x2 t08 03 inequalities & graphs (2013)
DESCRIPTION
TRANSCRIPT
![Page 1: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/1.jpg)
Inequalities & Graphs
![Page 2: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/2.jpg)
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
![Page 3: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/3.jpg)
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
![Page 4: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/4.jpg)
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
![Page 5: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/5.jpg)
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
![Page 6: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/6.jpg)
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
Oblique asymptote:
![Page 7: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/7.jpg)
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
2
42
2
2
xx
x
xOblique asymptote:
![Page 8: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/8.jpg)
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
2
42
2
2
xx
x
xOblique asymptote:
![Page 9: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/9.jpg)
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
2
42
2
2
xx
x
xOblique asymptote:
![Page 10: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/10.jpg)
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
![Page 11: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/11.jpg)
Inequalities & Graphs 1
2 Solve e.g.
2
x
xi
![Page 12: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/12.jpg)
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
![Page 13: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/13.jpg)
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
![Page 14: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/14.jpg)
(ii) (1990)
xy graph heConsider t
![Page 15: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/15.jpg)
(ii) (1990)
xy graph heConsider t
0 allfor increasing isgraph that theShow a) x
![Page 16: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/16.jpg)
(ii) (1990)
xy graph heConsider t
0 allfor increasing isgraph that theShow a) x
0 when increasing is Curve dx
dy
![Page 17: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/17.jpg)
(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
![Page 18: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/18.jpg)
(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
![Page 19: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/19.jpg)
(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
![Page 20: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/20.jpg)
(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
![Page 21: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/21.jpg)
(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
![Page 22: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/22.jpg)
n
nndxxn0
3
221
that;show Hence b)
![Page 23: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/23.jpg)
n
nndxxn0
3
221
that;show Hence b)
![Page 24: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/24.jpg)
n
nndxxn0
3
221
that;show Hence b)
curveunder Arearectanglesouter Area
;increasing is As
x
![Page 25: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/25.jpg)
n
nndxxn0
3
221
that;show Hence b)
curveunder Arearectanglesouter Area
;increasing is As
x
n
dxxn0
21
![Page 26: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/26.jpg)
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
![Page 27: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/27.jpg)
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
![Page 28: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/28.jpg)
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
![Page 29: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/29.jpg)
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
![Page 30: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/30.jpg)
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
1
1..
SHL
![Page 31: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/31.jpg)
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
1
1..
SHL
6
7
16
314..
SHR
![Page 32: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/32.jpg)
1 integers allfor 6
3421
that;show toinduction almathematic Usec)
nnn
n
Test: n = 1
1
1..
SHL
6
7
16
314..
SHR
SHRSHL ....
![Page 33: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/33.jpg)
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
![Page 34: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/34.jpg)
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
![Page 35: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/35.jpg)
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.
![Page 36: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/36.jpg)
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
![Page 37: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/37.jpg)
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
![Page 38: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/38.jpg)
Proof:
![Page 39: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/39.jpg)
Proof: 121121 kkk
![Page 40: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/40.jpg)
Proof: 121121 kkk
16
34
kk
k
![Page 41: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/41.jpg)
Proof: 121121 kkk
16
34
kk
k
6
16342
kkk
![Page 42: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/42.jpg)
Proof: 121121 kkk
16
34
kk
k
6
16342
kkk
6
1692416 23
kkkk
![Page 43: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/43.jpg)
Proof: 121121 kkk
16
34
kk
k
6
16342
kkk
6
1692416 23
kkkk
6
16118161 2
kkkk
![Page 44: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/44.jpg)
Proof: 121121 kkk
16
34
kk
k
6
16342
kkk
6
1692416 23
kkkk
6
16118161 2
kkkk
6
1611412
kkk
![Page 45: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/45.jpg)
Proof: 121121 kkk
16
34
kk
k
6
16342
kkk
6
1692416 23
kkkk
6
16118161 2
kkkk
6
1611412
kkk
6
161412
kkk
![Page 46: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/46.jpg)
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
![Page 47: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/47.jpg)
Hence the result is true for n = k +1 if it is also true for n =k
![Page 48: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/48.jpg)
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
![Page 49: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/49.jpg)
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)
![Page 50: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/50.jpg)
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
![Page 51: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/51.jpg)
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
![Page 52: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/52.jpg)
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
![Page 53: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/53.jpg)
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
![Page 54: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/54.jpg)
(iii) Prove x > sinx , for x > 0
![Page 55: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/55.jpg)
(iii) Prove x > sinx , for x > 0
1
y
x 2
y = x
y = sinx
![Page 56: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/56.jpg)
(iii) Prove x > sinx , for x > 0
1
y
x 2
y = x
y = sinx
( )
'( ) 1
f x x
f x
![Page 57: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/57.jpg)
(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
![Page 58: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/58.jpg)
(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
![Page 59: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/59.jpg)
(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
![Page 60: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/60.jpg)
(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
![Page 61: X2 t08 03 inequalities & graphs (2013)](https://reader033.vdocuments.net/reader033/viewer/2022051610/5481f58fb079591a0c8b4672/html5/thumbnails/61.jpg)
(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