Formule de algebr
Formule algebraEcuaia de gradul doi Ecuaia .Se calculeaz
Dac > 0 atunci ecuaia de gradul doi are dou rdcini reale diferite date de formula
x1, x2 =
Dac = 0 atunci ecuaia de gradul doi are dou rdcini reale egale date de formula
x1 = x2 =
Dac < 0 atunci ecuaia de gradul doi are dou rdcini complexe diferite date de formula
x1, x2 =
Relaiile lui Viete pentru ecuaia de gradul doi: :
Alte formule folositoare la ecuaia de gradul doi:
Funcia de gradul doi
f :R Rf (x) = ax2 + bx + c
abscisa, ordonataGraficul funciei de gradul doi este o parabol cu varful in punctul V
Dac a>0 atunci parabola are ramurile indreptate in sus.In acest caz valoarea minim a funciei este fmin=
Dac a 0,a 1,b > 0
= c Aceast echivalen transform o egalitate cu logaritm intr-o egalitate fr logaritm
,
Probabilitatea unui eveniment
Se calculeaz cu formula:
Legi de compoziie
Fie M o mulime nevid pe care s-a dat o lege de compoziie notat *.
Legea * este asociativ dac ( x y ) z = x ( y z) x, y, zM
Legea * este comutativ dac x y = y x x, yM
Legea * are element neutru e dac x e = e x = x xM
Un element xM se numete simetrizabil dac xM astfel inct x x = x x = e
Relaiile lui Viete pentru ecuaia de gradul trei
Dac are rdcinile atunci avem:
Relaiile lui Viete pentru ecuaia de gradul patru
Dac are rdcinile atunci avem:
Inversa unei matrici 1. = 1 =>
2. (linii coloane)3. (se taie linia 1 si coloana 1)4.
ex:
Formule - analiza matematica
Asimptote
Asimptote orizontale
Pentru a studia existenta asimptotei orizontale spre la graficul unei functii, se calculeaza .
Cazul 1. Daca aceasta limita nu exista sau este infinita atunci graficul nu are asimptota orizontala spre .
Cazul 2. Daca aceasta limita exista si este finita, egala cu un numar real , atunci graficul are asimptota orizontala spre , cu dreapta de ecuatie
Analog se studiaza existenta asimptotei orizontale spre
Asimptote obliceAsimptota oblica spre (daca exista) are ecuatia , unde m si n se calculeaza cu formulele:
Analog se studiaza existenta asimptotei oblice spre . Asimptote verticale
Se calculeaza si . Daca una din aceste limite este infinita atunci graficul are asimptota verticala dreapta de ecuatie .Derivata unei functii intr-un punct:
Tangenta la graficul unei functii in punctul de abscisa (poate fi ):
Functia continua Functia concava pe I ,
Domeniu de continuitate
f elementara => fct continua Functia convexa pe I ,
x cunoscut > fct cont
Reguli de derivare: Valori utile:
,
,
, ,
,
,
Tabel cu derivatele unor functii uzuale
,
Tabel cu integrale nedefinite
1. Formula de integrare prin parti pentru integrale nedefinite este:
2. Formula de integrare prin parti pentru integrale definite este:
3.
4. F continua => F admite primitive
F continua in x=a daca
5. => F primitiva lui f6. Aria: daca , daca
7. Volumul:
Formule de geometrie
1) Teorema lui PitagoraIntr-un triunghi dreptunghic are loc relatia:
2) Teorema lui Pitagora generalizata (teorema cosinusului)Intr-un triunghi oarecare ABC are loc relatia:
3) Aria unui triunghi echilateral de latura l este:
4) Aria unui triunghi oarecare (se aplica atunci cand se cunosc doua laturi si unghiul dintre ele):
5) Aria unui triunghi oarecare (se aplica atunci cand se cunosc toate cele 3 laturi):
formula lui Heron unde este semiperimetrul, iar si
6) Aria triunghiului dreptunghic este:
7) Teorema sinusurilorIntr-un triunghi oarecare ABC are loc relatia:
unde a,b,c sunt laturile triunghiului (desemn. de proiectiile punctului pe dreapta opusa) A,B,C sunt unghiurile triunghiului
R este raza cercului circumscris triunghiului
8) Distanta dintre doua puncte (lungimea unui segment)Daca si sunt doua puncte in plan atunci distanta dintre ele este:
9) Mijlocul unui segment:Daca si sunt doua puncte in plan atunci mijlocul segmentului AB este:
10) Centrul de greutate al unui triunghi:
Fie G centru de greutate in :
11) Vectorul de pozitie al unui punct:Daca atunci
12) Coordonatele punctelor de intersectie a functiilor Abcisa celor doua functii se obtine realizand ecuatia
13) Daca si sunt doua puncte in plan atunci vectorul este dat de formula
14) Ecuatia unei drepte care trece prin doua puncte date:Daca si sunt doua puncte in plan atunci ecuatia dreptei AB se poate afla cu formula:
sau cu formula
15) Ecuatia unei drepte care trece prin punctul si are panta data mEste data de formula:
16) Conditia de coliniaritate a trei puncte in planFie , , trei puncte in plan.
