Analisi Matematica 1.2 Formulario

8/22/2019 Analisi Matematica 1.2 Formulario

1/79


2/79

. . .


3/79


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5/79

pp p p

p q p q

pq p q

= p=q p p q

pq p q


6/79

p q p p q p q p=q pq

1 1 0 1 1 1 11 0 0 0 1 0 00 1 1 0 1 1 00 0 1 0 0 1 1

A={x1, x2, . . . , xn}={x|P(x)}

A B={x|xA xB}

A B ={x|xA xB}

A \ B={x|xA (xB)}

AB= A \ B B\ A

U Ac =cA= A={x|xU\ A}={x|(xA)}

P(A) = 2A ={E|EA}|P(A)|= 2|A|

(x, y) =x, y={{x}, {x, y}} AB={(x, y)|xAyB}A1 A2 An ={(x1, x2, , xn)|xiAii={1, 2, , n}}

(E F)c =Ec Fc(E F)c =Ec Fc


7/79

AA A A A BB A (A B) CA (B C) A BB A (A B) CA (B C) A AA A

A

A

A BA AA B A (B C)(A B) (A C) A (B C)(A B) (A C) A (A B)A A (A B)A (A B) A B (A B) A B A A (A A) (AB)(BA) (AB)(B A) A A A(BA) A(AB) ((AB) B) A (A A) A (AA)A ((AB)A)A ((AB) (BA)) A((AB)B) (A(BC))(B(AC)) ((AC) (BC))(A BC) ((AB) (AB))B (A

(B

C))

((A

B)

(A

C))

((AB) (BC))(AC) (A(BC))((A B)C)


8/79


9/79

[R, +, ] [R, +] [R\ {0}, ]

R

0

x R x

1

x R\{0} 1x = 1/x=x1

xyx + zy + zz R

xyxzyzz R, z0


10/79

A B a A, bB, ab

cR :abc

R = R{}

(a b)n =n

k=0

nk ankbk

(a b)2 =a2 2ab + b2 (a b)3 =a3 3a2b + 3ab2 b3 (a b c)2 =a2 2ab + b2 2bc + c2 2ca

P(x) =ann

i=0(x xi) =

n

i=0ai xi

P(x) = a(x x1) (x x2) (x xn)nN :xn yn = (x y) (xn1 + xn.2y+ + xyn2 + yn1)n= 2k +1, k Z :xn + yn = (x+ y) (xn1xn2y + xyn2+ yn1)

a2 b2 = (a b)(a + b)a3 b3 = (a b)(a2 ab + b2)

P(x) = ax2 + bx + c x1,2= b

b24ac2a

x1,2=

2ac

a := b2

A B=

A+A2B

2

AA2B

2


11/79

f(x)g(x)

g(x)0f(x)g2(x)

g(x)< 0f(x)0

f(x)< g(x)

f(x)0g(x)> 0f(x)< g2(x)

y 0, n N n y ny= y1/n =sup{xR :xn < y}= inf{x R :xn > y}

a0 = 1a1 =a

am+n =am anamn = a

m

an

(am)n =amn

am =emlog a

a1n =n

a

am = 1am

a= logb cba

=c

loga1 = 0loga a= 1loga(m n) = loga m + loga nloga

mn = loga m loga n

loga m = loga m R

loga b= logcblog

ca

loga b= 1logba


12/79

f : R R+0 = [0, +]f(x) =|x|= (x) = {x, x}=

x2

aR, |a| 0

|a|= 0a = 0

R, aR, | a|=|| |a|

a, b R, |a + b| |a| + |b|

|

n

i=1

ai|

n

i=1 |

ai|

||a| |b|| |a b|

nN, n! =f att(n) =n

i=1i=

= 1 2 (n 1) n

0! = 1n! =n (n 1)!


