unidad 12 - integración, problemas resueltos

8/10/2019 Unidad 12 - Integracin, problemas resueltos

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1 Bachillerato - Matemticas - Unidad 12 - Integracin

Pg.289 I a) as rectas concretas se cortan en ( )01 ! "or tanto la 1=x es ino"erante. #l rea triang$lar se limita "or las rectas% 0=y 33 = xy xx =

Base & 1x 'lt$ra & 33 x ( ) = xFrea ( ) ( )3312

1 xx

( ) ( )2123 = xxF ( ) ( ) = 11223

' xxF ( ) ( )13' = xxF

() 30 x a recta xy= "asa "or origen ( )00 #l rea triang$lar est limitada "or% 0=y xy= xx =

Base & 0x 'lt$ra & x ( ) xxxFrea =2

1

( ) 22

1xxF = ( ) = xxF 2

2

1' ( ) xxF ='

63


2/14


( ) 2344

1xxxxf += ( ) 23

2

1

3

1xxxg = [ ]( ) 234

2

3

3

4

4

1xxxxgf +=

,rea & [ ]( ) [ ]( ) [ ]( ) [ ]( )=+ 1301 fgfggfgf

[ ]( ) [ ]( ) [ ]( )== 3012 gfgfgf =

12

270

12

52 =

12

52

6

27

Pg.29 8 ( )x

yxf 1' = ( ) xyxg =' ( ) 2' =xxh +orte 'f :'g =x

x

1 1=x

,rea & ring$lo ra"ecio c$r3il4neo! am(os so(re e5e hori6ontal

ring$lo ( )=== 10 xyxy ,rea & = 112

1

2

1

ra"ecio

==== 02

11 yx

xyx ( ) = 2

1

xxg ,rea & ( ) ( )= 12 gg

4

3

,rea glo(al & =+4

3

2

1

4

5

Pg.297 9 a) ( ) =+ dxxx 132 24 Cxxx ++ 35

52

() ( ) =+ dxxxx 2123 =+++ ++ 121124

121

1

12

1

4

1xxx Cx

xx +++ 34

3

21

4

1

c) =

+ dxx

xx 223 =+

+

+++

xxx

12

11

1

123

1 11123 CLxxx

++ 22

12 2

d) ==+= + 61316767

13

6

167

1xxdxx Cx +6 13

13

6

Pg.297 10 a) = dxe x3 xe3 () =

dxx

x 43 CLx

Lx + 43

3

1

c) =

dx

xsenx

21

15 Cxx ++ arccoscos5

d) ( = dxxx cos3 ( ) =++= + Csenxxdxxx 1323/2

132

1cos Csenxx +3 4

4

3

Pg.29 11 a) ( ) ( ) =+++ dxttt 21

22.32 212

Ctt +++ 322

() =+

xx

e

e2

2

1

2

2

1( ) CeL x ++ 21

2

1

c) ( ) =+ dxxx 2125 202

( ) =+++ +

Cx 1202

1120

1

2

5

( ) Cx ++ 212

142

5

d) ( ) = dttLt 1

cos ( ) CLtsen +

e) ( ) =

+ dse

e

s

s 21

1( ) Cearctg s +

) ( ) =

dxx

x2

1

1

22 ( ) Cxarcsen +2

Pg.29 12 a) ( ) ( ) = dxxsenxx 2cos 242 ( ) =+

+

+Cx

142cos

14

1Cx + 25cos

5

1

2/10


3/14


()( )

( ) =++

dxx

xsen

23cos

23

( ) ( )( ) =++ dxxsenx 32323cos

1

3

1( ) CxL ++ 23cos

3

1

Pg.29 13 ( )=xf ' =x

tgx2

cos

12

x

senx3

cos

2( )

( )=

=

x

senxzxg

4

'

cos

cos2

x

senx3

cos

2

as deri3adas son id:nticas. as "rimiti3as se dierencian en $na constante.

