Multiple Choice Questions of Successive Differentiation (Calculus) for B.Sc. 1st Semester (Panjab University ) Mathematics students.There are 2o questions with answer keys.
Multiple Choice Questions of Successive Differentiation (Calculus) for B.Sc. 1st Semester (Panjab University ) Mathematics students.There are 2o questions with answer keys.
Successive Differentiation is the process of differentiating a given function successively times and the results of such differentiation are called successive derivatives. The higher order differential coefficients are of utmost importance in scientific and engineering applications.
Solucion de problemas de ecuaciones difrenciales hasta 19JAVIERTELLOCAMPOS
Este archivo contiene problemas resueltos del curso de ecuaciones diferenciales, cada problema esta resuelto paso a paso para su mejor comprensión del lector
Este archivo contiene problemas resueltos del curso de ecuaciones diferenciales, cada problema esta resuelto paso a paso para su mejor comprensión del lector
1. Fórmulas de Cálculo Diferencial e Integral (Página 1 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M.
Fórmulas de
Cálculo Diferencial
e Integral VER.6.8
Jesús Rubí Miranda (jesusrubim@yahoo.com)
http://www.geocities.com/calculusjrm/
VALOR ABSOLUTO
1 1
1 1
si 0
si 0
y
0 y 0 0
ó
ó
n n
k k
k k
n n
k k
k k
a a
a
a a
a a
a a a a
a a a
ab a b a a
a b a b a a
= =
= =
≥⎧
= ⎨
− <⎩
= −
≤ − ≤
≥ = ⇔ =
= =
+ ≤ + ≤
∏ ∏
∑ ∑
EXPONENTES
( )
( )
/
p q p q
p
p q
q
qp pq
p p p
p p
p
qp q p
a a a
a
a
a
a a
a b a b
a a
b b
a a
+
−
⋅ =
=
=
⋅ = ⋅
⎛ ⎞
=⎜ ⎟
⎝ ⎠
=
LOGARITMOS
10
log
log log log
log log log
log log
log ln
log
log ln
log log y log ln
x
a
a a a
a a a
r
a a
b
a
b
e
N x a
MN M N
M
M N
N
N r N
N N
N
a a
N
N N N N
= ⇒
= +
= −
=
= =
= =
=
ALGUNOS PRODUCTOS
ad+( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( )
( )
( )
2 2
2 2 2
2 2 2
2
2
3 3 2 2 3
3 3 2 2 3
2 2 2 2
2
2
3 3
3 3
2 2 2
a c d ac
