x2 t08 03 inequalities & graphs
DESCRIPTION
TRANSCRIPT
Inequalities & Graphs
Inequalities & Graphs 1
2 Solve e.g.
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Inequalities & Graphs 1
2 Solve e.g.
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Inequalities & Graphs 1
2 Solve e.g.
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Inequalities & Graphs 1
2 Solve e.g.
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Inequalities & Graphs 1
2 Solve e.g.
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xxi Oblique asymptote:
Inequalities & Graphs 1
2 Solve e.g.
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xxOblique asymptote:
Inequalities & Graphs 1
2 Solve e.g.
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xxOblique asymptote:
Inequalities & Graphs 1
2 Solve e.g.
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242
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xxOblique asymptote:
Inequalities & Graphs 1
2 Solve e.g.
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Inequalities & Graphs 1
2 Solve e.g.
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Inequalities & Graphs 1
2 Solve e.g.
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1or 2012
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xxxx
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Inequalities & Graphs 1
2 Solve e.g.
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1or 2012
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21or 2
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(ii) (1990)
xy graph heConsider t
(ii) (1990)
xy graph heConsider t0 allfor increasingisgraph that theShow a) x
(ii) (1990)
xy graph heConsider t0 allfor increasingisgraph that theShow a) x
0 when increasing is Curve dxdy
(ii) (1990)
xy graph heConsider t0 allfor increasingisgraph that theShow a) x
0 when increasing is Curve dxdy
xdxdy
xy
21
(ii) (1990)
xy graph heConsider t0 allfor increasingisgraph that theShow a) x
0 when increasing is Curve dxdy
xdxdy
xy
21
0for 0 xdxdy
(ii) (1990)
xy graph heConsider t0 allfor increasingisgraph that theShow a) x
0 when increasing is Curve dxdy
xdxdy
xy
21
0for 0 xdxdy
0,0at yx
(ii) (1990)
xy graph heConsider t0 allfor increasingisgraph that theShow a) x
0 when increasing is Curve dxdy
xdxdy
xy
21
0for 0 xdxdy
0,0at yx0,0when yx
(ii) (1990)
xy graph heConsider t0 allfor increasingisgraph that theShow a) x
0 when increasing is Curve dxdy
xdxdy
xy
21
0for 0 xdxdy
0,0at yx0,0when yx
0for increasing iscurve x
n
nndxxn0 3
221
that;showHence b)
n
nndxxn0 3
221
that;showHence b)
n
nndxxn0 3
221
that;showHence b)
curveunder Arearectanglesouter Area;increasing is As
x
n
nndxxn0 3
221
that;showHence b)
curveunder Arearectanglesouter Area;increasing is As
x
n
dxxn0
21
n
nndxxn0 3
221
that;showHence b)
curveunder Arearectanglesouter Area;increasing is As
x
n
dxxn0
21
nn
xxn
3232
0
n
nndxxn0 3
221
that;showHence b)
curveunder Arearectanglesouter Area;increasing is As
x
n
dxxn0
21
nn
xxn
3232
0
n
nndxxn0 3
221
1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnn
1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
11..
SHL
1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
11..
SHL
67
16
314..
SHR
1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
11..
SHL
67
16
314..
SHR
SHRSHL ....
1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
11..
SHL
67
16
314..
SHR
SHRSHL .... Hence the result is true for n = 1
1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
11..
SHL
67
16
314..
SHR
SHRSHL .... Hence the result is true for n = 1
integerpositiveais wherefor trueisresult theAssume kkn
1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
11..
SHL
67
16
314..
SHR
SHRSHL .... Hence the result is true for n = 1
integerpositiveais wherefor trueisresult theAssume kkn
kkk6
3421 i.e.
1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
11..
SHL
67
16
314..
SHR
SHRSHL .... Hence the result is true for n = 1
integerpositiveais wherefor trueisresult theAssume kkn
kkk6
3421 i.e.
1for trueisresult theProve kn
1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
11..
SHL
67
16
314..
SHR
SHRSHL .... Hence the result is true for n = 1
integerpositiveais wherefor trueisresult theAssume kkn
kkk6
3421 i.e.
1for trueisresult theProve kn
16
74121 Prove i.e.
kkk
Proof:
Proof: 121121 kkk
Proof: 121121 kkk 1
634
kkk
Proof: 121121 kkk 1
634
kkk
6
1634 2
kkk
Proof: 121121 kkk 1
634
kkk
6
1634 2
kkk
61692416 23
kkkk
Proof: 121121 kkk 1
634
kkk
6
1634 2
kkk
61692416 23
kkkk
6
16118161 2
kkkk
Proof: 121121 kkk 1
634
kkk
6
1634 2
kkk
61692416 23
kkkk
6
16118161 2
kkkk
6
161141 2
kkk
Proof: 121121 kkk 1
634
kkk
6
1634 2
kkk
61692416 23
kkkk
6
16118161 2
kkkk
6
161141 2
kkk
6
16141 2
kkk
Proof: 121121 kkk 1
634
kkk
6
1634 2
kkk
61692416 23
kkkk
6
16118161 2
kkkk
6
161141 2
kkk
6
16141 2
kkk
6
1746
16114
kk
kkk
Hence the result is true for n = k +1 if it is also true for n =k
Hence the result is true for n = k +1 if it is also true for n =kSince the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n
Hence the result is true for n = k +1 if it is also true for n =kSince the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n
hundrednearest theto1000021 estimate; toc) andb) Used)
Hence the result is true for n = k +1 if it is also true for n =kSince the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n
hundrednearest theto1000021 estimate; toc) andb) Used)
nnnnn6
3421 32
Hence the result is true for n = k +1 if it is also true for n =kSince the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n
hundrednearest theto1000021 estimate; toc) andb) Used)
nnnnn6
3421 32
100006
31000041000021 100001000032
Hence the result is true for n = k +1 if it is also true for n =kSince the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n
hundrednearest theto1000021 estimate; toc) andb) Used)
nnnnn6
3421 32
100006
31000041000021 100001000032
6667001000021 666700
Hence the result is true for n = k +1 if it is also true for n =kSince the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n
hundrednearest theto1000021 estimate; toc) andb) Used)
nnnnn6
3421 32
100006
31000041000021 100001000032
6667001000021 666700 hundrednearest theto6667001000021
Hence the result is true for n = k +1 if it is also true for n =kSince the result is true for n = 1 then it is also true for n =1+1 i.e. n=2, and since the result is true for n = 2 then it is also true for n =2+1 i.e. n=3, and so on for all positive integral values of n
hundrednearest theto1000021 estimate; toc) andb) Used)
nnnnn6
3421 32
100006
31000041000021 100001000032
6667001000021 666700 hundrednearest theto6667001000021
Exercise 10F