13/79

k N, (2k+ 1)!! =

ki=0(2i + 1)

(2k)!! =

1 k= 0k

i=1 k >0

(1)!! = 1

x R \ {0}, (x) = (x) = (x) =x|x| =

|x|x =

1 x >01 x


14/79

sinh : R Rsinh x= x= e

xex2

cosh : R [1, +)cosh x= x= e

x+ex

2

sinh2 x cosh2 x= 1

tanh : R Rtanh x= x= sinhxcoshx

coth : R \ {0} Rcoth x=

x= coshx

sinhx

:R R y= y = log(y+

y2 + 1)

: R R y= y = log(y+

y2 1)

: R R x= y= 1

2 log 1+x1x

ex = (x)

! f : R R

x1, x2 R, f(x1+ x2) = f(x1) f(x2)

f(1) =e e

xR f(x) = ex = (x)

f(x) =ax

=ex loga

an = a1+ (n 1)d

Sn =n

i=1

ai =n(a1+ an)

2 =

n(2a1+ (n 1)d)2


15/79

an = a1 qn1

Sn =n

i=1

ai = a11 qn

1 q = 1

q= 1 Sn =n

i=1

ai = n a1

Sk,n =n

i=k

ai = qk 1 q

nk+1

1 q = 1

x[2 , 2 ], | sin x| |x|

nN, a 1, (1+ a)n 1 +an

xR, ex x + 1

x >1, log(x + 1)x

a, b >0,p,q >1 : 1p + 1q = 1, a b 1p ap + 1q bq

ni=1

i= n(n + 1)

2n

i=1

i2 = n(n + 1)(2n + 1)

6n

i=1

i3 = (n(n + 1)

2 )2

ni=0

ni = (1 + 1)n = 2nn

i=0

(1)i

ni

= 0


16/79


17/79

sin2 x + cos2 x= 1

tan x= sinxcosx

x

=

2 + k

cot x= cosxsinxx=k

cot x= 1

tanxx=k 2

sec = 1cossec : R

\ {pi2

+ k,k

Z

} R

csc = 1sin

csc : R \ {k,k Z} R

sin( ) = sin cos cos sin cos( ) = cos cos sin sin tan( ) = tantan1tan tancot( ) = cot cot1cotcot


18/79

sin(2) = 2sin cos cos(2) = cos2 sin2 = 1 2sin2 = 2 cos2 1tan(2) = 2tan1tan2 sin(3) = 3sin 4sin3 cos(3) = 4 cos3 3cos tan(3) = 3tantan

3 13tan2

sin(

2 ) =1cos2cos(

2) =

1+cos

2

tan(2

) =

1cos1+cos

= sin1+cos

= 1cossin

tdef

= tan 2

sin = 2t1+t2

cos = 1t2

1+t2

tan = 2t1t2

sinp + sin q= 2 sin p+q2 cospq2

sinp sin q= 2 cos p+q2 sin pq2cosp + cos q= 2 cos p+q2 cos

pq2

cosp cos q=2sin p+q2 sin pq2

cosp sin q= 12 [sin(p + q) sin(qp)]sinp

sin q= 1

2

[sin(p

q)

cos(p + q)]cosp cos q= 12 [cos(p + q) + cos(p q)]


19/79

sin sin 1 cos2 tan1+tan2

cos 1 sin2 cos 11+tan2

tan sin1sin2

1cos2 cos

tan

cot

1sin2 sin cos1cos2 1tan

sec 11sin2

1cos

1 + tan2

csc 1sin 11cos2 1+tan2 tan

12

15624

6+2

4 2 3 2 + 3

8

2230

222

2+2

2

2 1 2 + 1

6

30 12

32

33

3

4

4522

22

1 13

6032

12

3

33

38 6730 2+22 222 2 + 1 2 1512

756+2

4

624

2 +

3 2 32

90 1 0 0

sin x cos x tan x cot x

x sin x

cos x

tan x

cot x

+ x sin x cos x tan x cot xx sin x cos x tan x cot x

2 x sin x cos x tan x cot x2 x cos x sin x cot x tan x2

+ x cos x sin x cot x tan x32

x cos x sin x cot x tan x32

+ x cos x sin x cot x tan x


20/79

S= 12

ab sin = 12

bc sin = 12

ac sin

S= 12a2sin sin sin(+) =

12b

2sin sin sin(+) =

12c

2sin sinsin(+)

