx
xyzW
10
31
0
2
++=++
=++
=+
zyzyx
xx
zxyx
-
8/2/2019 mth121
53/124
--
WFEKK
WxyzW
03
1052
363
223
=+
=+
=+
=++
zx
zyx
zyx
zyx
WKW
WK
WK
WF
KKE
KKW
K
WW
62)3(2)35.33745.3)2623)1 +==+=+ xxxxxxx
WEWW
362 = xx
WW
93 = x
WWW
33
9=
=x
-
8/2/2019 mth121
54/124
--
EW
435.375.3 =+ xxx
W70 =
WKEW
6622
6226
=+
+=
xx
xx
W
00 =WK
KW
W
WKWKKWKWW
WK
WKWK
WW
5.25.025.02543
52)352)252)1
===
=+=+=+
yxyxyx
yxyxyx
WEWyxW
xy 25 +=
-
8/2/2019 mth121
55/124
--
WW
25)25(43 =+ xx
WW
15
5
55
202583
258203
=
=
=
+=
=
x
x
xx
xx
WW
7)12(5
25
=+=
+=
y
xy
WW
71 == yx
EWyxW
xy 25 +=
WW
2)25(5.0 =+ xx
WW
5.40
5.22
25.2
=
+=
=
xx
xx
WKEWyxW
xy 25 +=
WW5.2)25(5.0 =+ xx
WW
00
5.25.2
5.25.2
=
+=
=
xx
xx
-
8/2/2019 mth121
56/124
--
WK
W
WxyW
222
111
cybxa
cybxa
=+
=+
1a2a1b2b1c2cKKD22
W
122122
11baba
ba
baD ==
22W
1221
22
11
1221
22
11
cacaca
caD
bcbcbc
bcD
y
x
==
==
WWWW
D
Dy
D
Dx
yx ==
WFEK
W
KWKW
EW
053843
12==
=D
-
8/2/2019 mth121
57/124
--
W
351550253
52
5)25(20
425
15
==
=
==
=
y
x
D
D
W
75
35
15
5
===
===
D
Dy
D
Dx
y
x
EW011
5.01
12==
=D
W
05.425.25.02
15==
=xD
xK
EW011
5.01
12==
=D
W
0555.21
52
0)5.2(5.25.05.2
15
==
=
==
=
y
x
D
D
K
-
8/2/2019 mth121
58/124
--
WW
1441353
482)31223)2432)1
=+==
=+=+=+
yxyxyx
yxyxyx
W
104631
42
5)9(413
34
053211
32)1
==
=
==
=
==
=
y
x
D
D
D
W
25
10
15
5
=
==
=
==
D
Dy
D
Dx
y
x
5760315
123
3823631
212
01910935
23)2
===
===
==
=
y
x
D
D
D
W
3
19
57
219
38
=
==
=
==
D
Dy
D
Dx
y
x
09611216414
84
08841
82)3
===
===
xD
D
K
-
8/2/2019 mth121
59/124
--
KW
K
WxyzW
3333
2222
1111
dzcybxa
dzcybxa
dzcybxa
=++
=++
=++
1a2a3a1b2b3b1c2c3c1d2d3dK
D33
W
333
222
111
cba
cba
cba
D =
33W
333
222
111
333
222
111
333
222
111
dba
dba
dba
D
cda
cda
cda
D
cbdcbd
cbd
D
z
y
x
=
=
=
WWWW
D
Dz
D
Dy
D
Dx z
yx ===
WFEK
-
8/2/2019 mth121
60/124
--
WK
WW
524213223
113211321132
6)36)26)1
===++
=++=++=++
=++=++=++
yxyxzyx
zyxzyxzyx
zyxzyxzyx
W
6262254243339
2
31
3
21
1323
1132611
1323
1132611
4241333261822
13
11
6
3
2
1
2133
1112
161
2133
1112
161
2221239221336
2
3
1
13
11
6
2213
1311
116
2213
1311
116
02429436
2
3
1
3
2
1
223
132
111
223
132
111
)1
=++===
=++===
=++===
=++===
z
y
x
D
D
D
D
F0DEW
326
22
4
12
2
===
=
==
=
==
DDz
D
Dy
D
Dx
z
y
x
-
8/2/2019 mth121
61/124
--
0180)12(122240
2
3
1
4
11
6
024
1311
116
024
1311
116
00)2()3(410
2
3
1
1
2
1
021
132
111
021
132
111
)2
=+=
=
=
==
=
=
xD
D
F00 = xDDKE
0)10()22()18(2411152
3
1
12
1
5211132
611
5211132
611
00)5()11(1060
5
11
6
1
2
1
051
1112
161
051
1112
161
00)12()15(2250
2
3
1
5
11
6
025
1311
116
025
1311
116
00)2()3(410
2
3
1
1
2
1
021
132
111
021
132
111
)3
==
=
=
==
=
=
==
=
=
==
=
=
z
y
x
