Guía Semana 8 - Docencia DIM-UChile

Guía
!"#!$#%&' (')#*+)$,' Semana 8
Ingeniería Matemática
FACULTAD DE CIENCIAS
FÍSICAS Y MATEMÁTICAS
UNIVERSIDAD DE CHILE
Cálculo Diferencial e Integral 08-1
-!$.#%/$0'0 0# 12$3#
Ejercicios
!" f :
45
Rp
#"$ %&! f !' (!)*+,*-" ,! (!)*.,. p/ 0)&!1! %&!
→
a+p
R
f (x) =
a
f (x) (")" #.,. a ∈
/
0
65
2"-!) &3" "'!4!)"-*+3 5!3!)"$ )!$"#*4" "
f (x)dx (")" f &3" 6&3-*+3 *7(")
−a
8 .#)" (")" f 6&3-*+3 (")/
75
Ra
Rb
9!7&!'#)! %&! '* f !' &3" 6&3-*+3 -.3#*3&" !3 [a, b] 8
!;*'#! &3 c !3 [a, b] #"$ %&! f (c) = 0/
85
2"$$")
Rb
Rb
a
95
f (x)g(y)dy
a
!
dx !3 #<)7*3.' ,!
Rb
Rb
f 8
a
Rx
f (x) = 0: !3#.3-!'
a
g/
a
2"$$") F 0 (x) '* F (x) = xf (t)dt/
0
:5
9!7.'#)") %&! '* f !' -.3#*3&" !3#.3-!'
Rx
f (u)(x − u)du =
0
;5
Rx Ru
0
f (t)dt du/
0
&(.35" %&! f !' *3#!5)"1$! !3 [a, b]/ 9!7.'#)") %&! !;*'#! &3 3=7!). x !3
[a, b] #"$ %&!
Rx
a
f=
Rb
f / 9!7.'#)"): -.3 &3 !>!7($.: %&! 3. '*!7()! !' (.'*1$!
x
!$!5*) x %&! !'#< !3 (a, b)/
<5
?"$-&$! $"' ,!)*4","' ,! $"' '*5&*!3#!' 6&3-*.3!'/
f (x) =
Rx2
sen(t4 )dt
f (x) =
1
Rx2
√
t2
1+t6 dt
f (x) =
cos(x)
R
(x − t) sen(t2 )dt
x3
x
Problemas
=45 >'?
!" f : [0, ∞[→ [0, ∞[ &3" 6&3-*+3 1*8!-#*4" 8 ,!)*4"1$! !3 ]0, ∞[/
@&!'#)! %&! g(x) =
Rx
f (t)dt +
0
fR(x)
f −1 (t)dt: '"#*'6"-! %&! g 0 (x) =
0
f (x) + f 0 (x)x/ ?.3-$&8" %&! g(x) = xf (x)/
=65 >'?
?.3'*,!)! $" 6&3-*+3 g(x) ,!A3*," (.) g(x) =
0
'! ,!A3! !3 -!). (.) -.3#*3&*,",/
>'?
9!7&!'#)! %&!B
R1
g(x)dx = g(1) −
0
>@?
Rx arctan(t)
R1
t
: ,.3,!
arctan(t)
t
arctan(t)dt
0
C#*$*D"3,. $. "3#!)*.): 7&!'#)! %&! B
R1
0
EF
g(x)dx = g(1) −
π
4
+
ln(2)
2 /
!"#!$#%&' (')#*+)$,'
-!$.#%/$0'0 0# 12$3#
456
!" g : → #$" %#$&'($ )'*!&+',"- .'%!/!$&'")0! * +"0 1#! g(0) = 02
!" f :
→] − 1, 1[ #$" %#$&'($ .'%!/!$&'")0!2 #34$5" 1#! f * g
6"+'6%"&!$7
g(x) =
g(x)
Z
f 2 (g −1 (x))dx + f (x).
0
4'6
8/#!)! 1#! f (x) = tanh(g(x))2
456
9"0&#0! 0" '$+!5/"0
Rx3
(tanh(t))2 dt2
0
!"#$%$#&!'
789 4'6
!" f (x) :=
:)6!/,! 1#! f (g −1 (x)) = tanh(x)2
x
Z
x ln(tx) dt- .!;$'." !$ ]0, +∞[2
Z
4:6 <$&#!$+/!
ln(t) * &"0&#0! f (2)2
1
=!>#!6+/! 1#! f 0 (x) = (4x − 1) ln(x) ∀x ∈]0, +∞[2
?6#>'!$.4 1#! 0" %#$&'($ g(t) = arc sen(arctan(t)) !6 &4$+'$#" !$
4;6
456
[0, tan(1)]- !$&#!$+/! 0" .!/',"." .! 0" %#$&'($ f (x) =
tan(x)
arc sen(arctan(t)) dt
0
3"/"
x ∈ [0, 1]2
7<9
Z
!" f : [a, b] → "&4+"." ! '$+!5/")0!- ,!/';&"$.4 1#! f ((a+b)−x) = f (x)
3"/" +4.4 x ∈ [a, b]2
4'6
8/4)"/ 1#!
Rb
xf (x) =
a
456
!" "@4/" g : [−1, 1] →
π
2
4,6
a+b
2
Rπ
Rb
f (x)
a
&4$+'$#"2 8/#!)! 1#!
Rπ
xg(sen(x)) =
0
g(sen(x))2
0
=!.#A&" 1#!
Rπ
0
x sen(x)
1+cos2 (x)
=
π
2
R1
−1
BC
1
1+x2
* &"0&#0! !0 ,"04/ .! 0" '$+!5/"02