1. Inequalities & Graphs

2. Inequalities & Graphs
  1
2
Solvee.g.
2

x
x
i

3. Inequalities & Graphs
  1
2
Solvee.g.
2

x
x
i

4. Inequalities & Graphs
  1
2
Solvee.g.
2

x
x
i

5. Inequalities & Graphs
  1
2
Solvee.g.
2

x
x
i

6. Inequalities & Graphs
  1
2
Solvee.g.
2

x
x
i Oblique asymptote:

7. Inequalities & Graphs
  1
2
Solvee.g.
2

x
x
i
2
4
2
2
2


 x
x
x
x
Oblique asymptote:

8. Inequalities & Graphs
  1
2
Solvee.g.
2

x
x
i
2
4
2
2
2


 x
x
x
x
Oblique asymptote:

9. Inequalities & Graphs
  1
2
Solvee.g.
2

x
x
i
2
4
2
2
2


 x
x
x
x
Oblique asymptote:

10. Inequalities & Graphs
  1
2
Solvee.g.
2

x
x
i

11. Inequalities & Graphs
  1
2
Solvee.g.
2

x
x
i

12. Inequalities & Graphs
  1
2
Solvee.g.
2

x
x
i
  
1or2
012
02
2
1
2
2
2
2






xx
xx
xx
xx
x
x

13. Inequalities & Graphs
  1
2
Solvee.g.
2

x
x
i
  
1or2
012
02
2
1
2
2
2
2






xx
xx
xx
xx
x
x
21or2
1
2
2



xx
x
x

14. (ii) (1990)
xy graphheConsider t

15. (ii) (1990)
xy graphheConsider t
0allforincreasingisgraphthat theShowa) x

16. (ii) (1990)
xy graphheConsider t
0allforincreasingisgraphthat theShowa) x
0whenincreasingisCurve 
dx
dy

17. (ii) (1990)
xy graphheConsider t
0allforincreasingisgraphthat theShowa) x
0whenincreasingisCurve 
dx
dy
xdx
dy
xy
2
1



18. (ii) (1990)
xy graphheConsider t
0allforincreasingisgraphthat theShowa) x
0whenincreasingisCurve 
dx
dy
xdx
dy
xy
2
1


0for0  x
dx
dy

19. (ii) (1990)
xy graphheConsider t
0allforincreasingisgraphthat theShowa) x
0whenincreasingisCurve 
dx
dy
xdx
dy
xy
2
1


0for0  x
dx
dy
0,0at  yx

20. (ii) (1990)
xy graphheConsider t
0allforincreasingisgraphthat theShowa) x
0whenincreasingisCurve 
dx
dy
xdx
dy
xy
2
1


0for0  x
dx
dy
0,0at  yx
0,0when  yx

21. (ii) (1990)
xy graphheConsider t
0allforincreasingisgraphthat theShowa) x
0whenincreasingisCurve 
dx
dy
xdx
dy
xy
2
1


0for0  x
dx
dy
0,0at  yx
0,0when  yx
0forincreasingiscurve  x

22.  
n
nndxxn
0
3
2
21
that;showHenceb)


23.  
n
nndxxn
0
3
2
21
that;showHenceb)


24.  
n
nndxxn
0
3
2
21
that;showHenceb)

curveunderArearectanglesouterArea
;increasingisAs

x

25.  
n
nndxxn
0
3
2
21
that;showHenceb)

curveunderArearectanglesouterArea
;increasingisAs

x

n
dxxn
0
21 

26.  
n
nndxxn
0
3
2
21
that;showHenceb)

curveunderArearectanglesouterArea
;increasingisAs

x

n
dxxn
0
21 
nn
xx
n
3
2
3
2
0






27.  
n
nndxxn
0
3
2
21
that;showHenceb)

curveunderArearectanglesouterArea
;increasingisAs

x

n
dxxn
0
21 
nn
xx
n
3
2
3
2
0





 
n
nndxxn
0
3
2
21 

28. 1integersallfor
6
34
21
that;showtoinductionalmathematicUsec)


 nn
n
n

29. 1integersallfor
6
34
21
that;showtoinductionalmathematicUsec)


 nn
n
n
Test: n = 1

30. 1integersallfor
6
34
21
that;showtoinductionalmathematicUsec)


 nn
n
n
Test: n = 1
1
1..

SHL

31. 1integersallfor
6
34
21
that;showtoinductionalmathematicUsec)


 nn
n
n
Test: n = 1
1
1..

SHL
 
6
7
1
6
314
..


SHR

32. 1integersallfor
6
34
21
that;showtoinductionalmathematicUsec)


 nn
n
n
Test: n = 1
1
1..

SHL
 
6
7
1
6
314
..


SHR
SHRSHL .... 

33. 1integersallfor
6
34
21
that;showtoinductionalmathematicUsec)


 nn
n
n
Test: n = 1
1
1..

