x2 t08 03 inequalities & graphs (2011)

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

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

Exercise 10F