MATEMATIKA I

[Pages:6]MATEMATIKA I

/rd/ 0

6

4

3

2

3 2

/?/ 0 30 45 60 90 180 270

sin

0

1 2

2

31 0

2

2

?1

cos 1

3 2

2 2

1 2

0

?1

0

tg 0 3 1

3

30

ctg 3 1

30

3

0

NEDOLOCENI IZRAZI:

,

0 0

,

0,

-,1

a x2 +bx +c = 0

x1,2

= -b ?

b2 2a

-4a c

x = -xx, x, x 0

s in(-) = -s in

cos-( ) = cos

tg(-) = -tg

ctg(-) = -ctg

s in(?) = s in

cos( ?) = -cos

tg(k?) = ?tg

ctg(k?) = ?ctg

s in(k2?) = ?s in

cosk(2?) = cos

s in(2?) = cos cos 2( ?) = ?s in tg(2 ?) = ctg ctg(2 ?) = tg

s in ?s in = 2 s in ?2 cos 2

cos+cos = 2 cos+2 cos-2

c o s-c

os

=

-2

s

in

+2 s

in

- 2

tg?tg

=

s in(?) c o sc o s

c tg? c tg

=

s in(?) s in s in

s

in

s

in

=

-

1 2

(cos( +)

-cos ( -))

c o sc

o s

=

1 2

(c

o

s( +)

+cos( -))

s

in

c o s =

1 2

(s

in

(+)

+s

in

(-))

s in(?) =s in cos?coss in

cos ( ?) = coscoss in s in

tg(?)

=

tg?tg 1 tgtg

c tg(?)

=

c tgc tg1 c tg?c tg

s in2=2s incos

cos2= cos2 -s in2

tg2=

1

2tg -tg2

c

tg2=

c

tg2-1 2c tg

s in3=3s in-4s in3

cos3= 4 cos3 -3cos

tg3=

3tg-tg3 1 -3tg2

c

tg3=

c

tg3 -3c tg 3tg2-1

s

in

2

=

?

1 - c o s 2

cos2 = ?

1 + c o s 2

tg

2

=

1

- c o s s in

=

?

1-cos 1 + c o s

ctg2

=

1+cos s in

=

s in 1-cos

=

?

1+cos 1 - c o s

s in2 +cos2 =1

1 + tg2 =

1 c o s2

1 + c tg2 =

s

1 in2

1

a

rc

s

inx

+a

rccoxs

=

2

a

rc

tgx+a

rcc

tg

x=

2

EULERJEVE FORMULE:

eix = cosx + is inx

ek2i = 1; k Z

s inx

=

eix

- e-ix 2

cosx

=

eix

+ e-ix 2

s inx = -s h(ix) cosx = ch(ix)

shx=

ex

- e-x 2

chx =

ex

+ e-x 2

thx

=

ex ex

- e-x + e-x

=

shx chx

=

1 c th x

cthx =

ex ex

+ e-x - e-x

=

chx shx

=

1 th x

c h2x -s h2x =1

s h(-x) = -s hx

ch(-x) = chx

th(-x) = -thx

cth(-x) = -cthx

( ) a rs hx= ln x + 1 + x2

a

rth x =

1 2

ln

1 1

+ -

x x

z =a +bi

z = a -bi

z = zz = a2 +b2

zw = z w

zw =zw z +w z +w

z -w z -w

z +w z -w

z = a +bi z = z (cos+i s in )

z = a2 +b2

= a rctgba +k; k = 0,1,2

a

= 0,

b

>0

=

2

a

= 0,

b

0, b = 0 = 0

a < 0, b = 0 =

z w = z w (cos(z +w ) +i s in(z +w ))

z w

=

z w

(cos(z -w ) +i s in(z -w ))

z-1 = z -1 (co s-i s in )

MOIVREOVAFO RMULA: zn = z n (cos+i s in)n = z n (co sn() +i s in (n))

xn -z =0

xk+1 = n z (cos+nk2+i s in +nk2) = x1 k (k = 0,1,2,...,n -1)

x1 = n z (cosn+i s in n)

=

c o s2n

+i

s

in

2 n

primitivnikore ne note

(a + b)n

=

k

n =0

nk

an

-k

bk

nk

=

n! k!(k -r)!

n N

kr

=

r

(r

-1)(r

-2)...(r k!