S=

p(p a)(p b)(p c)

a,b,c,d p= a+b+c+d

2 S=

(p a)(p b)(p c)(p d)

d= 0

AB= 2r sin

a

sin = a

sin = a

sin = 2R= abc

4S

a= b cos + c cos b= a cos + c cos c= a cos + b cos

a2 =b2 + c2 2bc cos b2 =a2 + c2 2ac cos c2 =a2 + b2 2ab cos

b= a sin = a cos c= a sin = a cos b= c tan = c cot c= b tan = b cot


21/79

r y= mx + n y= ax + by+ c m=ba n=ca

P1(x1, y1)P2(x2, y2)P3(x3, y3)

rP1 (y y1) = m(x x1)srP1 y1= mconstx1+ nrP1P2 :

y

y1

y2y1 = x

x1x2x1

mr = y2y1x2x1

rsmr =msrsmr = 1ms

P1P2= dist(P1, P2) =

(x1 x2)2 + (y1 y2)2dist(P1, r) =

|ax1+by1+c|a2+b2

P1P2= (x1+x2

2 ; y1+y2

2 )

P1P2P3= (x1+x2+x3

3 ; y1+y2+y3

3 )

: x2 + y2 + ax + by+ c= 0

C(x0, y0) r2 =x20+ y

20 c0

P(x, y)(P C) = r: (x x0)2 + (y y0)2 =r2

a=2x0b=2y0c= x20+ y

20 r2

x0=a2y0=b2r=

a2

4 + b

2

4 c

P :y = ax2 + bx + c

F(x0, y0); d: y = kP(x, y) P dist(P, F) = dist(P, d)

y0> k a >0 y0< k a


22/79

F( b2a , 14a )d: y =

1

4a )F( b2a , 4a )a: x = b2a =b2 4ac

P :x = ay2 + by+ c

F(x0, y0); d: y = kP(x, y)

P dist(P, F) = dist(P, d) x0> k a >0

x0< k a b e= ca a e= cb 1


23/79

I: x2

a2 y

2

b2 =1

c2 =a2 + b2, c0I x V1(0, b) V2(0, b)F1(0, c) F2(0, c); F1, F2yP(x, y) I |P F1 P F2|= 2aa1: y =

bax

a2: y =baxe= cb >1

I :x2 y2 =a2

c= a

2 a= bI x V1(a, 0) V2(a, 0)F1(a

2, 0) F2(a

2, 0); F1, F2x

P(x, y) I |P F1 P F2|= 2aa1: y = xa2: y =xe= 2

I: x2 y2 =a2

c= a

2 a= bI y V1(0, a) V2(0, a)F1(0, a

2) F2(0, a

2); F1, F2y

P(x, y) I |P F1 P F2|= 2aa1: y = xa2: y =xe=

2

I :xy = k

k >0 I bI,III :y = x V1(

k,

k) V2(

k, k)k >0 I bII,IV :y =x V1(

|k|,

|k|) V2(

|k|,

|k|)


24/79

leftrightarrow |P F1 P F2|= 2aa1: x = 0a2: y = 0e=

2

I :y = ax + bcx + d

I c= 0ad

bc

= 0

a1: x =dca2: y =

ac

e=

2

C : a11x2 + a22y2 + a12xy+ a13x + a23y+ a33 = 0

A= a11

12a21

12a13

12a12 a22

12a231

2a1312a23 a33

A=

a11

12

a2112

a12 a22

|A|= 0

|A| = 0

|A|> 0 |A|= 0 |A|< 0

|A

|= 0

|A| = 0

T :R R R R


25/79

(x, y)(x, y)

x= ax + by+py= cx + dy+ q

A=

a bc d

|A| = 0

T1 :R R R R(x, y)(x, y)

S

S =|A| x= d|A|x

+ b|A|y+ d|A|p +

b|A|q

y= c|A|x+ a|A|y

+ c|A|p +a|A|q

A1 =

d|A|

b|A|

c|A|

a|A|

T T T T T T


26/79

x= x +py= y + q

A=

1 00 1

|A|= 1

x= x cos y sin y= x sin + y cos

x= xcos + ysin y=xsin + ycos

A=

cos sin sin cos

|A|= cos2 + sin2 = 1

A1 =

cos sin sin cos

|A1

|= cos2 + sin2 = 1

x= ax byy= bx + ay a

2 + b2 = 1

:{ x

= x cos y sin y = x sin + y cos

:{ x = x +p

y= y + q

:{ x= x cos y sin +p

y= x sin + y cos + q

x= 2x0 xy= 2y0 y

(x0, y0)


