Inequalities & Graphs

Inequalities & Graphs 1

2 Solve e.g.

2

xxi

Inequalities & Graphs 1

2 Solve e.g.

2

xxi

Inequalities & Graphs 1

2 Solve e.g.

2

xxi

Inequalities & Graphs 1

2 Solve e.g.

2

xxi

Inequalities & Graphs 1

2 Solve e.g.

2

xxi Oblique asymptote:

Inequalities & Graphs 1

2 Solve e.g.

2

xxi

242

2

2

xx

xxOblique asymptote:

Inequalities & Graphs 1

2 Solve e.g.

2

xxi

242

2

2

xx

xxOblique asymptote:

Inequalities & Graphs 1

2 Solve e.g.

2

xxi

242

2

2

xx

xxOblique asymptote:

Inequalities & Graphs 1

2 Solve e.g.

2

xxi

Inequalities & Graphs 1

2 Solve e.g.

2

xxi

Inequalities & Graphs 1

2 Solve e.g.

2

xxi

1or 2012

022

12

2

2

2

xxxx

xxxx

xx

Inequalities & Graphs 1

2 Solve e.g.

2

xxi

1or 2012

022

12

2

2

2

xxxx

xxxx

xx

21or 2

12

2

xxxx

(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