#n eecto% ( ) ( ) ==== xx

x

xsen

x

xsen

xxfxg 2

2

2

2

2

2

2cos

cos

cos

1

coscos

11

Pg.;2 14 ( ) xf 42 += xy +orte *% =0y 2=x +orte & = 422

14

20


4/14


c) +orte% ( ) xf 12 ++= xxy ( ) xg 2=y 51=x 51+=x

( ) ( ) =++==+

dxxxdxgfrea 3251

51

=

++

+

51

51

233

2

1

3

1xxx 5

3

14

d) +orte% ( ) xf xy = ( ) xg xy 1= 1=x

=

+===

+

1

0

12121

1

0

1121

1

xdxxdxxS =

1

0

3

3

2x 3

2

[ ] ==== 121 2

1

2

1

2 LLLxdxx

S 2L =+= 21 SSrea 23

2L+ 36,1

e) +orte *% 0=y ( ) xf senxy= 0=x =x 2=x

( ) xg ( )= xxy 0=x =x

[ ] ( ) ( ) =+==

0

2

0

0 dxxxsenxdxgfS =

+

0

23

23

1cos xxx

62

3

+

[ ] ( ) ( ) ===

2 222 dxsenxxxdxfgS =

+

223 cos23

1 xxx 36

72 +

[ ] [ ] =+=

2

0 SSrea 34 +

f) Con un oscurantismo impropio de texto docente, no seespecifica una de las cuatro posibles soluciones:

Verde ABC, Marrn BCD, Rosa CDE, o bien la suma de estas tres.

as l4neas re"resentadas son%

@ecta A ( ) xf 3+= xy

@ecta A ( ) xg xy 3= Par(ola A ( ) xh 122 = xxy

os "$ntos de interseccin otros son%

A & ( )

+

+ 51

2

3

2

51( )85,461,1

B& ( )21 C& ( )4943 ( )25,275,0

D& ( )

+

+51

2

3

2

51( )85,162,0

E& ( )74 M ( )03 ? ( )30 P 021

( )041,0 ( )021 + ( )041,2 @ ( )10

( ) = dxxfF xx 32

1 2 + ( ) = dxxgG 2

2

3x ( ) = dxxhH xxx 23

3

1

,rea ABC& ( ) ( ) =+ C

B

B

A

dxFGdxHG24

11510 4734,0

,rea BCD& ( ) ( ) =+ D

C

C

BdxHGdxHF

19259049 + 3034,1

0/10


5/14


,rea CDE& ( ) ( ) =+ E

D

D

C

dxHFdxGF12

511213 7003,15

,rea s$ma CABDEC&64

521123 4770,17

Pg.;2 20 ( ) =+ dxxy 1 Cxx ++2

2

1 Bisectri6% =xy 22 == yx

( ) ( ) C++= 222

12

2

2=C 2

1

2

1 2 += xxy

Pg.;2 21 a) ( ) 2xxf = ( ) xexg = ( ) =dx Cex x +2

() ( ) Lxxf = ( ) senxxg = ( ) =dx CLxsenx +

c) ( ) xexf = ( ) senxxg = ( ) =dx Csenxex + d) ( ) xexf = ( ) xxg cos= ( ) =dx Cxex + cos e) ( ) xxf = ( ) Lxxg = ( ) =dx CLxx + ) ( ) xxf = ( ) Lxxg = ( ) =dx CLxx +