27/79

y= y0

x= xy= 2y0 y

|A|= 1 00 1

=1

x= x0 x= 2x0 xy= y

|A|= 1 00 1

=1

y= mx + q

x = 11+m2

[(1 m2)x + 2my 2mq]y= 1

1+m2[2mx + (m2 1)x + 2q]

|A|=

1m21+m2

2m1+m2

2m1+m2

m211+m2

=1

x= ax + hy= ay + k

|A|= a 00 a

=a2 >0

x= ax by+py= bx + ay+ q

|A|= a bb a

=a2 + b2 >0

|A|=

a bb a

=a2 b2


28/79

|k|> 1 |k|< 1

x = kxy= y

y

x = xy= ky

x = x + kyy= y

x = xy= y + kx

P T(P)

P Q

T(P)T(Q) P Q

T(P)T(Q) S S

|det(A)|


29/79

Mm A aA, M ama MA ={M R :aA, Ma}mA={mR :aA, ma}

aA,supAa >0, aA : supA asupA aA,infAa

>0, aA : infAainfA + A supA= +infA=

A

xR, aA : axxR, aA : ax

AR A M 0 :xA|x| M

aA,maxAa

maxAA aA, minAa

minAA


30/79

A R x, y : x < y

z : x < z < y, zA

A

(a, b) =]a, b[={xR :a < x < b}, a, bR

(a, b] =]a, b] ={xR : a < xb}, a

R, b R

[a, b) = [a, b[={xR : ax < b}, a

R, bR

[a, b] ={xR :axb}, a, bR

A A, A + B A A={y R :yA} A R A={ x: xA} A B A + B={xR :x = a + b, aA, bB}

sup(A+B) = supA+supB, inf(A+B) = infA + infB

sup(A) =infA, inf(A) =supA0 :sup( A) = supA, inf( A) = infA0 :sup( A) = infA,inf( A) = supA

a > 0, b N , n N :na > b

A R A

Q

Q

R

a, b R, a < b, (a, b)Q= (a, b)Q {x Q : a 0 x0 (x0 , x0+ ) 2

A R x0 A x0 A >0 : (x0 , x0+ )A (x0 , x0+ ) A=

A =intAA A


31/79

A R x0 R x0

A

>0, (x0 , x0+ ) A= (x0 , x0+ ) Ac

=

AR x0 R x0 A >0, (x0 , x0+ ) A \ {x} = r >0, yA : y=x : y(x r, x + r)

A A A

AR x0 R x0 A x0 A >0 : (x0 , x0+) A \ {x}=

A A = AA

A A= A x0A A x0 x0A, r >0 : (x0 r, x0+ r)A

A Ac A= A

AR A

R

A, B

R A

B

R x, y R x y xRy R

xA, xRx

xA, yB, xRyyRx

xA, yB, (xRy) (yRx)x = y


32/79

xA, y(A B), zB, (xRy) (yRz)(xRz)

R R

xA, yB, (xRy) (yRx)

R

P

(xP) (xP)

(x, yP)(x + y, x yP)

xyx yP xyyx > x > y

(x

y)

(x

=y) <

x < y(yx) (x=y)

f f

aA, !bB : af b

f :AB f :aAbB af b b a f b= f(a) a b f a= f1(b)

A B f A B

f bB, !!aA : f(a) = b a1, a2 A, f(a1) = f(a2) a1 = a2 a1, a2A : a1=a2f(a1)=f(a2)

f bB, aA : f(a) = b


33/79

f

f

b B, !a A : f(a) = b f1 f f1 =I dB, f1 f =I dA IdA IdB A B IdA : AA, xx IdB :BB, xx

f : AB

x, y A , x < y f(x) f(y)

x, y A, x < yf(x)f(y)

f :I R

I

f

f+ g f g fg f f g f g AR R

(f+ g)(x) : AR, xf(x) + g(x)

(f g)(x) :AR, xf(x) g(x)