-k

+1)

nk

=

n

n -k

nk +k n+1 = nk ++11

lim (1 +

n

1 n

)n

=e

lim n n = 1

n

lim n C = 1 C > 1

n

lim

n

s inx x

=

1

lim

n

cosx x

=

1

lim

n

tgx x

=

1

lim n 0

x

s

in

1 x

=

0

DIVERGENTNE VRSTE:

1

n =1 n

1

n =1 n

r R

2

KONVERGENTNE VRSTE:

1

n2

n =1

1

nk

n =1

k >1

(-1)n

n =1 n

(-1)n

nk

n =1

k >0

a qn q < 1, a 0

n =1

1

n =1 n(n +1)

1

n =2 n(n -1)

1

n =1 n!

TAYLORJEVE VRSTE:

ex

=

xn n= 0 n!

x

sinx

=

(- 1)n

n= 0

x2n + 1 (2n + 1)!

x

cosx

=

(- 1)n

n= 0

x2n (2n)!

x

shx=

n= 0

x2n+ 1 (2n + 1)!

chx=

x2n n= 0 (2n)!

ln(1+

x)

=

(- 1)n

n= 0

xn+ 1 n+1

x (- 1,1]

(1+

x)r

=

n=

0

r n

xn

x (- 1,1), r R

(a +

b)r

=

n=

0

r n

ar- n

bn

a> b

1

1- x

=

xn

n= 0

x (- 1,1)

GEOMETRIJSKA VRSTA:

a qn

n =0

sn

=

a

1- qn 1- q

,

q

> 1, a 0

n a, q = 1, a 0

s

=

a 1-

q

,

q

< 1, a 0

3

ODVODI:

(kx + n)'= k

(C)'= 0

(C f)'(x) = C f' (x)

n i=1

fi

(x)

=

n i=1

fi ' (x)

(f1 ... fn ) = f1 f2 ... fn +

f1 f2 f3 ... fn + ... + f1 f2 ... fn

(xn )' = n xn-1;n R

(ax )'= ax lna;a > 0

(ax )(n) = ax ln na;a > 0

(ex )' = ex

x = elnx

(ln x)' =

1 x

(ln x)(n )

=

(n - 1)! xn

(s inx)'= cosx

(cosx)'= -s inx

(tg x)' =

1 c o s2

x

(ctgx)'

=

-

s

1 in2

x

(a rcs inx)'= 1 1- x2

(a rccoxs)'= - 1 1- x2

(a

rc

tg)'

=

1 1+ x2

(a

rc

c

tg)'

=

-

1

1 + x2

(s hx)'= chx

(chx)'= s hx

(th x)' =

1 c h2 x

INTEGRALI:

(f ? g)dx = f dx ? gdx

C f dx= C f dx

u dv=uv - vdu P ERP ARTES

dx x

= ln

x

+C

f f

dx

=

ln

f

+C

xn

dx

=

x n +1 n +1

+

C

;n

1

axdx

=

ax ln a

+

C

exdx =ex +C

akxdx

=

k

ax lna

+

C

e

kxd

x

=

1 k

e

x

+C

s inx dx= -cosx + C

cosx dx= s inx + C

s hxdx= chx+ C

chxdx= s hx+ C

dx s in2 x

= -ctgx

+C

dx cos2 x

=

tg x

+C

dx = a rcs inx + C 1- x2

1

d +

x x

2

= a rctgx + C

dx a2 - x2

= a rcs in

x a

+C

dx a2 + x2

=

1 a

a

rctg

x a

+C

dx = ln x + x2 + k + C x2 + k

4

 p(n) (x)

dx = q(n-1) (x) a x2 + bx + c + A

a x2 + bx + c

dx a x2 + bx + c

(x2

Ax + B + px + q)n

dx =

T (2n-3) (x) (x2 + px + q)n11

+

Cx + D x2 + px +

q

d

x

(x

-

k)n

S(m) (x) a x2 +

bx

+

c

,

m

<

n

:

x

-

k

=

1 t

b

b

b

(f(x) ? g(x))dx = f(x)dx ? g(x)dx

a

a

a

b

b

C f(x)dx = C f(x)dx

a

a

b

c

b

f(x)dx = f(x)dx + f(x)dx a < c < b

a

a

c

b

a

f(x)dx = - f(x)dx

a

b

a

f(x)dx = 0

a

5

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