unidad 10 - derivadas.doc

8/11/2019 Unidad 10 - Derivadas.doc

1/22

1 Bachillerato - Matemticas - Unidad 10 - Derivadas

Pg.240 I a) ( )+= 323 xy 32 = xy

) ++

=++

10

1

23

2 yx +=+ 255 xy = 035yx

5

3

5

1= xy

c) ==3

330tgm ( )=+ 4

3

32 xy

3

634

3

3 += xy

Pg.240 II a) ( ) ( ) ( ) ===

3

21

03

0cos3cos

gTVM

6

) ( ) ( ) ( )

( ) ( ) ==

=2

0

11

1212 22

hTVM 0

Pg.240 1 a) ( ) =

++=

0

404

h

hfTVM

h

h 24+

) ( ) +

=1,0

241,01,00TVM 24845,0 ( )

+=

01,0

2401,001,00TVM 24984,0

( ) +=001,0

24001,0001,00TVM 24998,0 ( )== 0hlmTVI 25,0 = 41

c) ( ) ( ) ( ) ( )

=h

fhfhlmfTVI

00

+h

h 24 ( )( )( )

++

+++

24

2424

hh

hh

( )

++ 24hh

h

++ 241

h

+

++ 221

240

1

4

1

Pg.240 2 ( ) ( ) ( ) ( ) ( ) ( )

+++++

=h

hhhhlmfTVI

11312113120

22

h hh 72 2

72h 7

Pg.240 3 ( ) ( ) ( ) ( ) ( )

h

hsen

h

senhsenhlmfTVI

=

00

1,0

1,0sen993334,0

01,0

01,0sen999983,0

001.0

001,0sen999999,0

( ) 1=fTVI

Pg.241 4 a) ( ) ( )( ) ( ) +==h

hhlmfm

3311

01' ++ 332 hh 3

( ) 11 =f ( )11 ( )= 131 xy 23 = xy P!ntos" =x -2#$ -2 -1#$ -1 -0#$ 0 0#$ 1 1#$ 2 2#$

=y -1$#%& -' -&' -1 -0#& 0 0#1& 1 &' ' 1$#%&

) ( ) ( ) ( ) ( ) ( ) ( )

++++

==h

hhhlmfm

2252225202'

22

+1h 1

( ) = 82f ( )82 ( )+=+ 218 xy 6= xy P!ntos" =x -% -$ -4 -& -2 -1 0 1

=y 4 -2 -% -' -' -2 -2 4

1(22


2/22


c) ( ) ( ) ( ) ( )

+++

==h

hhlmfm

222202'

( )

( )

++

+

24

24 2

hh

h

++ 24

1

h4

1( ) =22f ( )22 ( )= 2

4

12 xy

2

3

4

1+= xy P!ntos" =x -2 -1#$ -1 -0#$ 0 1 2 &

=y

0 0#1 1 1#22 1#41 1#& 2 2#24

Pg.241 5 a) ( ) ( ) ( ) ( )

+

=h

fhfhlmf

2202' 97,12 + h 02,22 + h

03,0297,1 ==h ( ) ( ) ( )

03,0

297,12'

fff

03,0

7905,6 16,3 3

02,0202,2 ==h ( ) ( ) ( )

02,0

202,22'

fff

02,0

7059,7 95,2 3

) ( ) ( ) ( ) ( )

+

=h

fhfhlmf

4404' 01,44 + h 99,34 + h

01,0401,4 ==h ( ) ( ) ( )

01,0401,4

4'

ff

f

01,0

92,9

20

01,0499,3 ==h ( ) ( ) ( )

01,0

499,34'

fff

01,0

998,82

No# haida c!enta de tan dis*ares a*ro+imaciones reali,adas

Pg.242 6 ( )1'f Pendiente de tangente en *!nto de 1=x 2( )1'f Pendiente de tangente en *!nto de 1=x 0( )0'f Pendiente de tangente en *!nto de 0=x 7,0( )2'f Pendiente de tangente en *!nto de 2=x 2( )3'f Pendiente de tangente en *!nto de 3=x 0( )5'f Pendiente de tangente en *!nto de 5=x 5,1

