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