(fg )(x) :

{x

A : g(x)

= 0

} R, x

f(x)g(x)

R, ( f)(x) : AR, x f(x)

f :AB g : BC

(g f)(x) : AC, xg(f(x))

g f g f


34/79

f : I R :L > 0 :x, yI, |f(x) f(y)| L |x y|

L

f L

f : I R : L 0 : x, y I, |f(x)f(y)| L |xy| > 0 fC0,(I)

limxx0

f(x) = def

= >0,I(x0)

> 0 x, |x x0|< |f(x) |<

limxx0

f(x) =def=M >0,IM(x0)

M >0 x, |x x0|< M |f(x)> M

limx

f(x) =def

=

>0,

I()

N > 0 x, |x|> N |f(x) |<

limx f(x) =

def=M >0,

IM() NM>0 x, |x|> NM |f(x)|> M

(x0, x0+ )(x0 , x0)

limxx0

f(x) = limxx+0

f(x) = limxx0

f(x)

000 + 00

01


35/79

limx0

sin x

x = 1

limx+

(1 +1

x)x =e

limxP(n)Q(n) =

+ p > q ab >0 p > q ab


36/79

limx0

log(1 + x)

x = 1

limx0

loga(1 + x)

x =

1

log a

limx0

log(1 + x)

x =

limx0ex

1

x = 1

limx0

ax 1x

= log a

limx0

(1 + x) 1x

=, R

limx+

x

x = + > 1 , R

limx0+

x log x= 0 >0 R

limx+

(1 +n

x)x =en nR

limx+

log x

x = 0 >0 R

limx+ x

x

x = + R

limx+

n!

np = +, p R

limx+

n!

an = +, aR+0

limx+

n!

nn = 0


37/79

limx+

n!

nn en 2 n = 1

limx+

(2n)!!

(2n 1)!! 2 n = 1

+

+

= +

= =

R

+ = + = = = 0 = 0 = 0 = 0

+ = + > 0+ = 0 1= + 0< 1

f :A = (a, b) R R, x0 A, = limxx0f(x); =limxx0f(x); 1= limxx0g(x); 2= limxx0h(x) ,

, 1, 2 R

= !

A= (a, b), x0

[a, b]

=

0 I(x0) f(x) x0

( > 0 :x (x0, x0+ )\ {x0}, f(x) = g(x)) (1 = 1)

x(a, b) \ {x0}, f(x)g(x)h(x) = 21= = 2

x(a, b) \ {x0}, f(x)g(x) = +1= 1= +=


38/79

gf limy g(y) =1

>0 :x(x0 , x0+ ) \ {x0}, f(x)=

g() =1 g limxx0g f(x) = 1

f, g: (a, b)R, x0[a, b]

f, g (a, b)

x(a, b) \ {x0}, g(x)= 0

f(x0) =g(x0) = 0

limxx0 f(x)

g(x) =R

limxx0

f(x)g(x)

=

f, g : (a, b)R, a , bR

f, g (a, b)

x(a, b), g(x)= 0

limxa+f(x) =; limxa+g(x) =

limxa+ f(x)

g(x) =R

limxa+

f(x)

g(x) =

xb


39/79

= lim xx0f(x); 1 = lim xx0g(x); 2 = lim xx0h(x) , 1, 2 R ,,R aR+0\ {1} bR+0; nN x0= R x0=

limxx0 [f(x) + g(x)] = + 1

limxx0 [f(x) + g(x)] = + 1

limxx0 [f(x)]n =n >0

limxx01

f(x) = 1 = 0

limxx01

f(x) = 0limxx0f(x) =

limxx0f(x)g(x)

