VERIFIQUE LAS SIGUIENTES IDENTIDADES
a)
1
tan + cot
1
tan + cot
=
=
1
sec
csc
=
b)sec4
sec4
1
sec csc
+
1
1
=
sec2 + csc2
csc sec
1
=
2
sec + csc2
csc sec
csc
sec
csc sec
sec2 + csc2
=
csc sec
sec2 csc2
=
1
csc sec
sec2 = tan4 + tan2
sec2 = sec2 sec2
1 = sec2 tan2
2
2
= tan +1 tan = tan4 + tan2
= sec2
1 + 1 tan2
c)sec4
tan4 = sec2 + tan2
4
sec
sec2 = tan4 + tan2
4
! sec = tan4 + tan2 + sec2
! sec4
tan4 = tan2 + sec2
d)
cos4
sin4
cos + sin
= cos
sin
sin2
cos2 + sin2
cos + sin
cos2
sin2 (1)
(cos
sin ) (cos + sin )
=
=
cos + sin
cos + sin
cos4
sin4
cos + sin
=
cos2
= cos
sin
1
tan + cot
sin cos
sin cos =
=
sin2 + cos2
e)sin cos =
=
sin
cos
f)tan + cot =
+
cos
sin
sin2 + cos2
sin cos
1
=
sin2
sin cos
=
1
tan + cot
1
= (tan + cot )
1
sin cos
1
1
+
cos2
sin cos
1
sin cos =
1
tan + cot
! sin cos (tan + cot ) = 1
1
! tan + cot =
sin cos
csc
cos
g)
+
= 2 cot
sec
sin
cos
csc sin + sec cos
csc
+
=
sec
sin
sec sin
h)
=
1+1
2
=
tan
tan
tan2 + 1
= csc2
tan2
tan2 + 1
sec2
=
tan2
tan2
sec2
= 12
sec
csc2
=
sec2 csc2
sec2
= csc2
cot
sec
+
= sec2 csc
cos
cot
sec
cot2 + cos sec
cot2 + 1
cot
+
=
=
cos
cot
cos cot
sin
1
2
=
cot + 1 = csc sec2
sin
i)
r
1 + cos
sin
j)
=
1 cos s1 cos
s
r
1
1 + cos
(1 + cos ) (1 cos )
=
=
1 cos
(1 cos ) (1 cos )
(1
s
2
sin
sin
=
2 = 1
cos
(1 cos )
cos2
2
cos )
k)sin6 + cos6 = 1 2 sin2 cos2
3
3
sin6 + cos6 = sin2
+ cos6 = 1 cos2
+ cos6
2
4
6
= 1 3 cos + 3 cos
cos
+ cos6
2
4
= 1 3 cos + 3 cos = 1 3 cos2 1 cos2
= 1 3 cos2 sin2
l)sec6
sec6
tan6 = 1 + 3 sec2 tan2
tan6 = sec6 1 sin6 = sec6
= sec6 cos2
1 + sin2 + 1
2
1
sin2
cos2
1 + sin2 + sin4
2
= sec4
= sec4
= sec4
= sec4
= 3 sec4
1 + sin2 + 1 2 cos2 + cos4
2 2 cos2 + sin2 + cos4
2 1 cos2 + sin2 + cos4
2 sin2 + sin2 + cos4 = sec4 3 sin2 + cos4
sin2 + sec4 cos4 = 3 tan2 sec2 + 1
Veri…que las siguientes identidades
3 sin + 8 csc
3
i.
= + 2 csc2
4 sin
4
8 csc
3 2 csc
3
2
3 sin
+
= +
= +
4 sin
4 sin
4
sin
4 sin2
7 sec4
7 tan4
=7
sec2 + tan2
4
4
7 sec
7 tan
sec4
tan4
=
7
sec2 + tan2
sec2 + tan2
=
3
+ 2 csc2
4
ii.
= 7 sec2
=7
tan2
sec2 + tan2
sec2
sec2 + tan2
=7
1
sin2
cos2
csc3 + sin3
= cot2 + sin2
csc + sin
csc3 + sin3
= csc2
csc sin +sin2 = csc2
csc + sin
tan2
=7
iii.
