Unidad 11 - Parte 1 - Derivadas y representación gráfica

8/10/2019 Unidad 11 - Parte 1 - Derivadas y representacin grfica

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1 Bachillerato - Matemticas - Unidad 11 Derivadas y representacin grfica

Pg.!" I a# == 3

2log3 2 X

x x2log32 $# =x10log10 x10log2

1

10

c# =senxeelog ( )senxLne d# =x

ee 2log 2Lnxe

Pg.!" II a# ( ) =fLn ( )xLn4 $# ( ) ( )== eLnxfLn x

c# ( ) =fLn ( )xLnx d# ( ) =fLn ( ) ( )senxLnx + 3

Pg.!% 1 ( ) =xx 2 ( ) = 12 'xx ( )x

x2

1'=

Pg.!% 2 ( ) fInversaDeg ( )( ) =xxfg ( ) =++ xxxg 13

( ) =+++ 1131 23' xxxg ( )13

11

2

3'

+=++x

xxg

( )11'g =++ 1113 xx ( ) ( ) =++ 0522 2 xxx 2=x

( )( ) =

+

=123

111

2

'g =+ 143

1=

+112

1

13

1

Pg.!% 3 ( ) fInversaDeg ( )( ) =xxfg =++ xxxg 5

( ) =

+

+++ 152

115'

xxxg ( )

521

525'

+++=++x

xxxg

( ) 3'g =++ 35xx ( )( ) =++ 041 xx 1=x ( )NoVale 4=x

( ) =++

+=

5421

5423

'g =+ 121

12=

+

121

12

3

2

Pg.!% 4 ( ) =xx 55 ( ) ( ) = 15 '545 xx ( ) 5 4'

5

5

1

xx= =

=

5 432 325

1m

80

1

= 32x == 5 32y 2 ( )= 3280

12 xy

5

8

80+=

xy

Pg.!& 5 a# ( ) += 1

2

11

3

1

'

2

12

3

1xxxf ( )

xxxf

1

3

13 2

' +

=

$# ( ) +=

13

23

4

1 13

21

4

1

' xxxf ( ) 12

4

134 3

' +=

xxxf

c# ( ) +=

224

3

5

13

14

31

5

1

' xxxxf ( ) 224

3

5

345 4

'

+

= xxx

xf

Pg.!& 6

==

5 4

15

1'

5

1

5

1

xxym

=

5 45

13

x 4 515

1=x == 54 515

1y

4 15

1

P'ntos(

44 5 15

1

15

1

Pg.!& 7 a# ( ) =

4

7

xxf ( ) ==

4 11

4

11

'

4

7

4

7

xxxf ( ) 4 32

'

4

7

xxxf

=

$# ( ) ==

5

2

5

31

xxxf ( ) == 1

5

2

'

5

2xxf

5

3

.5

2x ( ) 5 3

'

5

2

xxf

=


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Pg.!& 8 a# ( ) ( ) += 31

35xxf ( ) ( ) =+=

5353

1 13

1' xxf

( )3 2353

5

+ x ( ) =

=

3 2

'

83

51f

12

5

$# += 353 xy =

5

33

yx ( )

5

331 = x

xf ( )( )5

3 2'1 xxf =

c# ( ) ( ) ==+= 33 83151f 2 ( ) ( )[ ] ( )[ ] === 5

2321

21'1 fff

5

12

S) por*'e( ( )[ ] ( ) [ ]( )21

112

1'''1

==

ffff 125

1512 =

Pg.!! 9 a# ( )=xf ' ( )56153 2

+ xe xx

$# ( )=xf ' ( ) ( )=+ 72372 xexxe xx ( )1052 xxex

c# ( )=xf ' =++2

322 13333

x

xexxxe xx ( )2

33 2133

x

xxex +

d# ( )=xf ' =+ xx exex21

x

xex

2

21+

Pg.!! 10 = xxey 6

2

( )626'

2

= xey xx =0'

'y = 062x 3=x

( ) ( ) += xx xxey 6" 2

6262( ) ( )52626

2

xxe xx


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Pg.!+ 13 ( ) =+

= xx

xf 24

12

'

4

22 +xx

( ) ( )

( )++

=22

2

"

4

242

x

xxf

( )

( )2

2

2

4

42

+

x

x

( ) =0' xf 0=x ( ) 00" >f ( ) MnimoLn = 40

Pg.!+ 14 a# ( ) = xLnxxfLn ( )