Pg.;2 22 ( ) 2321' 2 xxxf += y ( ) = dxxf ' Cxx ++ 25235

4

3

2

rigen% ( ) 00 ++= C000 0=C 535

4

3

2xxy +=

Pg.;2 23 ( ) ( )xfxx

x

x

xxg +=

++

++

++

112

1

12

12

12

22 ( ) ( )xfxg '' 0 += ( ) ( )xfxg '' =

Pg.; 24 a) =+ dxdxx 21

CxLx ++2

() == dxxsenxdxxsenx coscos ( ) CxL + cos

c) ( ) ( ) == dxxdxx 35331

53 22 ( ) =+ Cx 353

3

1

3

1( ) Cx + 353

9

1

d) == +

dxxdxx 417

4

53 =+

+

+Cx

14

17

1417

1=+ Cx 421

21

4Cx + 4 21

21

4

e) ( ) ( ) =

+=

+ 2222 1

2

2

1

1 x

xdx

x

x( ) Cxarctg + 2

2

1

) ( ) =+= Cxdxxsen 2cos21

222

1Cx+ 2cos

2

1

g) = dxex x2

22

1Cex +

2

2

1

h) = dxxxtg 23

cos

14

4

1Cxtg + 4

4

1

i)

Pg.; 25 =egCn la re"resentacin% +ada cuadroson 2 unidadesc$adradas =e c$entan $nos 15cuadros! 12 com"letos ms "or agr$"acin 1 "arcial = 25,15 ( )adradasUnidadesC31 ( ) = 3126c 5 ( ) == 3138d 7+ a res"$esta con menorse"aracin Derror a(sol$to) es c)D"or deecto)

7/10


6/14


Pg.; 26 ( ) ( ) ( )177

1

' ffdxxf = ( ) 273 = f ( ) =+= 237f 5

Pg.; 27 a) =

+=

+

3

1

12

13

1

2

1

2121

3

2

23 xxdx

xx [ ] = 313 22 xx 34

() [ ] [ ]=== 1232

0

3

2

0

3 eeeedee 35 ee

c) =

=

3

6

3

6

2 cos

1

cos

t

dtt

sent=

3

22 =

3

232

3

326

d) ( ) =+ dxxx204

0

212

2

1 ( )[ ] =++

4

0

2121

120

1

2

1x ( )117

42

1 21

Pg.; 28 ( ) ===

0

0

2

cos33 xdxsenxdxsenx ( )[ ] ==+ 23113 6

Pg.; 29 a) Verdadera% = !arf a"YEjeSi#etr$ ( ) ( ) =

a

a

dxxfdxxf0

0

( ) =

a

a

dxxf ( ) ( ) =+

a

a

dxxfdxxf0

0

( )a

dxxf0

2

() Verdadera% =continaf ' "or BarroE ( ) =7

0

' dxxf ( ) ( )07 ff

c) a!sa% a integral deinida es $n nCmero! 5ams de"ende de la 3aria(le.

d) Verdadera% ( ) ++

1

1

2

dxcbxax ( ) ( ) ++1

1

1

1

2

dxbxdxcax

=+ Fnci%n&A'cax2 ( ) ( )

+=+1

1

1

0

222 dxcaxdxcax

= A'Fnci%n()&bx ( ) 01

1

=

dxbx

e) a!sa% a $ncin ( )xf cam(ia de signo en inter3alo ( )30

) a!sa% a integral deinida es"eciicada es correcta Pero no mide :l rea al$dida! a F$e ( )xf cam(ia de signo en ( )30

al rea es% ( )[ ] ( ) =+ 3

1

2

1

0

2110 dxxdxx =

+

3

1

31

0

3

33x

xxx

3

22

Pg.; 30 a) ( ) xsenxf 3= es im"ar! centro ( ) += 20 x Integral & ; () ( ) senxxf = es im"ar! centro ( ) = 202 x Integral & ; c) ( ) senxxf = es im"ar! centro ( ) = 23 x Integral & ; d) ( ) senxxxf 2= es im"ar! centro ( ) += 2x Integral & ;

e) ( ) xxf 2

cos= es siem"re "ositi3aIn"e#ra! $%rea !imi"ada) 0 ) ( ) tgxxf = es im"ar! centro ( ) = 2110x Integral & ;