a b a b a b
a b a b a b a ab b
a b a b a b a ab b
x b x d x b d x bd
ax b cx d acx ad bc x bd
a b c d ac ad bc bd
a b a a b ab b
a b a a b ab b
a b c a b c ab ac bc
⋅ + =
+ ⋅ − = −
+ ⋅ + = + = + +
− ⋅ − = − = − +
+ ⋅ + = + + +
+ ⋅ + = + + +
+ ⋅ + = + + +
+ = + + +
− = − + −
+ + = + + + + +
1
1
n
n k k n n
k
a b a ab b a b
a b a a b ab b a b
a b a a b a b ab b a b
a b a b a b n− −
=
− ⋅ + + = −
− ⋅ + + + = −
− ⋅ + + + + = −
⎛ ⎞
− ⋅ = − ∀ ∈⎜ ⎟
⎝ ⎠
∑
( ) ( )
( ) ( )
( ) ( )
( )
2 2 3 3
3 2 2 3 4 4
4 3 2 2 3 4 5 5
( ) ( )
( ) ( )
( ) ( )
( ) ( )
2 2 3 3
3 2 2 3 4 4
4 3 2 2 3 4 5 5
5 4 3 2 2 3 4 5 6 6
a b a ab b a b
a b a a b ab b a b
a b a a b a b ab b a b
a b a a b a b a b ab b a b
+ ⋅ − + = +
+ ⋅ − + − = −
+ ⋅ − + − + = +
+ ⋅ − + − + − = −
( ) ( )
( ) ( )
1 1
1
1 1
1
1 impar
1 par
n
k n k k n n
k
n
k n k k n n
k
a b a b a b n
a b a b a b n
+ − −
=
+ − −
=
⎛ ⎞
+ ⋅ − = + ∀ ∈⎜ ⎟
⎝ ⎠
⎛ ⎞
+ ⋅ − = − ∀ ∈⎜ ⎟
⎝ ⎠
∑
∑
SUMAS Y PRODUCTOS
n
( )
( )
1 2
1
1
1 1
1 1 1
1 0
n k
k
n
k
n n
k k
k k
n n n
k k k k
k k k
n
k k n
k
a a a a
c nc
ca c a
a b a b
a a a a
=
=
= =
= = =
−
=
+ + + =
=
=
+ = +
− = −
∑
∑
∑ ∑
∑ ∑ ∑
∑
( )
1
( )
( )
( )
( )
( )
( )
( )
1
1
1
2
1
2 3 2
1
3 4 3 2
1
4 5 4 3
1
2
1
1 2 1
2
=
2
1
1 1
1
2
1
2 3
6
1
2
4
1
6 15 10
30
1 3 5 2 1
!
n
k
nn
k
k
n
k
n
k
n
k
n
k
n
k
n
a k d a n d
n
a l
r a rl
ar a
r r
k n n
k n n n
k n n n
k n n n n
n n
n k
n n
k
=
−
=
=
=
=
=
=
+ − = + −⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦
+
− −
= =
− −
= +
= + +
= + +
= + + −
+ + + + − =
=
⎛ ⎞
=⎜ ⎟
⎝ ⎠
∑
∑
∑
∑
∑
∑
∏
( )
( )
0
!
,
! !
n
n n k k
k
k n
n k k
n
x y x y
k
−
=
≤
−
⎛ ⎞
+ = ⎜ ⎟
⎝ ⎠
∑
( ) 1 2
1 2 1 2
1 2
!
! ! !
k
n nn n
k k
k
n
x x x x x x
n n n
+ + + = ⋅∑
CONSTANTES
9…3.1415926535
2.71828182846e
π =
= …
TRIGONOMETRÍA
1
sen csc
sen
1
cos sec
cos
sen 1
tg ctg
cos tg
CO
HIP
CA
HIP
CO
CA
θ θ
θ
θ θ
θ
θ
θ θ
θ θ
= =
= =
= = =
radianes=180π
CA
CO
HIP
θ
θ sin cos tg ctg sec csc
0 0 1 0 ∞ 1 ∞
30 1 2 3 2 1 3 3 2 3 2