D
D
D
D
F0==== zyx DDDDKE
KW
W
CAU=
AUC
-
8/2/2019 mth121
62/124
--
WKW
=
=
=
5.2
5
5.01
12)3
2
5
5.01
12)2
25
5
43
12)1
y
x
y
x
y
x
WKKW
=
=
=
5
11
6
021
132
111
)3
4
11
6
021
132
111
)2
13
11
6
223
132
111
)1
z
y
x
z
y
x
z
y
x
WWCAU =
AKAW
CAU1=
WW
W EWW
053843
12==
KWW
=
=
5
2
5
35
1
5
4
5
23
14
43
121
WW
=
=
=
7
1
25
5
5
2
5
35
1
5
4
25
5
43
121
y
x
-
8/2/2019 mth121
63/124
--
WW
71 == yx
EWW011
5.01
12==
EWW
0115.01
12==
WKW
EWW
02429436
2
3
1
3
2
1
223
132
111
223
132
111
=++==
KWW
=
=
++
+
++
=
=
=
5.05.05.25.05.05.0
102
223132
111
2
115
111
204
2
1)1()2(
)1()1(0
)5(14
2
223
132
111
2
223
132
111
223
132
111
1
1
tt
cofadj
WW
=
=
=
3
2
1
13
11
6
5.05.05.2
5.05.05.0
102
13
11
6
223
132
1111
z
y
x
-
8/2/2019 mth121
64/124
--
WW
321 === zyx
EWW
0
021
132
111
=
EWW
0021132
111
=
-
8/2/2019 mth121
65/124
--
WW
53
2)6)2(4)23(65)5
2
452)4
1
4
3
2)343)1(26)2
2
2)32()1
=+
++=+
=+
=
+=+
+=+
x
xxxx
xx
xxxx
xxx
WW
05.221810823
954)6954)5954)4
5225.004
042)3824)2132)1
=+=+=+
===+
===+
=+=+=
yxyxyx
yxyxyx
yxyxyx
yxyxyx
WW
122320
10113155
0)622)5132)4
1032612352432
32311202
12)31053)26)1
=+=+=
=+=+=+
=+=+=+
=++=+=+
=+=++=+
==+=++
yxzyyx
zyxzyxzy
zyzyxzx
zyxzyxzyx
yxzyxzyx
zxzyxzyx
-
8/2/2019 mth121
66/124
]
-
8/2/2019 mth121
67/124
--
WK
WW
oKoKoW
K
oK
WK
-
8/2/2019 mth121
68/124
--
KW
K
KW
WfXYX
Y1x2xXYW
)()( 2121 xfxfxx ==
)(xfxKXY)(xfx
fx)(xfy xf)(xfY=Y)(xfYKK
WYXf :
K
WW}3,2,1,0{=X}8,6,4,2{=Yf
XYW
6)3(,4)2(,2)1(===
ffffX
YKW
WYXf :Wxxf 2)( =
0
1
2
3
6
4
2
8
X Y
f
-
8/2/2019 mth121
69/124
--
WW}5,2,2,1,1,0{ =X}2,9,4,1,0{ =YfXYW
4)2()2(,1)1()1(,0)0( ===== fffff
fXYW
01122Y5
Y)5(f
Y
KKWW
YXf :
W2)( xxf =fW
( ) ( ) )()( 21222121 xfxfxxxx ===WWCitiesCountries
fCitiesCountriesWx)(xfKfxCities
CountriesKWxx
Countries
KW
KingdomUnitedLondonf
EgyptCairof
eriaAiersAf
ArabiaSaudiRiyadhf
=
=
=
=
)(
)(
lg)lg(
)(
)(AbhafCountriesAbhaK
0
1
-1
2
4
1
0
9
X Y
f
-2
5
-2
-
8/2/2019 mth121
70/124
--
WWCitiesCountriesgCitiesCountriesW)(xgxK
gxCitiesCountriesKW
ArabiaSaudiAbhag
KingdomUnitedLondong
EgyptCairog
eriaAiersAg
ArabiaSaudiRiyadhg
=
=
=
=
=
)(
)(
)(
lg)lg(
)(
WCountriesfCountriesCitiesW)(xfxK
WArabiaSaudiKK
WfFEf
KffDfRK
WKW
}4,1,0{}2,2,1,1,0{)2
}6,4,2{}3,2,1{)1
==
==
ff
ff
RD
RD
ECapitalsWCountriesRCapitalsD ff ==
CountriesRCitiesD gg ==)4WfWxxf 2)( =K
EfKEfKE)5(f)14(fKW
EfW
)()(22 212121 xfxfxxxx ===
-
8/2/2019 mth121
71/124
--
EfW=fDW},10,8,6,4,2{ L=fR
E1052)5( ==f28142)14( ==f.WRgRRWxxf 2)( =K
EgKEgKE)5.2(g)5(gKW
EgW
)()(22 212121 xgxgxxxx ===
Eg
WR=
g
D
RWR=gRW
=
==
222
22
yg
yyy
KW)5.1(3 g=)3.0(6.0 g=.E55.22)5.2( ==g1052)5( ==g.