1+sin2 = cot2 +sin2
cot3
tan3
= sec2 + csc2
1
cot
tan
cot3
tan3
= cot2 + tan cot + tan2 = cot2 + 1 + tan2
cot
tan
= cot2 + sec2
iv.
v.
cos A + cos B
= cos A cos B
sec A + sec B
cos A + cos B
cos A + cos B
cos A + cos B
=
=
1
1
cos B + cos A
sec A + sec B
+
cos A cos B
cos A cos B
=
cos A cos B (cos A + cos B)
= cos A cos B
cos A + cos B
3
1
= tan sec
csc
sin
1
sin
sin
=
=
csc
sin
cos2
1 sin2
vi,
vii,cos2
cos2
1 + cot2
1 + tan2
= cos2
viisin2
sin2
= sec tan
=1
= cos2
1
cos2
1 + cot2
1 + cot2
1 sin
cos cos
=
cos2
=
+ sin2
cos2
cos2
cos2
=1
=1
cos2
sin2
= sin2
1+
= sin2
1
sin2
= sin2
=
sin2
sin2
sin2
+ cos2
sin2
=1
sin
1 + cos
+
= 2 csc
1 + cos
sin
2
sin
1 + cos
sin2 + (1 + cos )
sin2 + 1 + 2 cos + cos
+
=
=
1 + cos
sin
(1 + cos ) sin
(1 + cos ) sin
1 + 1 + 2 cos
2 (1 + cos )
=
=
(1 + cos ) sin
(1 + cos ) sin
2
=
= 2 csc
sin
ix.
x. sin2 + cos2 = 1
sin2 + cos2 =
p
tan
2
+
1 + tan2
tan2
1
=
+
1 + tan2
1 + tan2
p
1
1 + tan2
1 + tan2
=
1 + tan2
MAXIMOS y Minimos
f (x) = x2 9
f 0 (x) = 2x ! 2x = 0 ! x = 0
f 00 (x) = 2 > 0 ! x = 0 minimo
f (0) = 0 9 = 9
4
2
=1
2
f (x) = 16 x2
f 0 (x) = 2x ! x = 0 punto critico
f 00 (x) = 2 < 0 ! x = 0 Maximo
f (0) = 16
f (x) = x2 + 2x + 1
f 0 (x) = 2x + 2 ! 2x + 2 = 0 ! 2x =
f 00 (x) = 2 > 0 ! x = 1 minimo
2
f ( 1) = ( 1) + 2 ( 1) + 1 = 0
2!x=
1
f (x) = 2x + 3
f 0 (x) = 2 no tine puntos criticos, siempre es creciente
no tiene maximos ni minimos
f (x) = 3 4x + x2
f 0 (x) = 4 + 2x ! 4 + 2x = 0 ! x = 2
f 00 (x) = 2 > 0 ! x = 2 Minimo
f (x) = x 1=2
f 0 (x) = 1 ! f uncion creciente
no tiene maximo ni minimo
f (x) = x3 + 3x2
25x
75
p
36 + 300
2p
= 1
21
f (x) = 3x + 6x 25 = 0 ! x =
6
3
2p
x=
21 1
3
00
f (x) = 6x + 6 ! 6x + 6 = 0
! x = 1 punto in‡eccion x<1 f(x) concaba hacia abajo
x>1 f(x) concava hacia arriba
2p
! 1+
21 minimo
3
p
2
! 1
21 maximo
3
p
2
f( 1 +
21) =
3
3
2
2p
2p
2p
=
1+
21 + 3
1+
21
25
1+
21
75
3
3
3
=-45,02760865
0
2
6
5
2p
21)
3
3
2
2p
2p
=
1
21 + 3
1
21
25
3
3
=69,02760865
3
2
f ( 1) = ( 1) + 3 ( 1)
25 ( 1) 75 = 12
f( 1
1
2p
21
3
f (x) = 64x 256 x3 + 4x2
2
2
f 0 (x) = 64
p 3x + 8x ! 3x + 8x + 64 = 0
p
8
64 + 768
4
!
=
1 + 13
6
3
p
p
8 + 832
4
!