Pg.242 7 =m 1 Pendiente de tangente en *!nto de2=x

=m 0 Pendiente de tangente en *!nto de1=x

=m 1 Pendiente de tangente en *!nto de0=x

=m 0 Pendiente de tangente en *!nto de 3=x

=m 2 Pendiente de tangente en *!nto de 5=xPg.242 8 ( )xf 0x ( ) xxxf += 2 Parola

>0x ( ) xxxf = 2 Parolamas *arolas son simtricas res*ecto al e/e angente en origen# con 0x 1=mangente en origen# con +0x 1=m No hay tangente3nica# en el origenP!ntos" =x -2#$ -2 -1#$ -1 -0#$ -0#$0 -0#2$ 0 0#2$ 0#$0 0#$ 1 1#$ 2 2#$

=y $ 2 0#$ 0 -0#1 -0#2$ -0#1 0 -0#1 -0#2$ -0#1 0 0#$ 2 $

Pg.24& 9 a) 5a 6!nci7n ( ) 12 + xxP 8*arola) es continuaen R 9# *or tanto# en ( )1

2(22


3/22


4/22


Pg.24 17 a) ( ) ( ) == 115' 4xxf ( ) 415 x ) ( ) ( ) =+= 3234' 3xxf ( )32312 +x

c) ( ) =xf' ( ) ( )143326 2523 ++ xxxxx d) ( )=xf'xx

x

3

2

2

13

Pg.24 18 a) ( ) ( ) ( ) ( )=++++ 4324412244 233243 xxxxxxx= ( ) ( )4297244 2433 ++ xxxxx

) ( ) = 2 1433432 xxx

xx

3438

xx

c) ( )

( ) =

2

2

12

2122

1

123

x

xxx

x

x( )312

12

2

3

+

x

xx

d) ( ) =

+452

51452

2

xxxx

452

51625 2

x

xx

Pg.24 19 a) ( )

( )=

1532

3535

5

4

x

x ( )

( ) 1532

515

5

4

x

x

) ( ) =

xx

xxx

2

22

2

123 ( ) xxx 212

2

3

c)

( ) ( ) ( ) ( ) ( ) ( )( )

=

+

++++++

63

22

34

33

323

3

2

23231222324

xx

xxxxxxxxxx

= ( ) ( )( )

=+

+++

43

43

2

2

3223

xx

xxx

( )( ) 43

2

2

2

23323

xx

xx

+

+++

d) ( ) ( ) ( )

( ) =

+

+

+

4

22

2

13

31321312

12

x

xxxxxxx

x

( )

( )=

+

++++=

32

222

132

1213212

xxx

xxxxxxxxx

( ) =+

++++++= 32

22222

132

1212122616

xxx

xxxxxxxxxxxx

=( )32

22

132

61116

+

++

xxx

xxxxx

Pg.24 20 a) ( ) ( ) ( )( )== 33 gfgf ( )=3f 2 ( )( ) ( )( ) ( )=== 322 fgfgf 2

) ( )3'f 1 ( )2'f 1 ( ) 3'g 1 ( )2'g 0 c) ( ) ( ) ( )( ) ( ) ( ) === 13'3'3'3' fggfgf ( )3'f ( ) =crecientef 3 ( ) 03' >f d) ( ) =22h ( ) ( ) ( ) ( )( ) ( ) ( ) ==== 02'2'2'2'2' fggfgfh 0 2=y

Pg.24 21 a) Falsa" =0'f = ... MnMxf i!ecreceiCrecef ,=

) er!a!era" >0'f > 0)(tgm Crecef =4(22


5/22


c) er!a!era" = !ecrecef ( ) ( )31 M:nimo ( )11 =

Pg.24 23 ( ) = xxxf 23' 2 ( ) ( )23' = xxxf xf

( )10P ( ) ( ) ( )== 02030'0 2fx ( ) 00' =f ?n ( )10P # la 6!nci7n ( )xf' *asa de creciente a decreciente# ha9 $%xi$o

Pg.24 24 a) ( ) ( ) ( )11444' 3 += xxxxxxf ( ) =0' xf 1=x 0=x 1=x 0' xf ( ) crecientexf =