WfgW
)()()2
)1
xgxfDx
DD
f
gf
=
=
Wgf =K
WfgW
Wgf WCitiesDCapitalsD gf ==
Wfg
WWgf W
R== gf DD
WW
g:
-
8/2/2019 mth121
72/124
--
W2)( xxg =fKgf =
WFEWgf DD ==W= fD393)3(632)3( 2 ==== gf
Wgf K
KW
WYXf
:WXDf =K
WW
W}8,6,4,2{}3,2,1,0{: fxxf 2)( =K}3,2,1,0{}3,2,1{ =fDF)0(fEK
W}2,9,4,1,0{}5,2,2,1,1,0{: f2)( xxf =K}5,2,2,1,1,0{}2,2,1,1,0{ =fDF)5(fE
KWCountriesCitiesf :)(xfxKK
CitiesCapitalsDf =F)(AbhafEK
WCountriesCitiesg :)(xgxK
CitiesDg=
K W:fxxf 2)( =K=fDK
WRR: gxxg 2)( =KR=gDK
Wg:2)( xxg =K
-
8/2/2019 mth121
73/124
--
N=gDK
WYXf :YDf f :K
FKE
WKW
W}8,6,4,2{}3,2,1{: fxxf 2)( =KKW}2,9,4,1,0{}2,2,1,1,0{: f2)( xxf =K
WCountriesCapitalsf
:)(xfxK KK
WYXf :FE1x2x
W2121 )()( xxxfxf ==XDf =K
WKW
W}8,6,4,2{}3,2,1{: fxxf 2)( =KW212121 22)()( xxxxxfxf ===KK
W}2,9,4,1,0{}2,2,1,1,0{: f2)( xxf =KKW)1()1(11)1( 22 ==== ffKK
WCountriesCapitalsf :)(xfxKKKW2121 )()( xxxfxf ==KK
WCountriesCitiesg :)(xgxKKW)()( AbhagArabiaSaudiRiyadhg ==K
W:fxxf 2)( =KKW212121 22)()( xxxxxfxf ===K
WRR: gxxg 2)( =KKW212121 22)()( xxxxxgxg ===K
-
8/2/2019 mth121
74/124
--
Wg:2)( xxg =KKW
21212
22
12
22
121 )()()()( xxxxxxxxxgxg =====K1x2x
K
WYXf :FEWYRf =
XDf =K
WKW
W}8,6,4,2{}3,2,1{: fxxf 2)( =KW}8,6,4,2{}6,4,2{ =fRKK
WFKKEW}2,9,4,1,0{}2,2,1,1,0{: f2)( xxf =K
W}2,9,4,1,0{}4,1,0{ =fRKK
WFKEWCountriesCapitalsf :)(xfxKKK
WCountriesRf =KKWCountriesCitiesg :)(xgxK
KWCountriesRg =KKW:fxxf 2)( =K
WN},10,8,6,4,2{= L
fRKK WKKWRR: gxxg 2)( =K
KWR=fRKWg:2)( xxg =K
KWN},25,16,9,4,1{ = LfRK
-
8/2/2019 mth121
75/124
--
KK
WYXf :ff RDf :K
K
WKW
W}6,4,2{}3,2,1{: fxxf 2)( =KW}4,1,0{}2,2,1,1,0{: f2)( xxf =K
W},10,8,6,4,2{:L
fxxf 2)(=
KK W},25,16,9,4,1{ Lg:2)( xxg =K
KWYXf :FE
K
WKKW
WE}6,4,2{}3,2,1{: fxxf 2)( =KECountriesCapitalsf :)(xfxKKKE},10,8,6,4,2{: Lfxxf 2)( =KERR: gxxg 2)( =K
E},25,16,9,4,1{ Lg:2)( xxg =K
KWRR: fW2)( xxf =KW
W)1()1(11)1( 22 ==== ffK
WfXYWEW)()( 2121 xfxfxx ==1x2xK
-
8/2/2019 mth121
76/124
--
EW)()( 2121 xfxfxx ==1x2xKXDf =
EW)()( 2121 xfxfxx ==1x2xKXDf =2121 )()( xxxfxf ==1x2xK
EW)()( 2121 xfxfxx ==1x2xKXDf =YRf =
EW)()( 2121 xfxfxx ==1x2xKXDf =2121 )()( xxxfxf ==1x2xKYRf =
KW
WYXf :ZYg :
fg oXZW[ ])()( xfgxfg =oWfg ogf oW
2
2
2
2
)(,RR:)(,RR:)4
)(,NN:)(,NN:)3
)(,NN:1)(,NN:)2
)(,RR:1)(,RR:)1
xxggxxff
xxggxxff
xxggxxff
xxggxxff
==
==
=+=
=+=
WERR: fg oW[ ] [ ] ( ) 1211)()( 22 ++=+=+== xxxxgxfgxfg o
RR: gf oW[ ] [ ] 1)()( 22 +=== xxfxgfxgf oENN: fg oW[ ] [ ] ( ) 1211)()( 22 ++=+=+== xxxxgxfgxfg o
NN: gf oW[ ] [ ] 1)()( 22 +=== xxfxgfxgf oENN: fg oW[ ] [ ] ( ) xxxgxfgxfg ==== 2)()(o
-
8/2/2019 mth121
77/124
--
NN: gf oW[ ] [ ] xxxxfxgfxgf ===== 22)()(oxK
ERR: fg oW[ ] [ ] ( ) xxxgxfgxfg ==== 2)()(oRR: gf oW[ ] [ ] xxxfxgfxgf ==== 22)()(ox
K
WK
WKEfggf oo W1)(12)( 22 +=++= xxgfxxxfg oo1=xK
Efggf oo W1)(12)(22
+=++= xxgfxxxfg oo1=xKEfggf oo =W)()( xgfxxfg oo ==KEfggf oo Wxxgfxxfg == )()( oo1=xK
KW
WK
WK W
WExFE)(xfy =KE),( yxKEx
K
-
8/2/2019 mth121
78/124
--
WWRR: fxxf =)(KW
WW25.200.275.150.125.100.175.050.025.000.0x
50.141.132.122.112.100.187.071.050.000.0xy =
WW
0.5 1 1.5 2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
WW