= 1 1= 0

limxx0 |f(x)|=||

limxx0loga f(x) = loga

limxx0bf(x) = b

limxx0 [f(x)]g(x) =1 >0

f : (a, b) R, a , b R,a < b, s0(a, b)

limx

x0

f(x) = {f(y) : y < x0}= {f(y) : y < x0}= f((a, x0))= f((a, x0))

limxx+0

f(x) = {f(y) : y > x0}= {f(y) : y > x0}= f((x0, b))= f((x0, b))

x0(a, b)

x0 R f, g x0 x0 limxx0f(x) = limxx0g(x) = 0


40/79

f g xx0 f

g

xx0

f=o(g, x0)

f=o(g)

limxx0

f(x)

g(x) = 0

0 xx0 o(1, x0)

limxx0f(x) = 0 >0 : limxx0

f(x)(xx0) = R \ {0} f

xx0 (x x0)

f1 = o(g, x0), f2 = o(g, x0)

f1+ f2= o(g, x)

k R, kf =o(kg,x0) =o(g, x0)

f1= o(g1, x0), f2= o(g2, x0)

f1+g1

f2+g2= g1

g2 x0

x,l >0, xk =o(xl, x0) =o(xl+k, xo)

f1 f2= o(g1 g2, x0)

f=o(g, x0), g= 0o(h, x0)

f=o(h, x0)

x0 R, f x0 x0 f(x) = f(x0) +a(xx0) + o(xxo, x0), a= 0 f1(y) =x0+

1a (y y0) + o(y y0, y0) y0= f(x0)

f, g x0 R : f, g= 0 x0 f g xx0 limxx0 f(x)g(x) =R \ {0} f g

fgf


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fg ghfh

f g( limxx0f(x) limxx0g(x))

y= f(x) x0 lim xx0f(x) =f(x0) >0 >0 x, |x x0|< |f(x) f(x0)|<

f, g x0 R f+g, f g, f

x0 g f x0 f1 x0

f :I R f(x0)> 0, >0 :x(x0 , x0+ )

[a, b] a, b R

f([a, b]) = [inf

x[a, b]f(x),sup

x[a, b]f(x)] c1, c2[a, b] :

f(c1) =inf

x[a, b]f(x), f(c2) =sup

x[a, b]f(x)

I

m

M x, y : f(x) < f(y), R : f(x) < < f (y) , zI :f(z) =

x 1 x2 ]x1, x2[ f(x) = 0 f : [a, b) R [a, b] a, b R, a < b :f(a) f(b)< 0 c(a, b) : f(c) = 0


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I x0 I x0

Rx0f(x) = f(x) f(x0)

x x0 I\ {0} f x0 lim xx0Rx0f(x)R

f(x) = D[f(x)] = dfdx

= f(x) = limh0 f(x + h) f(x)h = limxx0 Rx0f(x).

f : I R x0 f x0

f(x)f(x0)xx0 0f

)x0) + o(1, x0)f(x) f(x0) = f(x0) (x x0) + (x x0) o(1, x0)

f(x) =f(x0) + f(x0) (x x0) + o(x x0, x0) fI R, x0 I f x0 xI , L > 0 :f(x) = f(x=) + L (x x0) + o(x x0, x0)

k f(x0) = d2

dx2 = f(x) = ddxf

(x)f(k)(x0) =

dk

dxk = d

dxf(k1)

Ck f Ck, k N, k1 f k I I f(k) C0(I) =C(I) I C+(I) I k, dk

dxkfC(I)

C+(R) dkdxk

P =0 k > Pex C+(R) f :I R x0 I x0 L= f

(x)

f, g : I R, x0 I fg x0 f x0 g f

(x0) = g (x0)


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f :I R x0 x0 f(x0)0 0

f :I R x0I x0 f xI (x0 , x0+), >0 :f(x)f(x0) f(x0)

f : I R x0 I x0

f

x

I

(x0

, x0+ ),

>0 :f(x)< f(x0)

>f(x0)

x0 f xI , f(x) f(x0)

x0 f(xo) x0

f

f : I R x0 x0 I f(x0) = 0

f : IR

I I f

f : I R I I f

> 0 : f(x) 0() (x0

, x0) f(x)0(0) (x0, x0+) x0

f x0

f :I RC2, f(x0) = 0, f(x0)> 0(


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f f

f I x > x0, f(x)f(x0) + f+(x0) (x x0) x < x0, f(x)f(x0) + f(x0) (x x0)

f : (a, b)R x: 0 I :f (a, x0) (x0, b) x0 f x0 f x0

f : (a, +) R + limx+=

limx+

f(x)x

=a R \ {0}

limx+

f(x) a x= bR

y: a x + b

f, g If+ g I

f : [a, b] R [a, b] (a, b) f(a) =f(b) x0(a, b) f(x0) = 0

f : [a, b]R [a, b] (a, b) x0]a, b[ f(b)f(a)ba =f(x0)

f(x) g(x) [a, b]

