Formulario

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FORMULARIO
L´gica Matem´tica o a
Variables p V V F F q V F V F Negaci´n o ∼p F F V V Conjunci´n o p∧q V F F F Disyunci´n o inclusiva p∨q V V V F p → q V F V V p ↔ q V F F V Condicional Bicondicional Disyunci´n o exclusiva p∆q F V V F

Identidades trigonom´tricas e
1. sen2 A + cos2 A = 1 2. 1 + tan A = sec A 3. 1 + cot2 A = csc2 A 4. tan A = 5. cot A = sen A cos A
2 2

´ Angulo mitad
1.sen A = 2 1 − cos A , 2 1 − cos 2x 2 1 + cos A 2 1 + cos 2x 2 (A = 2x)

2. sen2 x = 3. cos A = 2

cos A sen A 1 6. sec A = cos A 1 7. csc A = sen A 1 8. cot A = tan A

4. cos2 x =

Suma y diferencia de dos ´ngulos a
1. sen(A ± B) = sen A cos B ± cos A sen B 2. cos(A ± B) = cos A cos B 3. tan(A ± B) = sen A sen B tan A ± tan B 1 tan A tan B

´ Angulo doble
1. sen 2A = 2 sen A cos A 2.cos 2A = cos2 A − sen2 A 3. tan 2A = 2 tan A 1 − tan2 A

Transformaci´n en producto o
A+B A−B cos 2 2 A+B A−B 2. sen A − sen B = 2 cos sen 2 2 1. sen A + sen B = 2 sen

1

2
3. cos A + cos B = 2 cos

Matem´tica IV a
A+B A−B cos 2 2 A−B A+B sen 4. cos A − cos B = −2 sen 2 2 l´ ım
n

Walter Arriaga D.
f (x) =
n

x→a

si n fuera par debe cumplirse que b ≥ 0

x→a

l´ f (x) = ım√ n

L,

n ∈ N,

F´rmulas: o 1. l´ sen x = 0 ım
x→0 x→0 x→0

Transformaci´n de producto en suma o
1 [sen(A + B) + sen(A − B)] 2 1 2. sen A sen B = [cos(A − B) − cos(A + B)] 2 1 3. cos A cos B = [cos(A + B) + cos(A − B)] 2 1. sen A cos B =

2. l´ cos x = 1 ım 3. l´ ım 4. 5. 6. 7. sen x =1 x tan x =1 l´ ım x→0 x 1 − cos x l´ ım =0 x→0 x 1 − cos x 1 l´ ım = x→0 x2 2 l´ arcsen x = 0 ımx→0 x→0+ x→0

Funciones hiperb´licas o
1. senh u = eu − e−u 2

eu + e−u 2. cosh u = 2 eu − e−u 3. tanh u = u e + e−u 4. coth u = 1/tanh u 5. sech u = 1/cosh u 6. csch u = 1/sinh u 7. cosh2 u − senh2 u = 1 8. 1 − tanh u = sech u 9. coth2 u − 1 = csch2 u
2 2

8. l´ arc cos x = ım arcsen x =1 x arctan x 10. l´ ım =1 x→0 x 9. l´ ım 11. 12.
x→+∞

π 2

l´ ım arctan x =

π 2 π 2

x→−∞l´ ım arctan x = − 1+ 1 x
x

13. l´ ım
x→0

x→∞

=e

14. l´ (1 + x)1/x = e ım 15. l´ ım ax − 1 = ln a, x→0 x a > 0, a = 1

L´ ımites

Propiedades: Si: l´ f (x) = L y l´ g(x) = M . Entonces: Derivadas ım ım x→a x→a
x→a

l´ c = c, ım

c constante. c con-

l´ [c f (x)] = c l´ f (x) = cL, ım ım x→a x→a stante.
x→a

l´ [f (x) ± g(x)] = l´ f (x) ± l´ g(x) = ım ım ım x→a x→a L±Mx→a

dy = y = f (x) dx f (x + h) − f (x) ∆y f (x) = l´ ım = l´ ım ∆x→0 ∆x h→0 h Propiedades: [kf (x)] = kf (x), k = constante [f (x) ± g(x)] = f (x) ± g (x) F´rmulas: o d (k) = 0 1. dx d 2. (x) = 1 dx

l´ [f (x).g(x)] = l´ f (x). l´ g(x) = L.M ım ım ım
x→a x→a

1 1 1 = = , l´ ım x→a g(x) l´ g(x) ım M
x→a

si M = 0

l´ f (x) ım f (x) L l´ ım = x→a = , x→a g(x) l´ g(x) ım M
x→a x→ax→a

si M = 0 n∈N

l´ [f (x)]n = [ l´ f (x)]n = Ln , ım ım

Walter Arriaga D.
3. 4. 5. 6. 7. 8. d n (x ) = nxn−1 dx d n (u ) = nun−1 u dx d u (a ) = u .au ln a dx d u (e ) = u .eu dx d v (u ) = vuv−1 u + uv v ln u dx d (u.v) = u.v + v.u dx u v.u − u.v = v v2

Matem´tica IV a
26. 27. 28. 29. 30. d (cosh u) = u .senh u dx d (tanh u) = u .sech2 u dx d (coth u) = −u .csch2 u dx d (sech u) = −u.sech u.tanh u dx d (csch u) = −u .csch u.coth u dx Regla de la cadena 31. (f ◦ g) (x) = f (g(x)).g (x) 32. Si y = f (u) y u = g(x), entonces: dy du dy = dx du dx Derivada Param´trica e x = x(t) 33. Si , entonces: y = y(t) dy dy = dt dx dx dt Derivada impl´ ıcita 34. Si F (x, y) = 0 adem´s y = f (x), entonces: a ∂F dy = − ∂x ∂F dx ∂y

3

d 9. dx

u d 10. (loga u) = loga e dx u d u (ln u) =dx u √ d u 12. ( u) = √ dx 2 u 11. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. d (sen u) = u . cos u dx d (cos u) = −u . sen u dx d (tan u) = u . sec2 u dx d (cot u) = −u . csc2 u dx d (sec u) = u . sec u. tan u dx d (csc u) = −u . csc u. cot u dx d u (arcsen u) = √ dx 1 − u2 −u d (arc cos u) = √ dx 1 − u2 u d (arctan u) = dx 1 + u2 d −u (arccot u) = dx 1 + u2 d u (arcsec u) = √ dx u u2 − 1...
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