x2 t08 03 inequalities & graphs (2012)
TRANSCRIPT
Inequalities & Graphs
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
Oblique asymptote:
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
2
42
2
2
xx
x
xOblique asymptote:
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
2
42
2
2
xx
x
xOblique asymptote:
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
2
42
2
2
xx
x
xOblique asymptote:
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
1or 2
012
02
2
12
2
2
2
xx
xx
xx
xx
x
x
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
1or 2
012
02
2
12
2
2
2
xx
xx
xx
xx
x
x
21or 2
12
2
xx
x
x
(ii) (1990)
xy graph heConsider t
(ii) (1990)
xy graph heConsider t
0 allfor increasing isgraph that theShow a) x
(ii) (1990)
xy graph heConsider t
0 allfor increasing isgraph that theShow a) x
0 when increasing is Curve dx
dy
(ii) (1990)
xy graph heConsider t
0 allfor increasing isgraph that theShow a) x
0 when increasing is Curve dx
dy
xdx
dy
xy
2
1
(ii) (1990)
xy graph heConsider t
0 allfor increasing isgraph that theShow a) x
0 when increasing is Curve dx
dy
xdx
dy
xy
2
1
0for 0 xdx
dy
(ii) (1990)
xy graph heConsider t
0 allfor increasing isgraph that theShow a) x
0 when increasing is Curve dx
dy
xdx
dy
xy
2
1
0for 0 xdx
dy
0,0at yx
(ii) (1990)
xy graph heConsider t
0 allfor increasing isgraph that theShow a) x
0 when increasing is Curve dx
dy
xdx
dy
xy
2
1
0for 0 xdx
dy
0,0at yx
0,0when yx
(ii) (1990)
xy graph heConsider t
0 allfor increasing isgraph that theShow a) x
0 when increasing is Curve dx
dy
xdx
dy
xy
2
1
0for 0 xdx
dy
0,0at yx
0,0when yx
0for increasing is curve x
n
nndxxn0
3
221
that;show Hence b)
n
nndxxn0
3
221
that;show Hence b)
n
nndxxn0
3
221
that;show Hence b)
curveunder Arearectanglesouter Area
;increasing is As
x
n
nndxxn0
3
221
that;show Hence b)
curveunder Arearectanglesouter Area
;increasing is As
x
n
dxxn0
21
n
nndxxn0
3
221
that;show Hence b)
curveunder Arearectanglesouter Area
;increasing is As
x
n
dxxn0
21
nn
xx
n
3
2
3
2
0
n
nndxxn0
3
221
that;show Hence b)
curveunder Arearectanglesouter Area
;increasing is As
x
n
dxxn0
21
nn
xx
n
3
2
3
2
0
n
nndxxn0
3
221
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
1
1..
SHL
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
1
1..
SHL
6
7
16
314..
SHR
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
1
1..
SHL
6
7
16
314..
SHR
SHRSHL ....
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
1
1..
SHL
6
7
16
314..
SHR
SHRSHL ....
Hence the result is true for n = 1
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
1
1..
SHL
6
7
16
314..
SHR
SHRSHL ....
Hence the result is true for n = 1
integer positive a is wherefor trueisresult theAssume kkn
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
1
1..
SHL
6
7
16
314..
SHR
SHRSHL ....
Hence the result is true for n = 1
integer positive a is wherefor trueisresult theAssume kkn
kk
k6
3421 i.e.
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
1
1..
SHL
6
7
16
314..
SHR
SHRSHL ....
Hence the result is true for n = 1
integer positive a is wherefor trueisresult theAssume kkn
kk
k6
3421 i.e.
1for trueisresult theProve kn
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
1
1..
SHL
6
7
16
314..
SHR
SHRSHL ....
Hence the result is true for n = 1
integer positive a is wherefor trueisresult theAssume kkn
kk
k6
3421 i.e.
1for trueisresult theProve kn
16
74121 Prove i.e.
k
kk
Proof:
Proof: 121121 kkk
Proof: 121121 kkk
16
34
kk
k
Proof: 121121 kkk
16
34
kk
k
6
16342
kkk
Proof: 121121 kkk
16
34
kk
k
6
16342
kkk
6
1692416 23
kkkk
Proof: 121121 kkk
16
34
kk
k
6
16342
kkk
6
1692416 23
kkkk
6
16118161 2
kkkk
Proof: 121121 kkk
16
34
kk
k
6
16342
kkk
6
1692416 23
kkkk
6
16118161 2
kkkk
6
1611412
kkk
Proof: 121121 kkk
16
34
kk
k
6
16342
kkk
6
1692416 23
kkkk
6
16118161 2
kkkk
6
1611412
kkk
6
161412
kkk
Proof: 121121 kkk
16
34
kk
k
6
16342
kkk
6
1692416 23
kkkk
6
16118161 2
kkkk
6
1611412
kkk
6
161412
kkk
6
174
6
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 =k
Since 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 =k
Since 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 the to1000021
estimate; toc) and b) Used)
Hence the result is true for n = k +1 if it is also true for n =k
Since 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 the to1000021
estimate; toc) and b) Used)
nn
nnn6
3421
3
2
Hence the result is true for n = k +1 if it is also true for n =k
Since 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 the to1000021
estimate; toc) and b) Used)
nn
nnn6
3421
3
2
100006
31000041000021 1000010000
3
2
Hence the result is true for n = k +1 if it is also true for n =k
Since 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 the to1000021
estimate; toc) and b) Used)
nn
nnn6
3421
3
2
100006
31000041000021 1000010000
3
2
6667001000021 666700
Hence the result is true for n = k +1 if it is also true for n =k
Since 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 the to1000021
estimate; toc) and b) Used)
nn
nnn6
3421
3
2
100006
31000041000021 1000010000
3
2
6667001000021 666700
hundrednearest the to6667001000021
(iii) Prove x > sinx , for x > 0
(iii) Prove x > sinx , for x > 0
1
y
x 2
y = x
y = sinx
(iii) Prove x > sinx , for x > 0
1
y
x 2
y = x
y = sinx
( )
'( ) 1
f x x
f x
(iii) Prove x > sinx , for x > 0
1
y
x 2
y = x
y = sinx
( )
'( ) 1
f x x
f x
( ) sin
'( ) cos
f x x
f x x
(iii) Prove x > sinx , for x > 0
1
y
x 2
y = x
y = sinx
( )
'( ) 1
f x x
f x
( ) sin
'( ) cos
f x x
f x x
for 0 , cos 12
x x
increases faster than siny x y x
sin , for 02
x x x
(iii) Prove x > sinx , for x > 0
1
y
x 2
y = x
y = sinx
( )
'( ) 1
f x x
f x
( ) sin
'( ) cos
f x x
f x x
for 0 , cos 12
x x
increases faster than siny x y x
sin , for 02
x x x
for , sin 12
x x
sin , for 2
x x x
(iii) Prove x > sinx , for x > 0
1
y
x 2
y = x
y = sinx
( )
'( ) 1
f x x
f x
( ) sin
'( ) cos
f x x
f x x
for 0 , cos 12
x x
increases faster than siny x y x
sin , for 02
x x x
for , sin 12
x x
sin , for 2
x x x
sin , for 0x x x
(iii) Prove x > sinx , for x > 0
1
y
x 2
y = x
y = sinx
( )
'( ) 1
f x x
f x
( ) sin
'( ) cos
f x x
f x x
for 0 , cos 12
x x
increases faster than siny x y x
sin , for 02
x x x
for , sin 12
x x
sin , for 2
x x x
sin , for 0x x x Exercise 10F