Pg.; 31 1) +ortes ( ) :2310 2xyxyxyx ==+== 0=x 1=x 71+=x

G/10


7/14


( ) ( )+

=

+

+

71

1

21

0

2

23

21 dx

xxdx

xx =

+

+

+ 71

1

321

0

32

623

62

xxx

xx

x

( ) ( ) ( ) =

+

+++

+=

6

1

2

13

6

71

2

717130

6

11

2

132

=

++

+++

=

6

1318

6

777373174733

6

163

3

1377

2) +ortes ( ):310 xyxyx =+== 0=x 1=x

( ) ( )[ ] =+1

0

13 dxxx ( ) =1

0

12 dxx =

1

0

2

22 xx 1

) +ortes ( ) :213 2xyxyxy =+== 1=x 71+=x 31+=x

( ) ( )[ ] ( )[ ]+

+

+

=+++31

71

2

71

1

2331 dxxxdxxx

=

++

+

+

+ 31

71

3271

0

2

63

222

xx

xx

x

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) =

+++

+

+++

+

+

++=

6

71713

2

71

6

31313

2

31

712

712

3232

2

=++++++++++=3

751173374

3

33533332722728

=3

773317 +

0) +ortes ( ) :023 2 === yxyxy 0=x 71+=x 3=x

( ) =+

+

+ 3

71

71

0

2

0302

dxxdxx

=

+

+

+ 3

71

271

0

3

23

6

xx

x

( ) ( ) ( ) =+++++=2

71713

2

99

6

7123

6

71447

7) +ortes ( ) :021 2 ==+= xxyxy 1=x 31=x 0=x

=

+=

+

31

1

3231

1

2

6221

xx

xdx

xx

( ) ( ) ( ) ( ) ( ) =

+

+

=

6

11

2

1

6

3131

2

31 3232

3

433

G) +ortes ( ) :0122 =+== yxyxy 1=x 31=x 0=x

( ) =++

0

31

231

1 2

1 dxx

dxx =

+

+

0

31

331

1

2

62

xx

x

( ) ( ) ( ) ( ) =

+

+

= 1

2

131

2

31 22

6

3611

Pg.;0 32 a $ncin "rimiti3a (F) crece c$ando s$ deri3ada (f) es "ositi3a #n "artic$lar! la $ncin (f) es "ositi3a en ( )0 a $ncin creciente en ( )0 es !a &unci'n C

/10


8/14


Pg.;0 33 #ntre las tres c$r3as! la corres(ondenciaes algo H#rosera. ?o o(stante% Parece III& crece / decrece! c$ando II& "ositi3a / negati3a A (((F ((f a restante de(e ser (f '

Pg.;0 34 a) areceF$e la grica (asa"or el "$nto ( )05 . $ego 14 == yx

> D0)% a integral es el rea entre 1=y 0=y 0=x 4=x #s decir! rectng$lo con Base & 0 'lt$ra &1 A =14 4

Je otro modo% ( ) [ ] ==== 041 404

0

4

0

xdtdttf 4

> D)% a grica incor"ora $n tramo de = )5(10 xy xy = 5

#l rea est entre 1=y 0=y 1=y xy = 5 0=x 7=x #sta com"$ta $na "ositi3a Dso(re 0=y ) otra negati3a D(a5o 0=y ) Por tanto% rea com"$tada & ( ) ( )=++ 121214 3

(ien% ( ) ( ) ( ) =

++

6

4

7

6

4

0151 dxdxxdx [ ] [ ] =+

+ 7

6

6

4

24

0 25 x

x

xx

( ) ( ) ( ) =+=+++= 10467121204 3

() a re"resentacin grica "edida! e3identemente! coincidecon la dada. ?o "arece necesaria eK"licacin alg$na.