0.5 1 1.5 2
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
KKK
WWEW)()( xfxf =0)()( =+ xfxffDxfDxKEW)()( xfxf =0)()( = xfxffDxfDxK
-
8/2/2019 mth121
79/124
--
WW
ERR: f1)( += xxfW
0211)1(1)()()(
0211)()()(
=+=++=
=+++=+
xxxxxxfxf
xxxfxf
RR: g2)( xxg =W
0)()()(2222 === xxxxxgxg
ENN: f1)( += xxfN= fDx
fDxKK
NN: g2)( xxg =KENN: fxxf =)(N= fDx
fDxKKERR: fxxf =)(
FKE
W
WFKE
WK
WK
WWRR: faxfy == )(aKKW
ER=fDKE}{aRf =KE)()( xfxf =KEK
-
8/2/2019 mth121
80/124
--
WWRR: f2)( =xfKW
-4 -2 2 4
-3
-2.5
-2
-1.5
-1
-0.5
0.5
1
WWRR: fxaxfy == )(0aKKW
ER=fDKER=fRKKE)()( xfxf =KEKK
WWRR: f
xxf 2)( =
K W
-3 -2 -1 1 2 3
-3
-2
-1
1
2
3
WWRR: fbxaxfy +== )(0a0bKK
WER=fDK
-
8/2/2019 mth121
81/124
--
ER=fRKKEK
EKK
WWRR: f5.12)( += xxfK
W
WWRR: fcxbxaxfy ++== 2)(0abcK
WER=fDKERfRKKE0=bK
E0== cbKWWRR: f5.12)( 2 += xxfKW
-2 -1 1 2
-3
-2
-1
1
2
WK
WRR: f1
12)(
+=
x
xxf
-2 -1 1 2 3
-3
-2
-1
1
2
3
-
8/2/2019 mth121
82/124
--
WE}1{R=fDK
E}2{R=fRKKEKE
K
WWRR: f
1
12)(
+=
x
xxfK
WW
WK
KFEFEW=180
W
KKWRR: f0>pW)()( pfpxf =+
fDxFpEK
WxWxsinxcosK
WWsinWRR:sin xy sin=KKK
WER=fDKE]1,1[=fRW1sin1 xKKKExx sin)sin( =K
-10 -5 5 10
-10
-7.5
-5
-2.5
2.5
5
7.5
10
-
8/2/2019 mth121
83/124
--
Exx sin)2sin( =+ 2KEK
- 2 p-
3 p
2
- p -p2
p2
p 3 p
2
2p
-1
-0.5
0.5
1
WWcosWRR:cos xy cos=KKK
WER=fDKE]1,1[=fRW1cos1 xKKKExx cos)cos( =KExx cos)2cos( =+ 2KEK
- 2 p-
3 p
2
- p -p2
p2
p 3p
2
2 p
-1
-0.5
0.5
1
WWtanWRR:tan x
xxy
cos
sintan ==KK
W
E},2
7,
2
5,
2
3,
2,
2
3,
2
5,{R LL
=fDK
ER=fRK
-
8/2/2019 mth121
84/124
--
Exx tan)tan( =KExx tan)tan( =+K
E??K
- 2 p-
3 p
2
- p -p
2
p2
p 3 p
2
2 p
-8
-6
-4
-2
2
4
6
8
WWcotWRR:cot xxxy sincoscot ==KKW
E},3,2,,0,,2,{R LL =fDKER=fRKExx cot)cot( =KExx tan)tan( =+K
Exx tan1
cot =KE??K
- 2 p-
3 p
2
- p -p2
p2
p 3 p
2
2 p
-8
-6
-4
-2
2
4
6
8
WWsecWRR:sec x
xycos
1sec ==KK
W
-
8/2/2019 mth121
85/124
--
E},2
7,
2
5,
2
3,
2,
2
3,
2
5,{R LL
=fDK
E),1[]1,()1,1(R == UfRW1sec x
1sec xKExx sec)sec( =KExx sec)2sec( =+ 2KE??K
WWcscWRR:csc xxy sin1csc ==KKW
E},3,2,,0,,2,{R LL =fDKE),1[]1,()1,1(R == UfRW1csc x1csc xKExx csc)csc( =KK
Exx csc)2csc( =+ 2KE??K
- 2 p -3 p
2
- p -p
2
p2
p 3 p2
2 p
-6
-4
-2
2
4
6
WWarcsinWRR:arcsin xy arcsin=yx sin=FKW1sinsinshiftKE
WExx =)arcsin(sinxx =)sin(arcsinK
- 2 p -3 p
2
- p -p
2
p2
p 3 p2
2 p
-6
-4
-2
2
4
6
-
8/2/2019 mth121
86/124
--
E]1,1[=fDK
E
=
2
,
2
fR
W2
arcsin2
xK
Exx arcsin)arcsin( =KEKKEK
WWarccos
WRR:arccos
xy arccos=yx cos=FKW1coscosshiftKE
WExx =)arccos(cosxx =)cos(arccosKE]1,1[=fDKE[ ],0=fRW xarccos0K
EKEKKEK
-1 -0.5 0.5 1
p4
p2
3 p
4
p
WWarctanWRR:arctan xy arctan=yx tan=FKW1tan tanshiftKE
W
-1 -0.5 0.5 1
-p
2
-p
4
p4
p2
-
8/2/2019 mth121
87/124
--
Exx =)arctan(tanxx =)tan(arctanER=fDK
E
=
2,
2
fR
W2
arctan2
-
8/2/2019 mth121
88/124
--
WWblogWRR:log bxy blog=ybx =1bKK
WExbxb =)(logxb xb =logKE),0( =fDKER=fRKKEKEWxxy elogln ==
71828.2e
KK xK
EWb
xxb
ln
lnlog =K
EWbxx eb ln=KEbKK
WWxy ln=K
W
5 10 15 20
-1
1
2
3
-
8/2/2019 mth121
89/124
--
WW
1)(,RR:)41)(,RR:)3
)(,NN:)2)(,NN:)1
33
33
=+=
==
xxffxxff
xxffxxff
WW
WFWE
1
1)(,RR:)6
1)(,RR:)5
1)(,RR:)41)(,RR:)3
)(,NN:)2)(,NN:)1
2
