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Ko‘rsatkichli tenglama va tengsizliklar




1. a f (x)  1  f (x)  0.



2. f (x)g ( x) 1
f (x)  1,
g(x)  R,


yoki
f (x)  0,
g(x)  0.




  1. a f ( x ) ag ( x)

 
f (x)  g(x) (a  0) .


  1. a f ( x ) bg ( x)

f (x)  g(x) loga b
(a,b  0) .




  1. a f ( x ) ag ( x)

0  a  1,


f (x)  g(x),


yoki
a  1,
f (x)  g(x).

 



6. a f (x)b, 0  a  1
b  0, a  1,


yoki
0  a  1, b  0,






a

a
f (x)  log b, f (x)  log b.





7. a f (x)b
b  0,
a  1,


yoki
b  0, 0  a  1,






a

a
f (x)  log b, f (x)  log b.

Logarifm va uning asosiy xossalari


b  loga N
( a  0,
a  1,
N  0 ) 
N ab .


  1. loga 1  0 , loga a  1,

alogaN
N ;

  1. log (bc)  log b  log

c , log
b  log b  log c ;




a a a
  a a

a c
 


  1. a
    log n

bm m log b ,
n a
loga b
1 ;
logb a

  1. log c

loga c ,

log
b logc b ;



a

c
ba 1  log b a log a


  1. b

    b
    loga b  logc d  loga d  logc b ,

alog c
clog a ;

  1. log10 x  lg x

  1. loge x  ln x
  • natural logarifm;


  1. Agar

a  1, 0  b  1
yoki 0  a  1,b  1
bo‘lsa loga b  0 ;

  1. Agar

a  1,
b  1
yoki 0  a  1,0  b  1
bo‘lsa loga b  0 ;

  1. Agar

b a  1 bo‘lsa, logb p  loga p
bo‘ladi ( p  0 );

  1. Agar 0  a b  1 bo‘lsa, logb p  loga p

bo‘ladi ( p  0 );


a b  0
bo‘lsin.

agar 0  p  1 bo‘lsa, log p a  log p b
bo‘ladi,

agar
p  1 bo‘lsa, log p a  log p b
bo‘ladi.



  1. loga

f (x)  b

Logarifmik tenglama va tengsizliklar





f (x)  ba .
f (x)  0, 0  a  1,





  1. log f ( x) a b

f (x)  1,


1
a  0,

f (x)  ab .




  1. log



f (x)  log


g(x)
0  a  1,
f (x)  0,

a a




0 
f (x)  g(x).
f (x)  1,

  1. log f ( x) g(x)  b


g(x) 
f b (x).




  1. logϕ ( x)

f (x)  logϕ ( x)


g(x)
0  ϕ(x)  1,


f (x)  g(x).
f (x)  0,





  1. log



f (x)  log
0  ϕ(x)  1,
g(x)  f (x)  0, yoki
ϕ(x)  1,
g (x)  0,

ϕ ( x)
ϕ ( x)

f (x)  g(x). f (x)  g(x).
 

0 
f (x)  1,



 
f (x)  1,

  1. log



f ( x)
g(x)  b
g(x)  0, yoki
g(x) 


f b (x).



g(x)  f b (x).

TRIGONOMETRIYA





α   180 α

π


рад ,
αрад
π
180
α 0



Sinα yα , Соsα xα ,

tgα


Sec
yα ,
xα
1 ,
xα
ctgα


С s сα
xα ,
yα
1 .
yα

1 rad  5701715; π  3,141592 . . .


Trigonometrik funksiyalarning choraklardagi ishoralari



Cosx Sinx tgx,ctgx


Asosiy trigonometrik ayniyatlar





    1. Sin2 x Cos2 x  1. 4. tgx

Sinx .
Cosx

    1. tgx ctgx  1. 5.

ctgx Cosx .
Sinx

3. 1  tg 2x
1



Cos2x
. 6. 1  ctg 2 x
1 .
Sin2x

Qo‘shish formulalari


sin(α β )  sinα cos β  cosα sin β ;


cos(α β )  cosα cos β ∓ sinα sin β ;



tg(α β ) 
tgα tgβ ;
1 ∓ tgαtgβ
сtg(α β ) сtgαсtgβ 1 .
сtgα сtgβ



Ikkilangan va uchlangan burchaklar


sin 2x  2sin x cos x ; sin 2x  2sin x cos x ;



tg 2x 2tgx ;
1  tg 2 x
ctg 2x
ctg 2x 1

;


2ctgx

sin 3x  3sin x cos2 x  sin3 x  3sin x  4sin3 x ;
cos 3x  cos3 x  3cos x sin2 x  4cos3 x  3cos x ;

tg3x tgx tg 2x
1  tgxtg 2x
3tgx tg3x
1  3tg 2x .



Yig‘indini ko‘paytmaga keltirish





  1. Sinx Siny  2Sin x y Cos x y .

2 2

  1. Cosx Cosy  2Cos x y Cos x y .

2 2

  1. Cosx Cosy  2Sin x y Sin x y .

2 2




  1. Cosx Sinx

2Sin π x  2Cos π x .

4   4
   




  1. Cosx Sinx

2Cos π x  2Sin π x .

4   4
   




  1. pCosx qSinx rSin(z x) , r



, Sinz
p ,Cosz q .
r r

  1. tgx tgy Sin x y ,

CosxCosy

  1. tgx ctgy Cos x y ,

CosxSiny
ctgx ctgy Sin x y .
SinxSiny
tgx ctgy   Cos x y .
CosxSiny

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