Pg.;0 35 ( ) ( )[ ] ( ) [ ] ( )=++=+=+=+= 1377.11 313

1

3

1

3

1

3

1

ddfdfdg 11

Pg.;0 36 a) ( ) ( ) ( ) =+=+= 714

2

2

0

4

0

dxxfdxxfdxxf 8

( ) ( ) ( ) === 211

0

2

0

2

1

dxxfdxxfdxxf 1

( ) ( ) ( ) === 271

0

4

0

4

1

dxxfdxxfdxxf 5

() ( ) 21

0

= dxxf 0> ( ) 12

1

= dxxf 0< #n ( )20 ha x F$e hace ( ) 0=xf

c) =i ( ) 5,3xf ( ) ( )


9/14


Pg.;0 39 ( ) ( ) ( )22 11 xxfx +++ ( ) ( ) ( ) ++

dxxdxxfdxx 21

1

1

1

211

( )1

1

31

1

1

1

3

33

+

+

xxdxxf

xx ( )

3

8

3

8 1

1

dxxf

( ) 67,23

81

1 dxxf Lalores "osi(les & 2 1 5,2

Pg.;0 40 a)dt

d+anace,araci% = cidadr-fica.e,o&endienteGa=

== 1201/ra#o 20 == 502/ra#o 0 == 4203/ra#o 5

() = +dxees!acio Sobre"0a.e,ocidadreaGr-fice =

=++=++= 40100102042

1205201

2

1e 160

Pg.;7 41 ( ) =

=

60

0

1,0

60

0

1,0

1,0

1414 tt etdte [ ] [ ] [ ]== 0660

0

1,0100410604104 eet t

[ ] [ ] +== 40992,02401040025,010604 244 ( )h#, 2/

Pg.;7 42 anC"1S/A1/EAce,eraci% ( ) [ ] === ++atadtt+ 0 at+ +0

( ) ( ) ( ) =

+=+==t

att+dtat+dtt+tx0

2

002

1 200

2

1att+x ++

Pg.;7 43 [ ] [ ]10303 2 xx a

= ( )3333 0120 =a 23 =a 3 2=a

Pg.;7 44 '(scisas corte D 12 == yxy )% 1=x 1=x

,rea inca & ( ) =

=

1

1

31

1

2

31

xxdxx =

+

3

11

3

11

3

4 @egin &

3

2

'(scisas corte D ayxy == 2 )% ax = ax =

,rea regin & ( ) aax

axdxxa

a

a

a

a 3

4

33

2 3

2 =

==

3 2

2

1=a

2

23=a

Pg.;7 45 ( ) 2xxxf ( ) =0xf 0=x 1=x


10/14


()( ) ( )

=+

=

+ 22

21

22

2

1

21

2

x

x

x

x L

L ( ) CarctgL

x +22

1

c) ( ) =+ dxxtgx 251

cos

13 ( ) =++

++

Ctgx 15

1

3151

1( ) Ctgx ++ 5 63

6

5

d) = dxLLx 22

2

1C

Lx + 2

2

1

e) ( ) =++= + Cdxxdxsenxx 133 cos13

1cos =+ Cx2cos21 C

x+

2cos2

1

)( ) =+ dxx 2211

( ) =

+ dxx 221

2

2

1( ) Cxarctg +2

2

1

Pg.;7 48 ( )( )( ) ( )

( )( )21

12

21

1

+

= xx

xBxA

xx ( ) ( ) 112 + xBxA

( ) ( ) 12 ++ BAxBA Identidad inde"endiente del 3alor x 0=+BA 12 =+BA 1=A 1=B

Pg.;7 49 ( )( ) = dxxx 123 = dxxx 11213 ( ) ( )[ ] =+ CxLxL 123

Cx

xL +

1

23

Pg.;7 50 0>b #n [ ]b0 es xx 33 =

=

bb xdxx

0

2

0 2

33

2

3 2b

Pg.;7 51 '(scisas de tramos 1=x 2=x 5=x 8=x 13=x

( ) ( ) ( ) ( ) ( ) =+++= 13

8

8

5

5

2

2

1

13

1

dxxfdxxfdxxfdxxfdxxf ( )=+Cadrantes/ri-ng,os

=

+

+

2

35

4

3

4

3

2

31 22

=2

126

( ) ( )=+= /ri-ng,oCadrantedxxf13

5

=

2

35

4

32

=

+

2

15

4

9

4

309 +

( ) ( )== Cadrantesdxxf8

2

=

4

3

4

3 22

0

Pg.;7 52 +$alF$ier "ar(ola se "$ede reerir a s$ "ro"io e5e D3ertical) "asando "or D 00 ) Por tanto! toda "ar(ola "$ede eK"resarse "or la ec$acin 2axy = =ea $n segmento "ara(lico de alt$ra h