( ) =+=

xxxLn

xf

xf 11

'

+ xLn1 ( ) xxLnxf += 1'

$# ( ) ( ) = 2LnxxfLn ( ) =x

Lnxf

f 12

'

= fx

Lnxf

2' ( )1' 2 = LnxxLnxf

c# = 2xe

xf = Lnxe

fLn

x

2 += xe

Lnx

e

f

f xx

1

22

'

( )

xex

xxLnx

e

f

+=

1

2

'

d# = LnxxfLn 2 +=x

xLnxxf

f 12 2

'

( ) ( )1' 2

21 ++= xxLnxf

e# ( ) += LnxxfLn 12 ( ) ++=x

xLnxf

f 1122

'

12' 21

2 +

++= xxLnx

xf

f# = LnxxfLn +=x

xLnxxf

f 1

2

1'

xxx

Lnxf

+= 1

21'

Pg.!, 15 ( ) = xxf cos' ( ) == 100' senf = 1m ( )= 010 xy xy=

Pg.!, 16 a# ( ) ( ) ( ) ++= xx

exexxf 222'

22cos ( ) ( )xx exexf

222'

cos2 ++=

$# ( ) ( )= senxx

xfcos2

1'

x

senxf

cos2

' =

c# ( ) =x

tgxxf2

'

cos

12

x

senxf

3

'

cos

2 =

d# ( ) ( ) ( )23cos323' += xxxsenxf

Pg.!, 17 ( ) == 22cos

12

'

xxf

x2cos

22 x

m2cos

22= Bisectri 1( 11 =m

< 12cos

22

x

>22cos 2 x > 22cos x 41,12cos >x Imposible

Ningn pntode la f'ncin c'mple la condicin especificada

Pg.!, 18 :rizontalTangenteHo =m 0 ( ) = xfm '' senxxcos= 0cos senxx 1=tgx 4=x 45=x

Pg.! 19 a# ( )=

= x

x

e

e

f2

'

1

1

x

x

e

e2

1$# ( )

=++

= xx

f 2.11

122

'

( )22112

x

x

++

c# 2'

1

1

arccos2

1

xxf

= d#

+

+=

2

'

1

1cos

1

xx

arcsenxsenxf

Pg.! 20 ( ) =arctgxxf ( ) 2'

1

1

xxfm

+= 0=tgm 0

1

12=+x

=x

Ningn pntode la f'ncin c'mple la condicin especificada


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Pg.! 21 =

+

++=

222

' 1

11

1

1

1

x

x

xf

( ) =

++ 22

2

2 11

1

xx

x

x=

+

+ 22 11

1

1

xx0

Pg.! 22 +2

cot

xarcarctgx ( ) =++

0cot1

1 '2

xarcx

( ) 2'

1

1cot

xxarc+=

Pg.+1 23 = 24 63 xxy ( )== 1121212' 23 xxxxy

( 13121236 22" = xxy = 0'y 1=x 0=x 1=x = 0"y 31=x

= 1x 3=y 0">y /n { }31 hay mnimorelativo=0x 0=y 0"y

/n{ }31

hay mnimorelativo

Pg.+1 24 ( ) =0" xf 1=x 3=x ( )doble 7=x1f 0'rva cncava hacia arriba

Pg.+1 25 =y12 +x

x

( ) ( )( ) =

+

+=

22

2

1

211'

x

xxxy

( )222

1

1

+

x

x


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( ) (

(

+=

2

22

" 12

x

xxy

( )323

1

62

+

x

xx

P'ntos infle3in( 00 == yx 433 == yx 433 == yx

/n p'nto infle3in) con a$scisa positiva(

4

33 =

8

1)3('ym

:Tangente ( )38

1

4

3= xy

8

33

8

1+= xy

Pg.+1 26 ( ) senxxxf = 2' ( ) xxf cos2" = 2cos =x Imposible

4a grfica no tienening5n p'nto de infle3in

Pg.+" 27 a# 652 += xxy par$ola) cncava hacia arri$a 0ortes 67( 0=y 0652 =+ xx 2=x 3=x 0orte 68( 0=x 6=y 9:rtice M;nimo a$sol'to#( 052' == xf 25=x

$# 32 += xxy par$ola) cncava hacia arri$a

0orte 67(=0y

032

=+xx

2o tiene ra;ces irreales# 0orte 68( 0=x 3=y

9:rtice M;nimo a$sol'to#( 012' = xf 21=x c# 92 += xy Par$ola) cncava hacia a$a ( ) ( ) ( )=


