Page 59 - DMTH202_BASIC_MATHEMATICS_II

Basic Mathematics-II

Notes f(a + h) = sin(a+h)
f(a + h) = sin(a+2h)
f(a + 3h) = sin) a + 3h)
h
( f a n  1 ) sin(a n  1 )



h
b


h
)
Now,  f ( ) dxlinh | (0)  ( f a h  ( f a  2 )  ( f a  3 ) ....  ( f a n  1 )|
x
f

h
h

a
b



h
h

h
  sinxdx  lim [sin a  sin(a h ) sin(a  2 ) sin(a  3 ) ... sin(a  (n a ) )


h
a
  n  1    nh 
sin a    h  sin  

  2    2 
 lim h
h 0  h 
sin  
 2 
h nh h  b a 



 lim h sin a   sin  
h 0  h  2  2 
sin  
 2 
2 b a h  b a 




 lim h 2sin a   sin  

h 
h  0 sin   2  2 
 2 



 2a b a   b a 
 2sin   sin  
 2   2 


 b a   b a 
 2sin   sin    cos  cosb
 2   2 
 /4
2.  cosxdx
0
i. f(x) = cos x,  = 0, b =/4, nh = b – a /4
ii. f() = f(0) = cos 0
iii. f(a + h) = f(h) = cos h
iv. f(a+2h) = f(2h) = cos = 2h
v. f(a+3h) = f(3h) = cos 3 h
( f a n  1 )  ( f n ah  cos(n a )h



h
)
b
h
)
h
f
x
a
h
Now,  f ( ) lim ( )  ( f a h  ( f a 2 ) ( f a 3 ) .... ( f a n 1 )|




a
b
h
f
h
h
h
  cosxdx  lim [ (0)  f ( )  f (3 ) .........  ( f n  1) ]

a
  n  1    nh 

cos 0     sin  
  2   2 
 lim h
h 

h 0 sin  
 2 
h nh h nh

 lim h cos   sin

h 
h 0 sin   2 2
 2 
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