os eKtremos de tal segmento son ahx = s$ longit$d es ah2

,rea segmento & ( ) =

=

ahahax

hxdxaxh0

3

0

2

322 =

a

hh

3

22

a

hh 2

3

2

Pg.;7 53 a) ( ) =+

++

+

+

11

1223

2

x

CBx

x

Axf

x

x ( )

1

2

+

+++x

x

ACxxBA 2=+BA 0=C 1=A

1=A 1=B 0=C ( ) =++

1

123

2

x

xxf

1

12 +

+x

x

x

() +


11/14


( )( ) 23

24

' 12

xx

xxxf

+

++= < 0 ( )xf es decreciente

( ) 0= fx,$# 's4ntota hori6ontal% 0=y ( ) += + fx,$# 0 's4ntota 3ertical% 0=x a grica se "arece a la de la ig$ra

c) ( ) =

++=

++

ee

xLLxdxx

x

x 1

2

112

1

1

12 ( ) =++ 22

1

11 2

LeL

++ 2

1

1

2e

L

Pg.;7 54 a) ( )xf ( )xg =on in3ersas entre s4 =im:tricas res"ecto xy= se cortan en ( )00 ( )11

=$ma integrales =$ma reas (a5o c$r3as Un rea la sim:trica de la otra dan +$adrado lado 1 1 () 'nlogamente! las $nciones Lxy= xey= son in3ersas

=on sim:tricas res"ecto (isectri6 xy= no se cortan

$ego! e

dxLx1 ,rea entre 0=x 1=x 0=y 1=

y @ectng$lo = e1e

Pg.;8 A1 a) &x =

21

2

x

xy

( ) ( )( )

=

=

22

2'

1

2212

x

xxxy

( )222

1

22

x

x

+

( ) = += 222

'

21222&& y#

910 ( )2

910

34 =+ xy 32910 = yx

() =0y = 320910x = 1032x = 516x ( )0516 c) 's4ntota "rKima 1=x = 329110 y = 922y ( )9221

Pg.;8 A4 ( ) dcxbxaxxf +++ 23 ( ) cbxaxxf ++= 23 2' ( ) baxxf 26" += ( ) 31 dcba +++= 3

( ) =01"

f 026 =+ ba

11/10


12/14


( )10 d=1 ( ) =00"f 0=c #l sistema acilita% 2=a 6=b 0=c 1=d

Pg.;8 A5 a) ( )

=

20

1

20

1

'

20

11000 eet1 ( )

=

20

11000 20' tet1

t

( ) =0' t1 20=t

( ) =01 000.25 ( ) 357.3220 =1 ( ) 673.25100 =1.%-imo& 357.32 20=t ./nimo& 000.25 0=t

()

( )

=

20

11000 20

"

t

et1

2

2050 20

te

t

( ) =0' t1 40=t

( ) 000.10'

=1 ( ) 13540'

=1 ( ) 27100'

=1=0t >0'1 crece1= == 100,40tt 0213"y )$ni#oy=

Jimensiones "ara coste m4nimo% = 3 21x 7932,0 ( )d# = 3 4h 5874,1

( )d#

Pg.;8 A7 a) Verdadera. #n [ ]31 la grica indica ( ) 0' >xf ( ) Crecientexf =() >alsa. a grica indica ( ) 00' >f ( )xf no tiene ah4 eKtremo relati3o c) Verdadera. ( ) = 10'f ( ) 1=ftg# xy# = tg a ( )xf "aralela a xy= d) >alsa. =egCn a) la ( ) crecexf = ! "ero "$ede! a la 3e6! ser negati3ae) >alsa. ( ) crcientexf = "$ede an$larse o no! en el inter3alo citado.

