x2 t08 03 inequalities & graphs
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TRANSCRIPT
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Inequalities & Graphs
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Inequalities & Graphs 1
2 Solve e.g.
2
xxi
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Inequalities & Graphs 1
2 Solve e.g.
2
xxi
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Inequalities & Graphs 1
2 Solve e.g.
2
xxi
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Inequalities & Graphs 1
2 Solve e.g.
2
xxi
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Inequalities & Graphs 1
2 Solve e.g.
2
xxi Oblique asymptote:
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Inequalities & Graphs 1
2 Solve e.g.
2
xxi
242
2
2
xx
xxOblique asymptote:
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Inequalities & Graphs 1
2 Solve e.g.
2
xxi
242
2
2
xx
xxOblique asymptote:
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Inequalities & Graphs 1
2 Solve e.g.
2
xxi
242
2
2
xx
xxOblique asymptote:
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Inequalities & Graphs 1
2 Solve e.g.
2
xxi
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Inequalities & Graphs 1
2 Solve e.g.
2
xxi
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Inequalities & Graphs 1
2 Solve e.g.
2
xxi
1or 2012
022
12
2
2
2
xxxx
xxxx
xx
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Inequalities & Graphs 1
2 Solve e.g.
2
xxi
1or 2012
022
12
2
2
2
xxxx
xxxx
xx
21or 2
12
2
xxxx
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(ii) (1990)
xy graph heConsider t
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(ii) (1990)
xy graph heConsider t0 allfor increasingisgraph that theShow a) x
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(ii) (1990)
xy graph heConsider t0 allfor increasingisgraph that theShow a) x
0 when increasing is Curve dxdy
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(ii) (1990)
xy graph heConsider t0 allfor increasingisgraph that theShow a) x
0 when increasing is Curve dxdy
xdxdy
xy
21
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(ii) (1990)
xy graph heConsider t0 allfor increasingisgraph that theShow a) x
0 when increasing is Curve dxdy
xdxdy
xy
21
0for 0 xdxdy
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(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
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(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
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(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
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n
nndxxn0 3
221
that;showHence b)
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n
nndxxn0 3
221
that;showHence b)
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n
nndxxn0 3
221
that;showHence b)
curveunder Arearectanglesouter Area;increasing is As
x
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n
nndxxn0 3
221
that;showHence b)
curveunder Arearectanglesouter Area;increasing is As
x
n
dxxn0
21
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n
nndxxn0 3
221
that;showHence b)
curveunder Arearectanglesouter Area;increasing is As
x
n
dxxn0
21
nn
xxn
3232
0
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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
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1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnn
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1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
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1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
11..
SHL
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1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
11..
SHL
67
16
314..
SHR
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1 integers allfor 6
3421
that;showtoinduction almathematic Usec)
nnnnTest: n = 1
11..
SHL
67
16
314..
SHR
SHRSHL ....
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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
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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
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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.
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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
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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
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Proof:
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Proof: 121121 kkk
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Proof: 121121 kkk 1
634
kkk
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Proof: 121121 kkk 1
634
kkk
6
1634 2
kkk
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Proof: 121121 kkk 1
634
kkk
6
1634 2
kkk
61692416 23
kkkk
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Proof: 121121 kkk 1
634
kkk
6
1634 2
kkk
61692416 23
kkkk
6
16118161 2
kkkk
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Proof: 121121 kkk 1
634
kkk
6
1634 2
kkk
61692416 23
kkkk
6
16118161 2
kkkk
6
161141 2
kkk
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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
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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
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Hence the result is true for n = k +1 if it is also true for n =k
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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
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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)
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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
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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
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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
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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
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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