Punctele A, B, C, sunt coliniare daca si numai daca
17) Aria unui triunghiFie , , trei puncte in plan.
Aria triunghiului ABC este data de formula
unde este determinantul anterior
18) Distanta de la un punct la o dreaptaDaca este un punct si o dreapta in plan atunci distanta de la punctul A la dreapta d este data de formula:
19) Panta unei drepteDaca si sunt doua puncte in plan atunci panta dreptei AB este data de formula:
sau (pentru fct simple)20) Conditia de coliniaritate a doi vectori in plan:Fie si doi vectori in plan. Conditia de coliniaritate a vectorilor si este:
21) Fie pct si dreapta .
22) Conditia de perpendicularitate a doi vectori in plan:Fie si doi vectori in plan. Avem:
(produsul scalar este 0);
23) Conditia de paralelism a doua drepte in planDoua drepte si sunt paralele daca si numai daca au aceeasi panta adica:
Altfel, daca dreptele sunt date prin ecuatia generala: si atunci dreptele sunt paralele daca .24) Conditia de perpendicularitate a doua drepte in planDoua drepte si sunt perpendiculare daca si numai daca produsul pantelor este egal cu -1 adica:
25) Punctul de intersectie dinte 2 drepteDreptele si sunt paralele daca , adica au pantele egale. Atunci dreptele si coincid daca si numai daca au coeficientii proportionali
26) Det. coord mijlocului unui segmentFie M mijlocul AB
27) Descompunerea unui vector
28) Aria paralelogramului
suma unghiurilor alaturate = 180, AD=BCsuma tuturor unghiurilor = 360
Legi de compozitie Fie structura (M, )
1. Asociativitate
2. Comutativitate
3. Element neutru
4. Element simetrizabil
Monoid (comutativ) = 1,3,(2) Grup (comutativ/abelian) = 1,3,4,(2)
Inel : 1. Grup comutativ
2. monoid
3. legea pct. distributiva fata de adunare
izomorfism de grupuri: a) fct bijectiva injectiva si surjectiva b) f izomorfism de grupuri
f injectiva
f surjectiva a.i.
Formule de trigonometrie
form. fundamentala a trigonom. Formule pt. transf. sumelor in produse
fct sin este impara
fct cos este para
Formule pt. transf produselor in sume
,
, ,
Tabelul trigonometric0304560
90
01234
01
2
sin0
1
cos1
0
tg0
1
/
ctg/
1
0
sin
, M(cos x, sin x) II I cos III IV
Cercul trigonom.
unde Ecuatii trigonometrice fundamentale1) Ecuatia are solutii daca si numai daca
In acest caz solutiile sunt
2) Ecuatia are solutii daca si numai daca
In acest caz solutiile sunt
3) Ecuatia are solutii
Solutiile sunt
4) Ecuatia are solutii
Solutiile sunt
Polinoame (2)
1. Radacinile polinoamelor. Teorema lui Bezout.
Definitia 1
Fie un polinom nenul cu coeficienti complecsi. Un numar complex, se numeste radacina a polinomului daca .Exemple
1. Numarul 2 este radacina pentru polinomul
pentru ca
2. Numarul
este radacina pentru polinomul pentru ca
Observatie
Pentru a afla radacinile unui polinom
se rezolva ecuatia ; spre exemplu, pentru a afla radacinile polinomului vom rezolva ecuatia si gasim radacinile polinomului , .
Teorema lui Bezout
Fie un polinom nenul. Numarul este radacina a polinomului daca si numai daca divide .
Exemplu
Polinomul avand radacinile , se va divide atat prin cat si prin .2. Radacini multiple1. este radacina unui polinom daca
2. ;
3. =>
=>
4.
5.
Sume remarcabile
Alte
Minkowski:
Cauchy-Buniakovski-Schwarz (CSB):
Nesbitt:
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