4. 03.
2
2
1
1
2
3
3
2 2 3
2 2
2
2 2 2
( ) 2. ; ; 4 2
2 0 (2 ) 0
/ 0
4 2 0
/ 2
( ) / 0 ; ; / 2
:
2 2
4 2 4 2
4
f t t t t t
D t t t t
D R t
D R
D t
D R t
Dom f t R t R R t
Rang o
x t t x t t
y t y t
z t z t
x y z
5. 04.
1
0
2
2 2
6
1 1
2
0 0
1 1
2 2
0 0
1 1
2 2
0 0
( ) 0,1 3 / ( ); ( )
1
( ) ; ;
1 4 5
4
:
( )
1
*
1 1
1
* *
1 1
t
t
t
x t
t t
t t t
t t
f t R f t halla f t
e t
f t
e t t
t
PARA X
e
X f t d t
e
e e d v
e e e
v d v
d v
v
e e
t
t
t
dv
dx v e
e
dv
e
dx
1
2
0
1
2 2
0
1
*
1
arc tan( ) ( )
arc tan( )
t
u
t
d u v e
u
u e u
e
6. 1 1 2
6
0 0
1 2
2
6
0
1
6
0
1
2
0
:
( )
4
1
*
3
4
1
3 * 4
1 1
2 2
PARA y
t
y f t y d t
t
t
d u
t
t
d u
t
d u
u
2
1
2
2
0
1
2
2
0
2
1
0
2
1
3 3
0
1
3 6
0
.
4 1 2
2 tan( )
2 sec ( )
1
* 2 sec ( )
2 tan ( ) 4
2 1
* sec ( )
2 tan ( ) 1
sec ( )
tan ( ) 1
1 1 1
ln tan arc tan( * ) arc tan( *
3 2 3
1 4
ln
3 2
POR SUST TRIGON OM ÉTRICA
a
a b
b
t u
d u
u
d v
u d u
u
u d u
u
u
u
t sen t
t t
3 2
1 ln* 4
lim ( ) 0 *
3 2
1 1 5
* ln
3 2
t t
t
y
7. 1 1
2
0 0
1
2
0
1
2
0
3
3 2
2 2
2
2
3
1
0
1
0
1
2
:
1
( )
4 5
1
( 2 ) 1
1
1
1
1
1
1
arc tan
tan 3 arc tan 2
: ( )
( ; ; )
1 5
1
( ) arc tan ; ln( ); arc tan 3 arc tan 2
4 2 3
z
PA RA Z
f t d t
t t
d t
t
d t
v
d u
v
v
p or form ula
Z arc t
halland o f t
x y z
f t
8. 3
/2
2 2 0
/2 /2 /2 /2
2 2
0 0 0 0
Sea f(t):[0; /2] R f(t)
cos
( ) ; cos ; 3 ; ( )
cos dt
* ( ) cos. 3 .
1
t
f t t t sen t HALLAR f t
a sen t
t
f t t dt sen t dt
a sen t
/2
2
0
/2
2
0
/2
0
2
cos
.
.
.
.
cos arccos( )
tan( )
arctan( )
t
dt
a sent
u
du
a sent sent
u
du
sent a sent
v t t u
arc a
a
a
a
/2
0
1
/2
0
/2
0
cos.
Por partes:
v=t v cos
( ) 1
. cos
( cos )
cos
cos 1
2 2 2 2
t dt
t
d
v t
dt
v sent dt t
tsent t
tsent t
sen
/2 /2
0 0
2 2 2
/2
0
2
/2
0
2
/2
0
3 . 3 ,
1 ( / 2) (0)
3 3. 3.
2 2 2
ˆ
3.
8
Por lo tanto: Hallar ( )
tan 2 3
ˆ ˆ
( ; ; ) ; ;
2 8
sen t dt sen t dt
sen sen sen
sen K
f t
ar a sen
J K
a
05.
9. 06.
1
2
1 2 2
1
1
Sea f(t) = (cost; ; )
2
Hallar: T(t); N(t); T(t).N(t)=?
1
*f ( ) ; ;cos
2
1
* ( ) ( ) (cos )
2
( )
( )
( )
1/ 2 cos
( ) ; ;
1 1 1
1 1 1
4 4 4
2 1 2
* ( ) . ; ; .
5 5 5
t
sent
t sent t
f t sent t
f t
T t
f t
sent t
T t
T t sent
1
1
1
2
1
2 2
1 2
1
cos
( )
2
( ) cos ;0;
5
2
( ) cos 0
5
2
( )
1/ 5
t
T t
N t
T t
T t t sent
T t sent
T t
2
cos ;0;
5
( )
2
5
* ( ) cos ;0:
( ) ( ). ( )
ˆ ˆ ˆ
2 2
( ) . .cos
5 5 5
cos 0
2 2
2
. cos
1/ 5 .cos ˆ ˆ
( ) 5 5 5
0 cos
2 / 5. / 5
cos 0
t sent
N t
N t t sent
B t T t N t
J K
B t sent t
t sent
sent t
t
B t J
sen t sent
sent
t
2 2
ˆ
1 2 2 1
ˆ ˆ
( ) . 0 . cos cos
5 5 5 5
1 2 1
( ) ; ; cos
5 5 5
K
B t sent sen t t J t
B t sent t
10. 07.
4 3 2 15
3 3 2
4 3 15
3 3 2
3 2
3 2
3 2
( 2 2 1)(1 8)
... 0
(2 1) ( 4 4)(2 1)
( 2 2 1)( 8) ...
(2 1) ( 4 4)(2 ) 0
( 4(2 1) (2 1)( 1) 0
1 ( 4)(2 1) (2 1)( 1) 0 1
( 4)(2 1) (2 1)( 1) 0
1 1
2
t t t
i
t t t t
t t t t t
t t t t t
t t t t
t t t t
t t t t
t
( )
4...
2
t
t t
Domf
Rango