45 1 2 1 2 1 1 2 2
60 3 2 1 2 3 1 3 2 2 3
90 1 0 ∞ 0 ∞ 1
[ ]
[ ]
sin ,
2 2
cos 0,
tg ,
2 2
1
ctg tg 0,
1
sec cos 0,
1
csc sen ,
2 2
y x y
y x y
y x y
y x y
x
y x y
x
y x y
x
π π
π
π π
π
π
π π
⎡ ⎤
= ∠ ∈ −⎢ ⎥
⎣ ⎦
= ∠ ∈
= ∠ ∈ −
= ∠ = ∠ ∈
= ∠ = ∠ ∈
⎡ ⎤
= ∠ = ∠ ∈ −⎢ ⎥
⎣ ⎦
Gráfica 1. Las funciones trigonométricas: sin x ,
cos x , tg x :
-8 -6 -4 -2 0 2 4 6 8
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
sen x
cos x
tg x
Gráfica 2. Las funciones trigonométricas csc x ,
sec x , ctg x :
-8 -6 -4 -2 0 2 4 6 8
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
csc x
sec x
ctg x
Gráfica 3. Las funciones trigonométricas inversas
arcsin x , arccos x , arctg x :
-3 -2 -1 0 1 2 3
-2
-1
0
1
2
3
4
arc sen x
arc cos x
arc tg x
Gráfica 4. Las funciones trigonométricas inversas
arcctg x , arcsec x , arccsc x :
-5 0 5
-2
-1
0
1
2
3
4
arc ctg x
arc sec x
arc csc x
IDENTIDADES TRIGONOMÉTRICAS
2
2
2 2
2 2
sin cos 1
1 ctg csc
tg 1 sec
θ θ
θ θ
θ θ
+ =
+ =
+ =
( )
( )
( )
sin sin
cos cos
tg tg
θ θ
θ θ
θ θ
− = −
− =
− = −
( )
( )
( )
( )
( )
( )
( ) ( )
( ) ( )
( )
sin 2 sin
cos 2 cos
tg 2 tg
sin sin
cos cos
tg tg
sin 1 sin
cos 1 cos
tg tg
n
n
n
n
n
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
θ π θ
+ =
+ =
+ =
+ = −
+ = −
+ =
+ = −
+ = −
+ =
( )
( ) ( )
( )
( )
sin 0
cos 1
tg 0
2 1
sin 1
2
2 1
cos 0
2
2 1
tg
2
n
n
n
n
n
n
n
n
π
π
π
π
π
π
=
= −
=
+⎛ ⎞
= −⎜ ⎟
⎝ ⎠
+⎛ ⎞
=⎜ ⎟
⎝ ⎠
+⎛ ⎞
= ∞⎜ ⎟
⎝ ⎠
sin cos
2
cos sin
2
π
θ θ
π
θ θ
⎛ ⎞
= −⎜ ⎟
⎝ ⎠
⎛ ⎞
= +⎜ ⎟
⎝ ⎠
( )
( )
( )
( )
( )
2 2
2
2
2
2
sin sin cos cos sin
cos cos cos sin sin
tg tg
tg
1 tg tg
sin 2 2sin cos
cos2 cos sin
2tg
tg2
1 tg
1
sin 1 cos2
2
1
cos 1 cos2
2
1 cos2
tg
1 cos2
α β α β α β
α β α β α β
α β
α β
α β
θ θ θ
θ θ θ
θ
θ
θ
θ θ
θ θ
θ
θ
θ
± = ±
± =
±
± =
=
= −
=
−
= −
= +
−
=
+
∓
∓
( ) ( )
( ) ( )
( ) ( )
( ) ( )
1 1
sin sin 2sin cos
2 2
1 1
sin sin 2sin cos
2 2
1 1
cos cos 2cos cos
2 2
1 1
cos cos 2sin sin
2 2
α β α β α β
α β α β α β