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$# 35 53 xxy = 0orte 68( =0x 0=y

0orte 67( =0y ( ) = 053 23 xx 0=x 35=x ( ) =xlm ( ) +=+xlm = 01515' 24 xxy 00 == yx 21 == yx

= 03060" 3

xxy

0=x 21=x

( ) 01" y ( ) ( )relativoMNIM"= 21

=0x === 0"' yyy ( ) HorizontalTangenteINL!XI#N ..00 = c# ( )( )( )412 += xxxy 863 23 + xxx 0orte 68( =0x 8=y ( ) =xlm 0orte 67( =0y 2=x 1=x 4=x = 0663' 2 xxy 31=x

= 066" xy 1=x = 31x 0"y MNIM"=+ 3631 = 1x 0=y 9' =y 0"=y ( ) INL!XI#N=01 d# xxxy 93 23 = (( (( 21532153 ++ xxx 0orte 68( 00 == yx ( ) =xlm 0orte 67( =0y ( ) 2153 =x 0=x ( ) 2153 +=x = 0963' 2 xxy 1=x 3=x

= 066" xy 1=x 51 == yx ( ) .51 MX= 273 == yx ( ) .273 M$N= 111 == yx ( ) .111 INL!X=

Pg.+" 29 a# 5366 23 ++= xxxy ( ) =xlm 0orte 67( ( ) 01 y =0y 1357,0x 0orte 68( =0x 5=y =+ 036123' 2 xxy alNox Re= 2i M7.) ni MC2.

= 0126" xy 2=x /n( ) .652 INLL!X=/n efecto( 0"2


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= 092102108" 2 xxy 56,0x51,1x

/n ( ) relativoMN.45,5625,1 = /n ( ) .10,2656,0 INL!X= /n ( ) relativoM)X.04,1033,0 =

/n ( ) .78,5951,1 INL!X= /n ( ) relativoMN.96,11333,2 = :x -1 -)& -) -1 )& 1 1)& " ")& :y -&1 -)"1 -%)! )! % ,)%% -1& -&,),1 -1 -"& &)" c# 5126103 234 +++= xxxxy ( ) +=xm 0orte 68( =0x 5=y 0orte 67( =0y 05126103 234 =+++ xxxx .imagx= =++ 012123012' 23 xxxy 44,0x

=+ 0126036" 2 xxy( ) 6135=x 43,1x 23,0x

/n y 0">y crecey= 0oncavidad hacia arri$a /n ( ) 96,723,0 .INL!Xy =

/n


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( ) 32 23 +++ xxxxf ( ) 143 2' ++ xxxf

( ) 46" + xxf =0'f 1=x

( ) 01"

s;ntota( 1=x=x )& )& )+& ) 1 1)1 1)& 1)& 1)+& )& )&=y )1" )&, 1),& &)+ I- -%)&& -1) )%& 1)&, )!+ "), &)!

$# ( )14 2

2

+

x

xxf ( ) ( )xfxf = =MP>? 0entro SIM&'()*A ( )00

( )144

1

4

12 +

=x

xxyxf

x y >s;ntota( xy4

1=

0orte 67( =0y 0=x 0orte 68( =0x 0=y

( )( )

( ) 2222

22

24'

14

34

14

34

++=

++

x

xx

x

xxy 0' =,iem-rey

Por tanto) ciente,iem-re'rey = 2o hay M7=M6@) ni MH2=M6@


9/12


=0'y 0=x 0= y ( ) ( )rizontalTangenteHoINL!XI#N=00=x )& )& )+& 1 1)& 1)& 1)+& " % &=y )1 )! )1" ) )+ )"% )% )%+ )+" ), 1)%

c# ( )2

92

2

+

xx

xxf ( )

( )( )12

9 2

+=

xx

xyxf

( ) 9. =xflm >s;ntota( 9=y( ) = 2. xflm >s;ntota( 2=x( ) = 1. xflm >s;ntota( 1=x

( )

( ) ( )22 12

49'

+

=xx

xxy = 0'y 0=x 4=x

'y A cam$ia de sentido en 0=x y 4=x ( ) .00 M)X= ( ) .84 M$N=

=x -" -)& -)& - -1)+& -1)& -1)& -1 -)+& -)& -)&=y )& ")1% &!), I- -%) -1!) -,)"" -%)& -)"1 -1) -)!=x )& )& )+& 1 1)1 1)& 1)& " % & 1 =y -)"" -1), -+)"! -I "&)1" 1+)"1 11)&+ ,)1 , ,)% ,)"" ,)!1

d# ( )1

12 +=

x

xyxf 012 +x contin*af =

0orte 67( =0y 0=x 0orte 68( =0x 1=y( ) = 0. xflm >s;ntota( 0=y

( ) 22

2

1

21'