3
2
3
33
33
+
=
+=
=+=
==
x
xxff
x
xxff
xxffxxff
xxffxxff
Wfg ogf oW
xxfgxxff
xxfgxxff
xxfgexff
xxfgx
xff
xxfgxxff
x
===+=
==
==
==
)(,RR:)(,RR:)5
)(,RR:1)(,RR:)4
)(,RR:)(,RR:)3
)(,RR:1
)(,RR:)2
)(,RR:sin)(,RR:)1
2
2
2
W
xxffexff
xxffxxff
x==
==
)(,RR:)4)(,RR:)3
1)(,RR:)22sin)(,RR:)1
W
WFWEWW
xxffexff
xxffxxff
x==
==
)(,RR:)4)(,RR:)3
1)(,RR:)22sin)(,RR:)1
WW
-
8/2/2019 mth121
90/124
--
)13.0arcsin()810arctan)72
1
arccos)6)1arcsin()5
56cot)44
tan)360cos)22
sin)1
WW
xxxxxx 3cos2tan)44
1cos2sin)3
4
1cos)22sin)1 +
W
W
W
WW50log)4360loglog)316log)210log)1
21023=
-
8/2/2019 mth121
91/124
]
-
8/2/2019 mth121
92/124
--
WKK
WWoKoKoK
WK
-
8/2/2019 mth121
93/124
--
FENapierJohn
KKK
K
KW
WxnxnWnxxxx n = L
W10 =x
WW342 )3()3)2()23)1
W
27333)3()3
162222)2()2
9333)1
3
4
2
==
==
==
W0xnxnW
n
n
xx
1=
WW342 )3()3)2()23)1
-
8/2/2019 mth121
94/124
--
W
27
1
27
1
)3(
1)3()3
16
1
)2(
1)2()2
9
1
3
13)1
3
3
4
4
2
2
=
=
=
=
=
==
WxynWnxy =n
yyn
1W
nWxyy nn ==1
nWxyy nn ==1
K
K
K
WW
4
1
3
1
3
1
16)3)27()28)1
W
21616)3
327)27()2
288)1
44
1
33
1
33
1
==
==
==
Wxq
pqxq
pW
( )pqqp
xx =
-
8/2/2019 mth121
95/124
--
WW
2
3
4
5
3
2
25)316)2)27()1
W
( )
( )
( ) 12552525)3
32
1
2
1
16
1
16
116)2
9)3(27)27()1
332
3
554
4
54
5
2233
2
===
====
===
WKWW
2)3)22)1
W
10000
986968696.9)3
10000
314161416.3)2
10000
141424142.12)1
2 =
=
=
Wxq
pq
xW
q
p
xx
KWW
2)23)1
2W
825.822)2
728.433)1
1416.3
14142.12
-
8/2/2019 mth121
96/124
--
WxyW
( )
+
=
=
=
=
xx
x
xxx
yxyx
xx
)4
)3
)()2
)1
WW6
4 2
3 3
235
126
4
3
4
32
4
23
)4)315
45
2
3)2
5
10)1
yx
xy
zyx
zyx
x
y
y
x
y
x
yx
yx
W
( )
( )
( )
( )
( )
( ) ( )
( )
( )
( ) ( )( )
x
yyxyx
yx
yx
yx
yx
yx
xy
yx
xy
yx
xy
yx
xy
z
yxzyxzyx
zyx
zyx
y
xyx
yx
yx
x
y
y
x
y
x
x
y
y
x
y
x
y
xyxyx
y
y
x
x
yx
yx
9
2
91
2
3632
2
32
6
62
2
32
6
62
2
32
23
4
62
3
63
6
4
12
3
13
6
4 2
3 3
3
321)3(256
235
126
3
3
642
3522
4
32
4
3
22
22
4
3
4
32
2
2224213
4
23
4
23
)4
)3
33
532
253
53
25
2
3
15
45
2
3)2
222
5
10
5
10)1
===
=
=
=
=
=
===
==
=
=
====
y4 2yx
2xyK
KW
W1bxbWxbxfy == )(
-
8/2/2019 mth121
97/124
--
WW
22)()3)()22)()1
x
xxxfyxfyxfy ======
W
E2
1Wx
x
xxfy
====
2
1
2
12)(
EK
E2W( )xx
xfy 22)( 2 ===
Wuv1bW
( ) vuvu
vu
v
u
vuvu
u
bb
bb
b
bbb
b
=
=
=>
+
)4
)3
)2
0)1
WW
( ) ( )xxx
xxx 2
3
2
4
13
39)3
125
255)222)1
+
+
W
( ) ( )( )
( )
( ) ( ) ( )xx
xx
x
xx
x
xx
x
x
x
x
xx
x
xx
x
xx
xx
x
999939)3
25
1
78125
1
5
1
5
1
5
1
5
1
55
1
5
15
5
5
5
55
5
55
125
255)2
282222)1
12
2727
2772
72
39
2
39
22
33
22
3
2
444
3
44
13
=
=
=
==
======
==
+
+
+
+
+
+
+
+
+
-
8/2/2019 mth121
98/124
--
KW
W1bx
bWxxfy blog)( ==Wybx =KFxbxb =logxb xb =logE
Wuv1bW
( )
1log)6
01log)5
loglog)4
logloglog)3
loglog)(log)2
0log)1
=
=
=
=
+=
>=
b
uvu
vuv
u
vuvu
uuy
b
b
bv
b
bbb
bbb
b
WW
)23(log)23(log6log)2log10log2)3(log)1 222333 xxxx ++++
W( )
( )
=++
+=
+
=++=++
6
)23)(23(log)23(log)23(log6log)2
3100log
103
loglog10log)3(loglog10log2)3(log)1
2222
3
2
33
2
33333
xxxx
x
x
x
x
xxxx
W
WK
Wxlog
WW
1000log01.0log1.0log)21000log001.0log100log)1 ++
W
63)2()1(10log10log10log1000log01.0log1.0log)2
433210log10log10log1000log001.0log100log)1
321
332
=+=+=+
==+=+