α β α β α β
α β α β α β
+ = + ⋅ −
− = − ⋅ +
+ = + ⋅ −
− = − + ⋅ −
( )sin
tg tg
cos cos
α β
α β
α β
±
± =
⋅
( ) ( )
( ) ( )
( ) ( )
1
sin cos sin sin
2
1
sin sin cos cos
2
1
cos cos cos cos
2
α β α β α β
α β α β α β
α β α β α β
⋅ = − + +⎡ ⎤⎣ ⎦
⋅ = − − +⎡ ⎤⎣ ⎦
⋅ = − + +⎡ ⎤⎣ ⎦
tg tg
tg tg
ctg ctg
α β
α β
α β
+
⋅ =
+
FUNCIONES HIPERBÓLICAS
sinh
2
cosh
2
sinh
tgh
cosh
1
ctgh
tgh
1 2
sech
cosh
1 2
csch
sinh
x x
x x
x
x
x x
x x
x x
x
e e
x
x e e
x
x e e
e e
x
x e e
x
x e e
x
x e e
−
−
−
−
−
−
−
=
+
=
−
= =
+
+
= =
−
= =
+
= =
−
x x
e e−
−
x
x
[
{ }
]
{ } { }
sinh :
cosh : 1,
tgh : 1,1
ctgh : 0 , 1 1,
sech : 0,1
csch : 0 0
→
→ ∞
→ −
− → −∞ − ∪ ∞
→
− → −
Gráfica 5. Las funciones hiperbólicas sinh x ,
cosh x , tgh x :
-5 0 5
-4
-3
-2
-1
0
1
2
3
4
5
senh x
cosh x
tgh x
FUNCIONES HIPERBÓLICAS INV
( )
( )
1 2
1 2
1
1
2
1
2
1
sinh ln 1 ,
cosh ln 1 , 1
1 1
tgh ln , 1
2 1
1 1
ctgh ln , 1
2 1
1 1
sech ln , 0 1
1 1
csch ln , 0
x x x x
x x x x
x
x x
x
x
x x
x
x
x x
x
x
x x
x x
−
−
−
−
−
−
= + + ∀ ∈
= ± − ≥
+⎛ ⎞
= <⎜ ⎟
−⎝ ⎠
+⎛ ⎞
= >⎜ ⎟
−⎝ ⎠
⎛ ⎞± −
⎜ ⎟= < ≤
⎜ ⎟
⎝ ⎠
⎛ ⎞+
⎜ ⎟= + ≠
⎜ ⎟
⎝ ⎠
2. Fórmulas de Cálculo Diferencial e Integral (Página 2 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M.
IDENTIDADES DE FUNCS HIP
2 2
sinh 1x x
( )
( )
( )
2 2
2 2
cosh
1 tgh sech
ctgh 1 csch
sinh sinh
cosh cosh
tgh tgh
x x
x x
x x
x x
x x
− =
− =
− = −
− =
− = −
− =
( )
( )
( )
2 2
2
sinh sinh cosh cosh sinh
cosh cosh cosh sinh sinh
tgh tgh
tgh
1 tgh tgh
sinh 2 2sinh cosh
cosh 2 cosh sinh
2tgh
tgh 2
1 tgh
x y x y x y
x y x y x y
x y
x y
x y
x x x
x x x
x
x
x
± = ±
± = ±
±
± =
±
=
= +
=
+
( )
( )
2
2
2
1
sinh cosh 2 1
2
1
cosh cosh 2 1
2
cosh 2 1
tgh
cosh 2 1
x x
x x
x
x
x
= −
= +
−
=
+
sinh 2
tgh
cosh 2 1
x
x
x
=
+
cosh sinh
cosh sinh
x
x
e x x
e x x−
= +
= −
OTRAS
( ) ( )
2
2
2
0
4
2
4 discriminante
exp cos sin si ,
ax bx c
b b ac
x
a
b ac
i e iα
α β β β α β
+ + =
− ± −
⇒ =
− =
± = ± ∈
LÍMITES
( )
1
0
0
0
0
1
lim 1 2.71828...
