Microsoft Word Formula kitobcha

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.

3. 
   _a f ( x ) _{ a}g ( x)
   

 
 f (x)  g(x) (a  0) .

3. 
   _a f ( x ) _{ b}g ( x)
   

 f (x)  g(x) log_a b
(a,b  0) .

3. 
   _a f ( x ) _{ a}g ( 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  log_a N
( a  0,
a  1,
N  0 ) 
N  a^b .

1. 
   log_a 1  0 , log_a a  1,
   

_alog_a^N
 N ;

2. 
   log (bc)  log b  log
   

c , log
^{ b }  log b  log c ;

a a a
  a a

a _c
 

3. 
   
   a
   log _n
   

b^m  ^m log b ,
_n a
log_a b 
1 _;
log_b a

4. 
   log c 
   

log_a c _,

log
_{b } log_c b _;

a

c
^ba 1  log b ^a log a

5. 
   
   b
   
   b
   log_a b  log_c d  log_a d  log_c b ,
   

_alog ^c
_{ c}log ^a _;

6. 
   log₁₀ x  lg x
   

• o‘nli logarifm;

7. 
   log_e x  ln x
   

• natural logarifm;

8. 
   Agar
   

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

9. 
   Agar
   

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

10. 
    Agar
    

b  a  1 bo‘lsa, log_b p  log_a p
bo‘ladi ( p  0 );

11. 
    Agar 0  a  b  1 bo‘lsa, log_b p  log_a p
    

bo‘ladi ( p  0 );

12. 
    

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. 
   log_a
   

f (x)  b

Logarifmik tenglama va tengsizliklar

^ f (x)  ^ba .
_{ } f (x)  0, 0  a  1,


2. 
   log _{f ( x)} a  b
   

_ f (x)  1,


_ 1
a  0,

^_ f (x)  a^b .

3. 
   log
   

f (x)  log


g(x)
_{ }0  a  1,
f (x)  0,

a a




_0 
^ f (x)  g(x).
f (x)  1,

4. 
   log _{f ( x)} g(x)  b
   

 _
g(x) 
f ^b (x).

5. 
   log_{ϕ ( x)}
   

f (x)  log_{ϕ ( x)}


g(x)
_0  ϕ(x)  1,


^{ } f (x)  g(x).
f (x)  0,

6. 
   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,

7. 
   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  y_α , Соsα  x_α ,

tgα 


Sec 
y_{α ,}
x_α
1 _,
x_α
ctgα 


С s сα 
x_{α ,}
y_α
1 _.
y_α

1 rad  57⁰17^15^; π  3,141592 . . .

Trigonometrik funksiyalarning choraklardagi ishoralari

Cosx Sinx tgx,ctgx

Asosiy trigonometrik ayniyatlar

1. 
   Sin² x  Cos² x  1. 4. tgx 
   

Sinx _.
Cosx

2. 
   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

2. 
   Cosx  Cosy  2Cos ^{x  y} Cos ^{x  y} .
   

2 2

3. 
   Cosx  Cosy  2Sin ^{x  y} Sin ^{x  y} .
   

2 2

4. 
   Cosx  Sinx 
   

2Sin^{ π}  x ^  2Cos ^{ π}  x ^ .

 ₄   ₄ 
   

5. 
   Cosx  Sinx 
   

2Cos ^{ π}  x ^  2Sin^{ π}  x ^ .

 ₄   ₄ 
   

6. 
   pCosx  qSinx  rSin(z  x) , r 
   

, Sinz 
^p ,Cosz  ^q .
r r

7. 
   tgx  tgy  ^{Sin x  y } ,
   

CosxCosy

8. 
   tgx  ctgy  ^{Cos  x  y } ,
   

CosxSiny
ctgx  ctgy  ^{Sin x  y } .
SinxSiny
tgx  ctgy   ^{Cos  x  y } .
CosxSiny