]a, b[ g(x)= 0x[a, b] x0]a, b[ f(b)f(a)g(b)g(a) =f(x0)

g

(x0)

f : I R x0, x1 I : x0 < x1 x[x0, x1], f(x)f(x0) + f(x1)f(x0)x1x0 (x x0)

def= rx0x1(x);

x[x0, x1], f(x)rx0x1(x)

f : I R I r x0 < x1 0 I = 1a Ax

1

dx+ 1a Bx

2

dx = Aa log |x 1| +Ba log |x 2| + c

P2(x) = 0 I = 1a

Axdx +

1a

B(x)2dx =

1aA log |x |

A+Ba(x) + c

P2(x)< 0 I=

gx+hax2+bx+c

dx =

=gs d(ax2+bx+c)

x2+bx+c + ht

dx(kx+j)2+1 =

=gs log |ax2 + bx + c| + ht arctan(kx +j) + c

x= g(t) f(x) dx= f(g(t)) g(t) dt

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

n 0 e 0


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b

a

f(x) dx

b a

n [f(x0) + f(x1) +

+ f(xn

1)]

e (b a)2

2n M |f(x)| M

n 0 e 0 ba

f(x) dx b a2n

[f(x0) + f(xn) + 2 [f(x1) + + f(xn1)]]

e (b a)3

12n2 M |f(x)| M

ba

f(x) dx b a3n

[f(x0) + f(xn) + 4 [f(x1) + f(x3) + ] + 2 [f(x2) + f(x4) + ]]e (b a)

5

180n4 M |fiv(x)| M

=

x2

x1 1 + [f(x)]2 dx

x= x(t)y= y(t)

= t2t1

[x(t)]2 + [y(t)]2 dt

V= ba

[f(x)]2 dx

S S S

23

Slaterale = 2 ba

f(x)

1 + [f(x)]2 dx

n N, P = Q = n : f(x) = P(x) + o((x x0)n, x0) =Q(x)o((x x0)n, x0)P Q


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f : (a, b)R, x0(a, b)

f

n1

(a, b)

n

x0

f

f(x) =n

k=0

f(k)(x0)

k! (x x0)k + o((x x0)n, x0)

n x0

Pn,x0f=n

k=0

f(k)(x0)

k! (x x0)k

x0= 0

f : (a, b)R, x0(a, b), f n+1 x0

c= c(x)( {x0, x}, {x0, x}) :

f(x) = Pn,x0f(x) +f(n+1)(c(x))

(n + 1)! (x x0)n+1

f :I R, x0I , fCn(I)

f(x) Px0n f(x) = 1

n! xx0

f(n+1)(t)(xt)n dt

f(x) = Pn,0f+ o(xn, 0) =

nk=0

f(k)(x0)

k! (x x0)k + o(xn, 0)

ex = 1 + x +x2

2 + + x

n

n! + o(xn, 0)

log(1 + x) =x x2

2 +x3

3 + + (1)n1 xn

n + o(xn, 0)

sin x= x x3

3! +

x5

5! + + (1)n x

2n+1

(2n + 1)!+ o(x2n+2, 0)

cos x= 1 x2

2 +

x4

4! + + (1)n x

2n

(2n)!+ o(x2n+1, 0)

tan x= P6,0tan +o(x6, 0) = x +

x3

3 +

2 x515

+ o(x6, 0)

arcsin x= x +1

2 x

3

3 +

3

8 x

5

5 + +(2n 1)!!

(2n)!!

x2n+1

2n + 1+ o(x2n+2, 0)


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arccos x=

2x

1

2

x3

3

+

(2n 1)!!