1
lim 1
sen
lim 1
1 cos
lim 0
1
lim 1
1
lim 1
ln
x
x
x
x
x
x
x
x
x
x e
e
x
x
x
x
x
e
x
x
x
→
→∞
→
→
→
→
+ = =
⎛ ⎞
+ =⎜ ⎟
⎝ ⎠
=
−
=
−
=
−
=
DERIVADAS
( )
( ) ( )
( )
( )
( )
( )
( )
0 0
1
lim lim
0
x
x x
n n
f x x f xdf y
D f x
dx x x
d
c
dx
d
cx c
dx
d
cx ncx
dx
d du dv dw
u v w
dx dx dx dx
d du
cu c
dx dx
∆ → ∆ →
−
+ ∆ − ∆
= = =
∆ ∆
=
=
=
± ± ± = ± ± ±
=
( )
( )
( ) ( )
( )
2
1n n
d dv du
uv u v
dx dx dx
d dw dv du
uvw uv uw vw
dx dx dx dx
v du dx u dv dxd u
dx v v
d du
u nu
dx dx
−
= +
= + +
−⎛ ⎞
=⎜ ⎟
⎝ ⎠
=
( )
( )
( )
( )
12
1 2
(Regla de la Cadena)
1
donde
dF dF du
dx du dx
du
dx dx du
dF dudF
dx dx du
x f tf tdy dtdy
dx dx dt f t y f t
= ⋅
=
=
=⎧′ ⎪
= = ⎨
′ =⎪⎩
DERIVADA DE FUNCS LOG & EXP
( )
( )
( )
( )
( )
( ) 1
ln
log
log
log
log 0, 1
ln
ln
a
a
u u
u u
v v v
u
dx u u dx
d e du
u
dx u dx
ed du
u a
dx u dx
d du
e e
dx dx
d du
a a a
dx dx
d du dv
u vu u u
dx dx dx
−
= = ⋅
= ⋅
= ⋅ >
= ⋅
= ⋅
= + ⋅ ⋅
1d du dx du
a ≠
DERIVADA DE FUNCIONES TRIGO
( )
( )
( )
( )
( )
( )
( )
2
2
sin cos
cos sin
tg sec
ctg csc
sec sec tg
csc csc ctg
vers sen
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u u
dx dx
d du
u u u
dx dx
d du
u u
dx dx
=
= −
=
= −
=
= −
=
d du
DERIV DE FUNCS TRIGO INVER
( )
( )
( )
( )
( )
( )
( )
2
2
2
2
2
2
2
sin
1
1
cos
1
1
tg
1
1
ctg
1
si 11
sec
si 11
si 11
csc
si 11
1
vers
2
u
dx dxu
d du
u
dx dxu
d du
u
dx dxu
d du
u
dx dxu
ud du
u
udx dxu u
ud du
u
udx dxu u
d du
u
dx dxu u
∠ = ⋅
−
∠ = − ⋅
−
∠ = ⋅
+
∠ = − ⋅
+
+ >⎧
∠ = ± ⋅ ⎨
1d du
− < −⎩−
− >⎧
∠ = ⋅ ⎨
+ < −⎩−
∠ = ⋅
−
∓
DERIVADA DE FUNCS HIPERBÓLICAS
2
2
sinh cosh
cosh sinh
tgh sech
ctgh csch
sech sech tgh
csch csch ctgh
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u
dx dx
d du
u u u
dx dx
d du
u u u
dx dx
=
=
=
= −
= −
= −
d du
DERIVADA DE FUNCS HIP INV
1
2
-1
1
-12
1
2
1
2
1
1
12
senh
1
si cosh 01
cosh , 1
si cosh 01
1
tgh , 1
1
1
ctgh , 1
1
si sech 0, 0,11
sech
si sech 0, 0,11
u
dx dxu
ud du
u u
dx dx uu
d du
u u
dx u dx
d du
u u
dx u dx
u ud du
u
dx dx u uu u
−
−
−
−
−
−
−
= ⋅
+
⎧+ >± ⎪
= ⋅ > ⎨
− <− ⎪⎩
= ⋅ <
−
= ⋅ >
−
⎧− > ∈⎪
= ⋅ ⎨
+ < ∈− ⎩
∓
1d du
1
2
1
csch , 0
1
d du
u u
dx dxu u
−
⎪
= − ⋅ ≠
+
INTEGRALES DEFINIDAS, PROPIEDADES
Nota. Para todas las fórmulas de integración deberá
agregarse una constante arbitraria c (constante de
integración).