-
8/2/2019 mth121
99/124
--
WFEeW
71828.21
1lim
+=
n
n n
e
WxlnK
WW
2ln)315.3ln)210ln)1
W
3466.02ln)3
1474.115.3ln)2
3026.210ln)1
W1bxW
b
xx
eb
b
bxx
ln
lnlog)2
)1ln
=
=
KWW
5log)32log)210log)1252
W
643.43466.0
6094.1
2ln
5ln5log)3
215.06094.1
3466.0
5ln
2ln2log)2
322.36931.0
3026.2
2ln
10ln10log)1
2
5
2
==
==
==
-
8/2/2019 mth121
100/124
--
KW
WuvW
>
>=
=
0
0lnln
v
u
vu
vu
K
WW
16
25
5
4
)392
1
)245)1
12
32
13
23
2
=
=
=
+
+
xx
x
x
x
W
954.03
25ln
4ln
25ln
4ln3
5ln
4ln234ln5ln)23(4ln5ln45)1 2323
+
=+=
====
xx
xxxx
018.0
)9ln22
1ln3(
2
1ln9ln3
2
1ln9ln3)9ln2
2
1ln3(
2
1ln9ln39ln2
2
1ln39ln39ln2
2
1ln
2
1ln3
9ln)32(2
1
ln)13(9ln2
1
ln92
1
)2
32
13
32
13
==
==+
=+=
=
+
+
xx
xxxx
xxx
x
x
x
1010)1(0120212
212
5
4ln
5
4ln2
12
5
4ln
5
4ln
12
5
4ln
5
4ln)12(
16
25ln
5
4ln
16
25
5
4)3
222
22
2
2
2
2
1212 22
==+=+=++=++
=+
=+
=+
=+=
=
++
xxxxxxx
xxxxxx
xx
xxxx
-
8/2/2019 mth121
101/124
--
WW
2ln)3ln()2ln()35)53ln()22ln)1 =++== xxxx
W
389.70
lnln2ln)1 22
2 =
>
=== ex
x
exexx
138.513
553
053
53ln)53ln(5)53ln()2
55
55
+=+=
>
===
exex
x
exexx
>+
>
=+
>+
>
=+
>+
>
=+
>+
>
=+
>+
>
=+
=++
03
02
08
03
02
026
03
02
26
03
02
2)3)(2(
03
02
2ln)3)(2ln(
2ln)3ln()2ln()3
222
x
x
xx
x
x
xx
x
x
xx
x
x
xx
x
x
xx
xx
W082 =+ xxKW033)814(14 22 >=== acb
W
372.212
331
2
372.312
331
2
2
1
+=
+=
==
a
bx
a
bx
FWEW021 xKW022 >x032 >+xKK
WW372.22 331 +
=x
WW
1)3(log)2(log)35)53(log)22log)1 2263 =++== xxxx
W
90
33lnln3ln2ln2
3ln
ln2log)1
22
3 =
>
===== x
x
xxx
xx
-
8/2/2019 mth121
102/124
--
67.25933
7781577763777653053
653
6ln)53ln(6ln5)53ln(56ln
)53ln(5)53(log)2
5
56
==+==
>
=
===
=
xxxx
x
xxx
x
>+
>
=+
>+
>
=+
>+
>
=+
>+
>
=+
>+
>
=+
=++
03
02
2)3)(2(
03
02
2)3)(2(
03
02
2ln)3)(2ln(
03
02
12ln
)3)(2ln(
03
02
1)3)(2(log
1)3(log)2(log)3
2
22
x
x
xx
x
x
xx
x
x
xx
x
x
xx
x
x
xx
xx
W372.212
331
+=xK
WW5.351.44 3)32)22)1 === xxxx
W
( )
( ) 346.13lnln7.33lnln
3lnln7.33lnln3333)3
184.12
lnln1.4
2lnln2lnln1.42lnln2)2
189.122
lnln
4
2lnln2lnln42lnln2)1
7.3
1
7.3
1
3ln7.3
3ln
7.3
3ln
7.37.35.32.05.35
1
5.35
1.4
1
1.4
12ln
1.4
2ln
1.4
2ln
1.41.4
444 2ln
4
2ln
4
2ln
44
=====
======
==
======
===
======
xexexexx
xxxxxxxxx
xex
exexxxxx
xxex
exexxxxx
{{
KK
-
8/2/2019 mth121
103/124
--
WW
6 8.1
52.4
3 5.14.5
4 2.14.27.1
6.4
7.23.5
2
2.31.4
2
32
63
32
632
3
5
6 182
3 62
)5)4
10
2)3
10
3
15
2)2)1
x
zy
xy
zx
y
xz
z
yx
zyx
y
yx
zxyx
y
x
yx
yx
WW
( ) ( )2323
2
21
1
1
21)38
263)2
22)1 xx
x
x
x
x
x
ee +
+
+
WW
)1(log)1(log)4loglog)3
ln4)1ln(2)3ln()2)12(log)1(log)1
32
92
31
5
222
+
++++++
xxee
xxxxxx
xx
WW26353 2 2)32)24)1 === xxxxx
WW26.135.03 5.2 2)32)24)1 === xxxxx
WW
5)436
2)333)252)1 31
21 ==== ++
++ xx
x
xxxxx e
WW
)12ln()6ln()34ln)2ln()3ln()20)2ln(ln)1 =+=+++=+ xxxxxx
WW
0)2(log)3(log)53log)4)32log()6log()3
2log)1(log)5(log)20)2(loglog)1
52
33322
=++=+=
=+++=+
xxxxx
xxxx
-
8/2/2019 mth121
104/124
]
-
8/2/2019 mth121
105/124
--
WK
WW
oKoKoKoK
WK
-
8/2/2019 mth121
106/124
--
EulerLeonhardKWK
KW
KW
012 =+x
12 =xK
W1=x1=x
1K
Wj012 =+xW
1=j
KWW
432 )3)2)1 jjj
W
1)1)(1()1()1()1()3
)1()1()1()1()2
1)1()1
2244
233
22
====
====
==
j
jjj
j
WzW
jbaz +=
ab1=jKW