+

+=x

xxy

= 0'y

21=

x 21+=

x'y A cam$ia de signo en 41,021 =x ( ) M$NIM"= 21,141,0'y A cam$ia de signo en 41,221 +=x ( ) M)XIM"= 21,041,2=x -" - -1 -)%1 )& 1 1)& )%1 " & + 1=y -)% -! -1 -1)1 -1 -% )1& ) )1 ) )1& )1% )

Pg.+" 34 F'ncin racional 0ociente de polinomios del mismo grado >s;ntota 2=y 0ociente de coeficientes de mayor grado 2 2'merador tipo( ( ) baxxxN ++ 22

>s;ntotas( 2=x 4=x

Denominador( ( ) ( ) ( )=+ 42 xxxD 822

xx ( ) eciente,im-reDecrxf = ( ) = 2. xflm ( ) += +2. xflm ( ) = 4. xflm ( ) += +4. xflm

/n los e3tremos del intervalo ( )+ 42 ( ( )xf tiene signos contrarios /n este intervalo) tiene *'e ha$er 'na ra; *'e haga ( ) 0=xf 0orte 67

>l pasar de I) en + 2 ) a -) en 4 ) ha$r 'na ( ) 0" =xf =2F4/7=J2

/n este mismo intervalo) tiene *'e ha$er otra ra;( ) 0=xf

0orte 67


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11/12


( ) 04" s;ntota( 0=y 0orte 68( =0x 1=y

( ) 2

2' xxexf

( ) ( ) 22" 42 xexxf + ( ) =0' xf 0=x ( ) 00" ? / A 68 ( ) { }11 += +fD > 1x ( ) ntin*aDefinida'oxf =

( ) = +1. xflm >s;ntota( 1=x oLadoDerec/

( ) +=+xflm. 2o hay as;ntota horiontal 0orte 67( =0y = 112x 41,12 =x

( )1

22

'

x

xxf

( )( 2

2

"

1

12

+

x

xxf

/n ( )fD ) siempre es ( ) 0' >xf 2o hay M7=M6@ ni MH2=M6@

@iempre se tiene *'e ( ) 0"


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$# ( )xf senxxy += ( ) +fD = ( ) contin*axf = ( ) ( )xfxf = =MP>? 0entro @=M/O?H>( ( )00 0orte 67( =0y =+ 0senxx 0=x ( ) =+ 0cos1' xxf ( )12 += 0x /n estos p'ntos hay tangente horiontal

( ) 0' xf

( ) ciente,iem-re'rexf =

( ) = 0" senxxf0xx=

/n estos p'ntos cam$ia signo ( )xf " ,INL!XI"N! =x N, N% "N, N &N, "N% +N, N, &N%

=y 1)+, 1)% )1 )&+ ), ")! ")1" ")1% ")1& ")

=x 11N, "N 1", +N% 1&N, =y ")% ")+1 %)1, %)+ &)&1 !),

c# ( )xf xexy /2= ( ) contin*axf = ( ) +fD = 0orte 67( = 0y 0=x 0orte 68( =0x 0=y ( ) +=xflm. ( ) 0. =+xflm ay) p'es) as;ntota horiontal( 0=y /aHaciaDerec ( ) ( ) = 02 2' xexxxf 0=x ( ).M$N 2=x ( ).M%X

( ) ( ) =+ 024 32" exxxf

22 =x INL!X . =x - -1)& -1)& -1 -)& )& ), 1 " ")%1 % & =y )&! 1), &)%& )+ )%1 )1& )1 )"+ )&% )%& )", )

)1+

d# ( )xf ( )xtgy 2= ( ) -eri(dicaxf =

Per;odo( x2 2

1=T

Oiene as;ntotas verticales( 0x2

1

4

1 +=

/s la grfica de tgx comprimida con factor 2

=x )1 ) )& )" )% )& )! )+ )+& ), ) 1 =y )+" "), I- -"), -+" )+" "), I -"), -+"

Unidad 11 - Parte 1 - Derivadas y representación gráfica

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Transcript of Unidad 11 - Parte 1 - Derivadas y representación gráfica