( ) ( ){ } ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( ) ( ) ( )
( ) [ ]
( ) ( )
( ) ( ) [ ]
( ) ( )
0
, , ,
,
si
b b b
a a a
b b
a a
b c b
a a c
b a
a b
a
a
b
a
b b
a a
b b
a a
f x g x dx f x dx g x dx
cf x dx c f x dx c
f x dx f x dx f x dx
f x dx f x dx
f x dx
m b a f x dx M b a
m f x M x a b m M
f x dx g x dx
f x g x x a b
f x dx f x dx a b
± = ±
= ⋅ ∈
= +
= −
=
⋅ − ≤ ≤ ⋅ −
⇔ ≤ ≤ ∀ ∈ ∈
≤
⇔ ≤ ∀ ∈
≤ <
∫ ∫ ∫
∫ ∫
∫ ∫ ∫
∫ ∫
∫
∫
∫ ∫
∫ ∫
INTEGRALES
( ) ( )
( )
( )
1
Integración por partes
1
1
ln
n
n
adx ax
af x dx a f x dx
u v w dx udx vdx wdx
udv uv vdu
u
u du n
n
du
u
u
+
=
=
± ± ± = ± ± ±
= −
= ≠ −
+
=
∫
∫ ∫
∫ ∫ ∫ ∫
∫ ∫
∫
∫
INTEGRALES DE FUNCS LOG & EXP
( )
( )
( ) ( )
( )
( )
2
2
0
1ln
1
ln ln
1
ln ln ln 1
1
log ln ln 1
ln ln
log 2log 1
4
ln 2ln 1
4
u u
u
u
u
u
u u
a
a a
e du e
aa
a du
aa
a
ua du u
a a
ue du e u
udu u u u u u
u
udu u u u u
a a
u
u udu u
u
u udu u
=
>⎧
= ⎨
≠⎩
⎛ ⎞
= ⋅ −⎜ ⎟
⎝ ⎠
= −
= − = −
= − = −
= ⋅ −
= −
∫
∫
∫
∫
∫
∫
∫
∫
INTEGRALES DE FUNCS TRIGO
2
2
sin cos
cos sin
sec tg
csc ctg
sec tg sec
csc ctg csc
udu u
udu u
udu u
udu u
u udu u
u udu u
= −
=
=
= −
=
= −
∫
∫
∫
∫
∫
∫
tg ln cos ln sec
ctg ln sin
sec ln sec tg
csc ln csc ctg
udu u u
udu u
udu u u
udu u u
= − =
=
= +
= −
∫
∫
∫
∫
( )
2
2
2
2
1
sin sin 2
2 4
1
cos sin 2
2 4
tg tg
ctg ctg
u
udu u
u
udu u
udu u u
udu u u
= −
= +
= −
= − +
∫
∫
∫
∫
sin sin cos
cos cos sin
u udu u u u
u udu u u u
= −
= +
∫
∫
INTEGRALES DE FUNCS TRIGO INV
( )
( )
2
2
2
2
2
2
sin sin 1
cos cos 1
tg tg ln 1
ctg ctg ln 1
sec sec ln 1
sec cosh
csc csc ln 1
csc cosh
udu u u u
udu u u u
udu u u u
udu u u u
udu u u u u
u u u
udu u u u u
u u u
∠ = ∠ + −
∠ = ∠ − −
∠ = ∠ − +
∠ = ∠ + +
∠ = ∠ − +
= ∠ − ∠
∠ = ∠ + + −
= ∠ + ∠
∫
∫
∫
∫
∫
∫
−
INTEGRALES DE FUNCS HIP
2
2
sinh cosh
cosh sinh
sech tgh
csch ctgh
sech tgh sech
csch ctgh csch
udu u
udu u
udu u
udu u
u udu u
u udu u
=
=
=
= −
= −
= −
∫
∫
∫
∫
∫
∫
( )
( )1
tgh lncosh
ctgh ln sinh
sech tg sinh
csch ctgh cosh
1
ln tgh
2
udu u
udu u
udu u
udu u
u
−
=
=
= ∠
= −
=
∫
∫
∫
∫
INTEGRALES DE FRAC
( )
( )
2 2
2 2
2 2
2 2
2 2
tg
1
ctg
1
ln
2
1
ln
2
du
u a a a
u
a a
du u a
u a
u a a u a
du a u
u a
a u a a u
= ∠
+
= − ∠
−
= >
− +