),( baz =K
-
8/2/2019 mth121
107/124
--
azbKW
)Im()Re( zbza ==
KK
K
W
W
16)55.02)42)30)23)1 + jj
WW
)4,0(411611616)5
)5.0,2(5.02)4
)2,0(202)3
)0,0(000)2)0,3(033)1
====
=+
=+=
=+==+=
j
j
jj
jj
K
KW
WWFE
FEFKE
WW
)4()4()6)68(62)521)5()4
)21()3)5()23()2)47()32()1
jjjjjj
jjjjj
++++
+++
a
bbjaz +=
x
jy
-
8/2/2019 mth121
108/124
--
W
000)11()44(44)4()4()6
606)66()82(6862)68(62)5
36)21()15(21521)5()4
21)21()338)12()53(523)5()23()2
5)43()72(4732)47()32()1
=+=++=++=+
=+=++=++=++
=++=+=+
+=
+=++=++=+
=++=++=++
jjjjjj
jjjjjj
jjjjjj
jjjjjjjj
jjjjjj
WK
W
W FKE
WW
)4)(4()4()6)68)(62()5)21)(5()4
)5)(21()3)5)(23()2)47)(32()1
2jjjjjjj
jjjjjj
=+
++
W
jjjjj
jjjjjjj
jjjjj
jjjjjj
jjjjj
jjjjjj
jjjjj
jjjjjj
jjjjj
jjjjjjjjjjj
jjjjjj
81518164416
)()4()4()44()4)(4()4()6
602036601636481216
)66()86()62()82()68)(62()5
11321152105
)2()1()25()15()21)(5()4
11321152105
)2()52()1()51()5)(21()3
131321315210315
)2()52()3()53()5)(23()22921229141221814
)43()73()42()72()47)(32()1
2
2
2
2
2
2
2
==+=
+++==
+=++=++=
+++=+
==+=
+++=
==+=
+++=
+=++=++=
+++=+
+=++=++=
+++=+
WK
-
8/2/2019 mth121
109/124
--
WW
jzjzjz 181432 321 ==+=
W31323221 )2)1 zzzzzzzz +
W
jjjjjjj
jjjjjjzzzzzzz
57271557121536012)153)(4(
)18132)(4())181(32)(4()()1
2
3123221
==++==
+=++=+=+
jjjjjjj
jjjjjjzzzzzzz
104787210467210846)46)(181(
)324)(181())32(4)(181()()2
2
1233132
==++==
+=+==
W
WK
WW
2
36
2
24)3
2
3624)2
2
32)1
++
++ jjjjj
W
jjjjj
jjjjj
jj
5.015.1322
36
2
24)3
5.015.1322
3624)2
5.112
32)1
=++=
++
=++=
+
+=+
W
Wjbaz +=jbaz =K
-
8/2/2019 mth121
110/124
--
WW
)1()2()325)7()232)3()1 jjjjjj ++++
W
jjjjjjjjj
jjjjj
jjjjj
3113222)1)(2()1()2()3
)27()5(25725)7()2
2132332)3()1
2 +=++=++=+=
++=++=+
+=++=+++
WFKE
WW
)22()22()2)3()3()1 jjjj++
W
64242)2()2()22)(22()22()22()2
1019339)3)(3()3()3()1
222
2
=+===+=
=+=+=+=++
jjjjjj
jjjjjjj
W
WFEK
WW
j
j
j
j
j
+
6
32)3
32
5)2
43
1)1
W
jjj
j
jjj
j
j
j
j
j
j
jjj
j
j
j
jjj
j
j
j
j
j
j
jjj
j
j
j
j
j
j
jj
37
16
37
15
37
1615
136
31612
6
318212
6
6
6
32
6
32)3
13
17
13
7
13
177
94
177
94
31710
)3(2
321510
32
32
32
5
32
5)2
25
4
25
3
25
43
169
43
169
43
)4()3(
43
43
43
43
1
43
1)1
22
2
222
2
222
=
=+
+=
+=
+
+
=
+=+
=+
+=
+=
+++=
+
+
+=
+
+=+
=+
+=
+=
+=
+
+
=
-
8/2/2019 mth121
111/124
--
KW
Wjbaz +=
W22
baz +=
WW
jjjjjj 5432)45432)343)243)1 +++++
W
411354)3(25432)4
1024026265432)3
5254)3(43)2
525)4()3(43)1
2222
22
22
22
+=+++=++
==+=+=++
==+=+
==+=
jj
jjj
j
j
WK
WW)23)(23( jj +W
132323)23()23()23)(23(222
=+===+ jjjjjWz
W
)sin(cos jrz +=
rz
zzKK
W= rz
WW
905.0)41353)3302)200)1
=+= rjrz )sin(cos
zr=
jy
xcosr
sinr
-
8/2/2019 mth121
112/124
--
W
jjj
jjj
jjj
jj
5.0)10(5.0))90sin()90(cos(5.0905.0)4
2
23
2
23
2
1
2
13)135sin135(cos31353)3
321
232)30sin30(cos2302)2
0)11(0)0sin0(cos000)1
==+=
+=
+=+=
+=
+=+=
=+=+=
Wjbaz +== rz
W22 bazr +==KKKW0=r0=00=zF= 0zKE
0r0b
=
r
aarccosKK
0r0
-
8/2/2019 mth121
113/124
--
WnKW
nrr nn = )(
K
WW1010 )1()2)1()1 ++ jj
WEWj+1W
=
=
==
=+=+=
==
452
1arccosarccos01
0211
1,1
2222
r
ab
bar
ba
W==+ 4521 rjWW
===+ 450324510)2()452()1( 101010j
WWjjjj 32)10(32)450sin450(cos3245032)1( 10 =+=+==+
EW
=+ 4521 j
WW
===+ 45032