Pg.;8 A8 = axaxy 22 +ortes con 0=y 0=x ax 2=

12/10


13/14


,rea encerrada =

a

dxxa

xa

2

0

2

32

12=

ax

a

x

a

2

0

3

3

2

23

1

2

2=

a

a

x

a

x 2

0

3

3

2

2

3

( ) ( )==

3

3

2

2

3

22

a

a

a

a =3

84

3

4 Inde(endien"ede a

'(scisa 3:rtice "ar(ola% =x ( )=+ a202

1 a =

=

a

a

x

ay 2

2 a

1

L:rtice% ax = a

y 1

= Dec$aciones "aram:tricas del l$gar geom:trico

#c$acin cannica Deliminando el "armetro)% 1=yx #l l$gar geom:trico es $na i(ro!a eui!%"erade as4ntotas e5es coordenados

Pg.;9 B1 a) a $ncin com"$esta "or H"arte entera de 3aria(le!am%s es con"inua =iem"re est constit$ida! como m$cho! "or Htramos! sino "or "$ntos () enteroax = ( ) ( ) [ ] ==+= aaaaafxf 0

=on "$ntos aislados en *#n [ ]44 son -0! -! -2! -1! ;! l! 2! ! 0

1oEnteroax = ( )1+ aax( ) ( ) ( ) =++=+ 1aaxxxf 1

#s la recta 1=y DeKce"to a(scisas Henteras)

Pg.;9 B2 ( ) +

16

3277

xx,$#

0

0

( (( )

=

++1

2

16

1632

x

xx ( (7

1632

++

x

xx 0

0

( ) ( )

( ) ( ) =

++

+

+

327

1632 22

xx

xx [ ]( )( ) ( )

=+++

327

167

xx

xx

32

16

++

+

x

x =

6

2

3

1

Pg.;9 B3 Primero! se tra6a la recta as4ntota 32 += xy ( )2 ( ) crecientexf = ( ) ntotaDerechaAs$xf =

( ) 10' =f = 1tg# angente en origen% xy= ( ) 02' =f MKimo relati3o% 2=x ( ) 13' =f angente en 3=x 3+= xy ( ) 04' =f M4nimo relati3o% 4=x ( ) 65' =f angente en 5=x 306 = xy

Pg.;9 B4 a) #n [ ]102 es ( ) t& ( )2610 += xy ( )2610 = xy

Par(ola 3ertical a(ierta hacia arri(a! de 3:rtice ( )106 ( ) 62 ( ) decrecet& = ( )106 ( ) crecet& =

#n ( )1210 es ( ) tf 9228 = xy xy 2512

128 =

#K"onencial decreciente% ( )106 ( ) decrecet& = () ( ) 102t ( ) ( )62' = ttf ( ) 1210t ( ) 22 9' Ltf t =

( ) =0' tf

6=t

( )=2f 26

( ) =6f 10

( )=10f 26

( ) =12f 20

#n [ ]122 ha .%-imoD2G) en 2=t 10=t ./nimoD1;) en 6=t

1/10


14/14


c) >12t ( ) 9228 = tt& ( ) =0t& +=2

289

L

Lt 8,13 ( )a2os

Pg.;9 B5 a) ( ) iaAbajoAbiertaHac,Eje.ertica&ar-bo,atf ,,= +ortes *% ( ) =0tf 0=t / 3=t Fina,(nicio/

)(reni%n

( ) ttf 600900' ( ) =0

'

tf 23=t ( ) 230 ( ) crecetf = ( ) 323 ( ) decrecetf = () +a"acidad mKima en 3:rtice "ar(ola% 23=t c) #s la "ar(ola re"resentada al margen

Pg.;9 B6 ( )#Ladox= ( )#A,tray= ( ) ++= 22 50410720 xxyxCoste ( ) xxy 2336 2= ( ) ( )23365,0 xxxf. = ( ) ( )2' 9365,0 xxf = ( ) = 0' xf 2=x

>0. 12

unidad 12 - integración, problemas resueltos

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