+
= <
− −
∫
∫
∫
1 u
INTEGRALES CON
( )
( )
2 2
2 2
2 2
2 2 2 2
2 2
2
2 2 2 2
2
2 2 2 2 2 2
sin
cos
ln
1
ln
1
cos
1
sec
sen
2 2
ln
2 2
du u
aa u
u
a
du
u u a
u a
du u
au a u a a u
du a
a uu u a
u
a a
u a u
a u du a u
a
u a
u a du u a u u a
= ∠
−
= −∠
= + ±
±
=
± + ±
= ∠
−
= ∠
− = − + ∠
± = ± ± + ±
∫
∫
∫
∫
∫
∫
MÁS INTEGRALES
( )
( )
2 2
2 2
3
sin cos
sin
cos sin
cos
1 1
sec sec tg ln sec tg
2 2
au
au
au
e a bu b bu
e bu du
a b
e a bu b bu
e bu du
a b
udu u u u u
−
=
+
+
=
+
= + +
∫
∫
∫
au
ALGUNAS SERIES
( ) ( ) ( )( )
( )( )
( )
( )( )
( ) ( ) ( )
( )
( )
( )
( )
( )
2
0 0
0 0 0
0 0
2
2 3
3 5 7 2 1
1
2 4 6
''
'
2!
: Taylor
!
'' 0
0 ' 0
2!
0
: Maclaurin
!
1
2! 3! !
sin 1
3! 5! 7! 2 1 !
cos 1
2! 4!
nn
n n
n
x
n
n
f x x x
f x f x f x x x
f x x x
n
f x
f x f f x
f x
n
x x x
e x
n
x x x x
x x
n
x x x
x
−
−
−
= + − +
−
+ +
= + +
+ +
= + + + + + +
= − + − + + −
−
= − + − ( )
( )
( ) ( )
( )
2 2
1
2 3 4
1
3 5 7 2 1
1
1
6! 2 2 !
ln 1 1
2 3 4
tg 1
3 5 7 2 1
n
n
n
n
n
n
x
n
x x x x
x x
n
x x x x
x x
n
−
−
−
−
−
+ + −
−
+ = − + − + + −
∠ = − + − + + −
−
3. Fórmulas de Cálculo Diferencial e Integral (Página 3 de 3) http://www.geocities.com/calculusjrm/ Jesús Rubí M.
ALFABETO GRIEGO
ayúscula Minúscula NombreM Equivalente
Romano
1 Α α Alfa A
2 Β β Beta B
3 Γ γ Gamma G
4 ∆ δ Delta D
5 Ε ε Epsilon E
6 Ζ ζ Zeta Z
7 Η η Eta H
8 Θ θ ϑ Teta Q
9 Ι ι Iota I
10 Κ κ Kappa K
11 Λ λ Lambda L
12 Μ µ Mu M
13 Ν ν Nu N
14 Ξ ξ Xi X
15 Ο ο Omicron O
16 Π π ϖ Pi P
17 Ρ ρ Rho R
18 Σ σ ς Sigma S
19 Τ τ Tau T
20 Υ υ Ipsilon U
21 Φ φ ϕ Phi F
22 Χ χ Ji C
23 Ψ ψ Psi Y
24 Ω ω Omega W
NOTACIÓN
Seno.sin
cos Coseno.
tg Tangente.
sec Secante.
csc Cosecante.
ctg Cotangente.
vers Verso seno.
arcsin sinθ θ= Arco seno de un ángulo θ .
( )u f x=
sinh Seno hiperbólico.
cosh Coseno hiperbólico.
tgh Tangente hiperbólica.
ctgh Cotangente hiperbólica.
sech Secante hiperbólica.
csch Cosecante hiperbólica.
, ,u v w Funciones de x , , .( )u u x= ( )v v x=
Conjunto de los números reales.
{ } Conjunto de enteros., 2, 1,0,1,2,= − −… …
Conjunto de números racionales.
c
Conjunto de números irracionales.
{ }1,2,3,= … Conjunto de números naturales.
Conjunto de números complejos.