=
1
13
6
3
f 00 (x) = 6x + 8 ! 6x + 8 = 0 ! x = 4=3 punto in‡exion
para x<4/3 la funcion es concava hacia arriba
para x>4/3 es concava hacia abajo
p
4
1 + 13 punto maximo
3
p
4
1
13 punto minimo
3
f (x) = x2
f 0 (x) = 2x ! x = 0
f 00 (x) = 2 > 0
! x = 0 Minimo
f (0) = (0)2 = 0
f (x) = x
f 0 (x) = 1 no tiene maximo ni minimo
la funcion es creciente
f (x) = x2
f 0 (x) = 2x ! 2x = 0 ! x = 0
f 00 (x) = 2 < 0
! x = 0 Maximo
f (0) = (0)2 = 0
f (x) = x
f 0 (x) = 1 no tiene maximo ni minimo
la funcion es decreciente
f (x) = x2
16
6
75
f 0 (x) = 2x ! 2x = 0 ! x = 0
f 00 (x) = 2 > 0 !la funcion es concava hacia arriba
! x = 0 es minimo
f (x) = 3x + 1
f 0 (x) = 3 no tiene maximo ni minimo
la funcion es decreciente
f (x) = x3 + x2 9x 9
f 0 (x) = 3x2 + 2x ! x (3x + 2) = 0
x = 0; x = 2=3
0
f 0 (x) = 6x + 2 ! 6x + 2 = 0
! x = 1=3 punto in‡exion
para x>-1/3 f(x) es concava hacia arriba
para x<-1/3 f(x) es concava hacia abajo
Hallese
y = cos
1
y0 =
2x
1 x2
p
x2 ! cos y = x2 !
dy
dx
dy
2x
dy
sin y = 2x !
=
dx
dx
sin y
y = cos 1 (1=x) ! cos y = (1=x)
dy
1
dy
1
!
sin y =
!
= 2
2
dx
x
dx
x sin y
y = sec 1 5x ! sin y = 5x
dy
5
dy
cos y = 5 !
=
!
dx
dx
cos y
y = sec 1 (5x) ! sec y = 5x
dy
dy
5
!
tan y sec y = 5 !
=
dx
dx
tan y sec y
p
p
y = tan 1 x ! tan y = x
dy
1
dy
1
p
!
csc2 y = p !
=
2
dx
dx
2 x
2 csc y x
y = cot 1 (1 x) ! cot y = (1 x)
dy
dy
1
( csc y) = x !
=
= sin2 y
dx
dx
csc2 y
y = cos 1 (2x) ! cos y = 2x
7
!
dy
2
=
=
dx
sin y
y = 5 tan
!
1
sec2
5
1
2 csc y
3x ! tan
1
y
5
1
y
5
= 3x
dy
dy
=3!
=
dx
dx
y = sen 1 (1 x) ! seny = 1
dy
dy
x
cos y
= x!
=
dx
dx
cos y
15
1
sec2
y
5
= 15 cos2
1
y
5
x
y = csc 1 x2 + 1 ! csc y = x2 + 1
dy
sin2 y
cos y dy
=
2x
!
=
2x
=
dx
cos y
sin2 y dx
2x sin y tan y
x
x
y = sin 1
! sin y =
2
2
dy
1
dy
1
cos y
= !
= sec y
dx
2
dx
2
1
x
x
tan 1
! tan (3y) =
3
3
3
dy
1
dy
1
3
2
= !
= cos (3y)
cos2 (3y) dx
3
dx
9
y=
p
p
y = csc 1 x ! csc y = x
cos y dy
dy
1
= p !
=
2
dx
dx
2
x
sin y
sin2 y
p
=
2 x cos y
p
x sin y tan y
2x
p
p
y = cot 1 x 1 ! cot y = x 1
p
1 dy
1
dy
sin2 y
x 1 sin2 y
p
p
=
!
=
=
2 dx
dx
2 (x 1)
2 x 1
2 x 1
sin y
p
csc y + sec y = 2 x
dy
dy
1
1
( csc y cot y + sec y tan y)
=p !
=p
dx
dx
x
x (sec y tan y
x 1
x 1
1
y = sin
! sin y =
x+1
x+1
dy
x+1 x+1
2
cos y
=
=
2
2
dx
(x + 1)
(x + 1)
dy
2 sec y
!