14510)2()452()1( 101010j
WW
jjjj32
1)10(
32
1))450sin()450(cos(
32
1450
32
1)1(
10 ==+==+
W sincos je j +=W jrer =.K
K
-
8/2/2019 mth121
114/124
--
WW
)90sin90(cos5.0
)30sin30(cos2)2)905.0)(302()1
+
+
j
j
W
jj
eeeejjjjj
2
3
2
1)120sin120(cos1
15.025.02)905.0)(302()1 12090309030
+=+=
=== +
jjj
eee
e
j
j jjjj
j
3222
3
2
1
4))60sin()60(cos(4
45.0
2
5.0
2
)90sin90(cos5.0
)30sin30(cos2)2
609030
90
30
=
=+=
===+
+
KW
WnKW
1...,,0360
)(
11
=
+= nkn
k
nrr nn
nkK
WW
1)2)1 63 == zjz
WEW
3
1
)( jz =
jK
WjW
==
=
=
=
-
8/2/2019 mth121
115/124
--
WW
2,1,0120301
2,1,0
3
360
3
901)901()( 3
1
3
1
3
1
=+=
=
+
===
kkz
kk
jz
WkW
jjzk
jjjzk
jjzk
2
1
2
3)210sin210(cos12101)2120(301:2
10)90sin90(cos1901)1120(301:1
2
1
2
3))30sin()30(cos(1301)0120(301:0
2
1
0
=+==+==
=+=+==+==
=+==+==
EW6
1
1=z
1KW1W
==
=
=
==
=+=+=
==
011
01
1
arccosarccos00
0101
01
2222
r
r
a
b
bar
ba
WW
5,4,3,2,1,0601
5,4,3,2,1,06
360
6
01)01(1 6
1
6
1
6
1
==
=
+
===
kkz
kk
z
WkW
-
8/2/2019 mth121
116/124
--
jjzk
jjzk
jzk
jjzk
jjzk
jzk
2
3
2
1)300sin300(cos130015601:5
2
3
2
1)240sin240(cos124014601:4
1)180sin180(cos118013601:3
2
3
2
1)120sin120(cos112012601:2
2
3
2
1)60sin60(cos16011601:1
1)0sin0(cos1010601:0
5
4
3
2
1
0
=+====
=+====
=+====
+=+====
+=+====
=+====
KWKK
Wz0abcW
02 =++ czbza
W
a
acbbz
a
acbbz
2
4
2
4 2
2
2
1
+=
=
WWW
06)301)2052)1222 ==++=++ zzzzzz
W
jj
a
acbbz
jj
a
acbbz
cba
212
42
2
)116(2
2
1162
2
162
12
)514(22
2
4
212
422
)116(22
1162
2
162
12
)514(22
2
4
521)1
22
2
22
1
+=+
=+
=+
=
+=
+=
+=
====
=
=
=
===
-
8/2/2019 mth121
117/124
--
jj
a
acbbz
jj
a
acbbz
cba
2
3
2
1
2
31
2
)13(1
2
131
2
31
12
)114(11
2
4
2
3
2
1
2
31
2
)13(1
2
1312
31
12
)114(11
2
4
111)2
22
2
22
1
+=+
=+
=+
=
+=
+=
+=
=
=
=
=
=
=
=
===
32
6
2
51
2
251
12
)614()1()1(
2
4
22
4
2
51
2251
12)614()1()1(
24
611)3
22
2
22
1
==+
=
+=
+=
+=
=
=
=
=
==
===
a
acbbz
aacbbz
cba
-
8/2/2019 mth121
118/124
--
WW
)3
12(2
3
1)3)35.0(6)44()2)2()52(2)1 jjjjjjjj ++++
WW
)1()3()2()3)42()532()2)65)(32()1 2 jjjjjjjj ++++
WW
)24)(31()33)22)1 jjjj ++
WW
jzjz =+= 32 21
W
21212
21 )2()2)()1 zzjzzjzz ++
WW
( )
)25)(2(31
)325
1
)21
3)1
2jj
j
j
j
j
j
j
j
j++
++
+
WW
jjj 22)41)33)24)1
WW
jjjj 33)425)3322)231)1 +
WW
( ) 675 )2()3)2()231)1 jjj ++WW
1)38)216)1 534 === zjzz
WW
022)301)2032)1 222 =++=++=++ zzzzzz
-
8/2/2019 mth121
119/124
--
E
JK
E
K
3) Gwyn Davies and Gordon Hick, Mathematics for scientific and technical students,
Addison Wesley Longman, Harlow, England, 1998.
4) Anders Hald, A History of Probability and Statistics and Their Applications before
1750, John Wiley and Sons, New York, 1989.
5) Alexander Schrijver, Theory of Linear and Integer Programming, John Wiley & Sons,
Chichester, England, 1986.
6) Seymour Lipschutz and Marc Lipson, Discrete Mathematics, McGraw-Hill, New York,
1997.
7) Peter Tebbutt, Basic Mathematics, John Wiley & Sons, Chichester, England, 1998.
-
8/2/2019 mth121
120/124
WKK
K
cbxax ++2
K
K
W
-
8/2/2019 mth121
121/124
K
2233
K
K
K
K
K22
W
K
K
K
-
8/2/2019 mth121
122/124
KK
W
KK
KK
W
K
K
K
K
W
K
K
-
8/2/2019 mth121
123/124
3K
K
K
-
8/2/2019 mth121
124/124