SHL
 
6
7
1
6
314
..


SHR
SHRSHL .... 
Hence the result is true for n = 1

34. 1integersallfor
6
34
21
that;showtoinductionalmathematicUsec)


 nn
n
n
Test: n = 1
1
1..

SHL
 
6
7
1
6
314
..


SHR
SHRSHL .... 
Hence the result is true for n = 1
integerpositiveaiswherefortrueisresulttheAssume kkn 

35. 1integersallfor
6
34
21
that;showtoinductionalmathematicUsec)


 nn
n
n
Test: n = 1
1
1..

SHL
 
6
7
1
6
314
..


SHR
SHRSHL .... 
Hence the result is true for n = 1
integerpositiveaiswherefortrueisresulttheAssume kkn 
k
k
k
6
34
21i.e.

 

36. 1integersallfor
6
34
21
that;showtoinductionalmathematicUsec)


 nn
n
n
Test: n = 1
1
1..

SHL
 
6
7
1
6
314
..


SHR
SHRSHL .... 
Hence the result is true for n = 1
integerpositiveaiswherefortrueisresulttheAssume kkn 
k
k
k
6
34
21i.e.

 
1fortrueisresulttheProve  kn

37. 1integersallfor
6
34
21
that;showtoinductionalmathematicUsec)


 nn
n
n
Test: n = 1
1
1..

SHL
 
6
7
1
6
314
..


SHR
SHRSHL .... 
Hence the result is true for n = 1
integerpositiveaiswherefortrueisresulttheAssume kkn 
k
k
k
6
34
21i.e.

 
1fortrueisresulttheProve  kn
1
6
74
121Provei.e. 

 k
k
k

38. Proof:

39. Proof: 121121  kkk 

40. Proof: 121121  kkk 
1
6
34


 kk
k

41. Proof: 121121  kkk 
1
6
34


 kk
k
 
6
1634
2


kkk

42. Proof: 121121  kkk 
1
6
34


 kk
k
 
6
1634
2


kkk
6
1692416 23


kkkk

43. Proof: 121121  kkk 
1
6
34


 kk
k
 
6
1634
2


kkk
6
1692416 23


kkkk
  
6
16118161 2


kkkk

44. Proof: 121121  kkk 
1
6
34


 kk
k
 
6
1634
2


kkk
6
1692416 23


kkkk
  
6
16118161 2


kkkk
  
6
161141
2


kkk

45. Proof: 121121  kkk 
1
6
34


 kk
k
 
6
1634
2


kkk
6
1692416 23


kkkk
  
6
16118161 2


kkkk
  
6
161141
2


kkk
  
6
16141
2


kkk

46. Proof: 121121  kkk 
1
6
34


 kk
k
 
6
1634
2


kkk
6
1692416 23


kkkk
  
6
16118161 2


kkkk
  
6
161141
2


kkk
  
6
16141
2


kkk
   
   
6
174
6
16114




kk
kkk

47. Hence the result is true for n = k +1 if it is also true for n =k

48. 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

49. 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
hundrednearesttheto1000021
estimate;toc)andb)Used)
 

50. 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
hundrednearesttheto1000021
estimate;toc)andb)Used)
 
n
n
nnn
6
34
21
3
2 
 

51. 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
hundrednearesttheto1000021
estimate;toc)andb)Used)
 
n
n
nnn
6
34
21
3
2 
 
 
 
10000
6
3100004
10000211000010000
3
2 
 

52. 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
hundrednearesttheto1000021
estimate;toc)andb)Used)
 
n
n
nnn
6
34
21
3
2 
 
 
 
10000
6
3100004
10000211000010000
3
2 
 
6667001000021666700  

53. 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
hundrednearesttheto1000021
estimate;toc)andb)Used)
 
n
n
nnn
6
34
21
3
2 
 
 
 
10000
6
3100004
10000211000010000
3
2 
 
6667001000021666700  
hundrednearesttheto6667001000021  

54. (iii) Prove x > sinx , for x > 0

55. (iii) Prove x > sinx , for x > 0
1
y
x 2
y = x
y = sinx

56. (iii) Prove x > sinx , for x > 0
1
y
x 2
y = x
y = sinx
( )
'( ) 1
f x x
f x



57. (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



58. (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 1
2
x x

  
increases faster than siny x y x  
sin , for 0
2
x x x

  

59. (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 1
2
x x

  
increases faster than siny x y x  
sin , for 0
2
x x x

  
for , sin 1
2
x x

 
sin , for
2
x x x

  

60. (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 1
2
x x

  
increases faster than siny x y x  
sin , for 0
2
x x x

  
for , sin 1
2
x x

 
sin , for
2
x x x

  
sin , for 0x x x  

61. (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 1
2
x x

  
increases faster than siny x y x  
sin , for 0
2
x x x

  
for , sin 1
2
x x

 
sin , for
2
x x x

  
sin , for 0x x x   Exercise 10F