=
2
dx
(x + 1)
y = csc
1
p
x + sec
1
p
x
8
csc y cot y)
Calcular los siguientes limites
2
(x + h)
x2
((x + h) x) ((x + h) + x)
= lim
h!0
h!0
h
h
(h) (2x + h)
= lim
= lim (2x + h) = 2x
h!0
h!0
h
h
i
2
2
3
((x
+
h)
x)
(x
+
h)
+
(x
+
h)
x
+
x
3
(x + h)
x
lim
= lim
h!0
h!0
h h
i h
2
2
(h) (x + h) + (x + h) x + x
2
= lim (x + h) + (x + h) x + x2
= lim
h!0
h!0
h
2
= (x) + (x) x + x2 = 3x2
lim
3
2
4
((x + h) x) (x + h) + (x + h) x + (x + h) x2 + x3
(x + h)
x4
= lim
lim
h!0
h!0
h
h
3
2
(h) (x + h) + (x + h) x + (x + h) x2 + x3
= lim
h!0
h
3
2
= lim (x + h) + (x + h) x + (x + h) x2 + x3 = x3 + x2 x + xx2 + x3 = 4x3
h!0
p
p
p
p
x
x+h+ x
p
lim
= lim
p
h!0
h!0
h x+h+ x
(x + h x)
h
= lim p
p = lim p
p
h!0 h
h!0
x+h+ x
h x+h+ x
1
1
1
p = p
= lim p
p =p
h!0 x + h +
x+ x
2 x
x
p
x+h
h
x3 + 3x2
x!5
x
lim
p
x
25x
5
75
x+h
= lim
(x
x!5
5) x2 + 8x + 15
x 5
x3 + 3x2 + 25x 40x 75
25 + 40 + 15 = x3 +
x3 + 3x2 25x 75=x 5
= lim x2 + 8x 15 = 52 + 8 5
15 = 80
x!5
lim
x3
x!4
(x
x2 16x + 16
= lim
x!4
x 4
= lim x2 + 3x
x!4
4
4 = 16 + 12
4) x2 + 3x
x 4
4
=
4 = 24
(x a) x + ax2 + a2 x + a3
x
a
= lim
x!a x
x!a
a
x a
= lim x3 + ax2 + a2 x + a3 = a3 + aa2 + a2 a + a3 = 4a3
4
3
lim
x!a
9
(x 3) x2 + 3x + 9
x3 27
= lim
x!3
x!3 x
3
x a
= lim x2 + 3x + 9 = 9 + 9 + 9 = 27
lim
x!3
(x + 2) x2 2x + 4
x3 + 8
lim
= lim
x! 2 x + 2
x! 2
x+2
= lim x2 2x + 4 = 4 + 4 + 4 = 12
x! 2
x3 + 2x2
x2 (x + 2)
x+2
2
lim 4
=
lim
= lim 2
=
=
3
2
2
2
x!0 x + x
x!0 x (x + x
6x
6) x!0 x + x 6
6
1
3
Derivadas
f ( ) = tan3 (2 ) ! f 0 ( ) = 3 tan2 (2 ) sec2 (2 ) (2) = 6 tan2 (2 ) sec2 2
f ( ) = sec6 (3 ) ! f 0 ( ) = 6 sec5 (3 ) sec (3 ) tan (3 ) 3
f 0 ( ) = 18 sec6 (3 ) tan (3 )
f ( ) = sin5 (2 ) ! f 0 ( ) = 5 sin4 (2 ) cos (2 ) 2 = 10 sin4 (2 ) cos (2 )
f( ) =
p
3
cos4 (3 ) ! f 0 ( ) =
f( ) =
p
5
3 csc2 (3 )
cot(3 ) ! f 0 ( ) = p
5
cot2 (3 )
4
cos1=3 (3 ) ( sin (3 )) 3
3
p
= 4 cos1=3 (3 ) sin (3 ) = 4 sin (3 ) 3 cos (3 )
p
2
f ( ) = 3 csc2 (2 ) ! f 0 ( ) = csc 1=3 (2 ) ( csc (2 ) cot(2 ) 2
3
p
4
4
2=3
=
csc (2 ) cot(2 ) =
cot(2 ) 3 csc2 (2 )
3
3
p 3
f ( ) = e tan (2
)
! f 0( ) =
p 3
3p 3
tan (2 ) sec2 (2 )e tan (2
2
2
2
f ( ) = esec (5 ) ! f 0 ( ) = 2 sec(5 ) (sec (5 ) tan (5 )) esec
2
= 2 sec2 (5 ) tan (5 ) esec (5 )
q
)
(5 )
1
2=3
p
sin2 (2 ) ! f 0 (x) = q
2 cos(2 )
3
3
2
sin(2
)
sin (2 )
= 4=3 cot(2 )
f ( ) = ln
3
10
3
f ( ) = ln ecos
=
(3x)
! f 0 (x) =
9 cos (3x) sin (3x)
1
ecos3 (3x)
11
3
ecos
(3x)
3 cos2 (3x) ( 3 sin (3x))
0