(2n)!!

x2n+1

2n + 1

+ o(x2n+2, 0)

arctan x= x x3

3 +

x5

5 + + (1)n x

2n+1

2n + 1+ o(x2n+2, 0)

sinh x= x +x3

3! +

x5

5! + + x

2n+1

(2n + 1)!+ o(x2n+2, 0)

cosh x= 1 +x2

2 +

x4

4! + + x

2n

(2n)!+ o(x2n+1, 0)

tanh x= P6,0tanh +o(x6, 0) = x x

3

3 +

2 x515

+ o(x6, 0)

x= x 1

2x3

3 +

3

8x5

5 + + (1)n

(2n

1)!!

(2n)!!

x2n+1

2n + 1+ o(x

2n+2

, 0)

x= 1

2 log1 + x

1 x =x +x3

3 +

x5

5 + + x

2n+1

2n + 1+ o(x2n+2, 0)

(1 + x) = 1 + x + ( 1)2

x2 + + !( n)! n! x

n + o(xn, 0)

1

1 + x = 1 x + x2 + + (1)n xn + o(xn, 0)

limxx0f(x) =

limx

f(x) = den= num

limx f(x) = den= num 1 m= limx

f(x)x

n= limx[f(x) mx]A.Ob.: y = mx + n

+

0

+

0


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f(x) [a, b] f(a)f(b)< 0 f(x) x0]a, b[ f(a+b

2 )

]a, a+b2

[[ f(a) a+b2

< 0 ]a+b2

, b[[ a+b2 f(b) < 0

f(x) [a, b] f(a) f(b)< 0 f(x) x0]a, b[

(a, f(a)) (b, f(b)) x: y = 0 c

f(c)

]a, c[[f(a) f(c)< 0

]c, b[[f(c) f(b) < 0

f(x) [a, b] f(a) f(b)< 0 f(x) x0]a, b[

(a, f(a)) (b, f(b)) x : y = 0 c]a, b[ f(c) ]a, c[[f(a) f(c)< 0 ]c, b[[f(c) f(b)< 0


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n! = 1 2 (n 1) n=n

i=1

i

0 ! = 1n ! =n (n 1) ! n1

n

k

=

n !

(n k) ! k !

nk = n

n kn

k

=

n 1k 1

+

n 1

k

Pn,k = n!(nk)!k! =nk

Cn,k =

n+k1

k


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Pn = n!Dnk1,k2,,kn = n!k1!k2!kn!

Dn,k =Cn,k Pk = n!(nk)!

Dn,k

= nk

p= fn

p= limn

f

n

p() = 0

p() = 1

0fn0 fn 10p1

p(Ac) = p(A) = 1 p(A)

p(A \ B) = p(AB)p(B)

A B = p(A B) =p(A) +p(B)

A B= p(A B) = p(A) +p(B) p(A B)


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p(A) =p(A\B p(A B) = p(A) p(B)

p(A)=p(A \ B p(A B) = p(A) p(B\ A)

p(Hi\ E) = p(Hi) p(E\ Hi)n

1p(Hi) p(E\ Hi)

pn,k =

n

k

pk (1 p)nk

M(X) =n1

xipi


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N

N=anbn + an1bn1 + . . . + a1b + a0

0

ai < b i= 1, . . . , n

R

R= anbn + an1bn1 + . . . + a1b + a0+

i=1

aibi

0ai < b in

c= b x


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b|c

a= 0

b= aq+ r,0r 1

n > 1

n= pn11 pn22 pnmm

p1, p2, . . . pm n1, n2, . . . nm 1 n

a b c

c|a c|b

a b

D|a D|b x|a x|b x|D

a, b D= (a, b)

a b

h A + k B


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a b D = (a, b) m n

ma + nb= D

a b m

a|m b|m n a|n b|n=m|n

a b m m|(a b)

ab mod m

m

a aa mod m

ab mod m ba mod m

ab mod m bc mod m ac mod m

m m m m a

[a]m

[a]m+ [b]m = [a + b]m [a]m [b]m = [a b]m

a[0]m


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a m a0 modm

p

(p 1)! 1 modp

(m) m m r

1r < m (r, m) = 1

(b, m) = 1 b(m) 1 modm

ap1 1 modp

ap a mod p


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Pn n

P0

nN Pn n+1 Pn nN


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=b2 4ac

106

3, 141592653589793238462643383279... 1, 618...

1, 618... (n) (V)


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ex = (x)


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Analisi Matematica 1.2 Formulario

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