@arece# *or tanto# de m+imos o m:nimos

Pg.24 25 10 M:nimo > 1

) ( ) = 32' xxf = 032x 2

3=x [ ]52

4

9

2

3=

f ( ) 22 =f ( ) 105 =f M+imo > 10 M:nimo >

4

9

c) ( ) = tttf 63' 2 ( ) = 023 tt 0=t [ ]41 2=t [ ]41( ) 41 =f ( ) 00 =f ( ) 42 =f ( ) 164 =f M+.> 16 M:n.> 4

d) ( ) ++= 183' 2 xxxf =++ 0183 2 xx 4

134 =x 53,2 [ ]43

4

134 +=x 13,0 [ ]43 ( ) 88,053,2 =f ( ) 07,613,0 =f

( ) 03 =f ( ) 1264 =f M+imo > 126 M:nimo 08,6 e) ( ) += 607515' 24 xxxf =+ 045 24 xx 2=x [ ]25

1=x [ ]25 ( ) 65505 =f ( ) 381 =f ( ) 162 =f ( ) 162 =fM+imo > 38 M:nimo > 16

$(22


6/22


6) = xxf 124' 3 ( ) = 034 2xx 0=x [ ]22 3=x [ ]22( ) 82 =f 93 =f ( ) 00 =f M+imo > 0 M:nimo > 9

Pg.2$1 27 A3meros > yx, 8*ositivos) 25=yx M"IM#yx$ =+ x

x$ 25

+=

= 225

1'x

$ = 025

12x

5=x 15=x ( )+ 0 5== yx

5a s!ma es m:nima c!ando amos n3meros son ig!ales.

Pg.2$1 28 ;e *ide el m+imo de ( ) 562 += xxxf en el intervalo [ ]5,22 ( ) += 62' xxf =+ 062x 3=x [ ]5,22 ( ) 32 =f ( ) 75,35,2 =f M+imo > 75,3 500.2=x ( )camisetas

Pg.2$1 29 5ados rectng!lo > yx, 2002 =+ xy yx% = ( ) 222002200 xxxx% == = x% 4200' = 04200 x 50=x 100=y rea m+ima > =10050 ( )2000.5 m Dimensiones > ( )m10050

Pg.2$4 30 =x 0 1 0#$ 0#& 0#1 -0#1 ( )=xf 1 2 1(414214 1(231144 1(071773 0(933033

[ ]

=

05,0

1414214,15,00TVM 8284,0 [ ]

=03,0

1231144,13,00TVM 7705,0

[ ][ ]

=

01,0

1071773,11,00TVM 7177,0 [ ]

+

=1,00

933033,0101,0TVM

6697,0

( ) +

2

6697,07177,00TVI 6937,0

( ) = 22' &xf x ( ) === 221220 0 &&&TVI 6931,0

Pg.2$4 31 5a derivada# en 3=x # es la *endiente de la tangente en dicho *!nto 3=x ?sta tangente *arece

5a *endiente de dicha tangente es# *!es# 1=m ( ) 13' =f

Pg.2$4 32 a) ?ntre D9 ) ) ?ntre *9 + c) ?ntre +9 ,

Pg.2$4 33 = 34

3

Vr ( ) ( ) == rhlmrTVI 0 ( )

h

hhlm

33

310001000

0

4

3 +

=1h

1

10001001

4

3 333

1

10003332,10620351.0 00206715,0

= 1,0h

1,0

10001,1000

4

3 33

3

00206776,0 ( ) 2067,0=rTVI

Pg.2$4 34 a) ( ) ( ) ( ) ( ) ( ) ( ) ++=h

hhhlmf

1312131201'

22

h

hh 72 2

72h 7

%(22


7/22


( ) ( ) ( ) ( ) ( ) ( )

++

=h

hhhlmf

2322232202'

22

+h

hh 52 2

+ 52h 5

) ( ) ( ) ( ) ( ) [ ] ++++=h

hhhlmf

50050000'

23

+h

hh3+12h 1

( ) ( ) ( ) ( ) [ ]

++++

=h

hhhlmf

55555505'

33

++ 76152 hh 76

c) ( ) ( ) ( ) ( ) +=h

hhlmf

525202'

22

4h 4

( ) ( ) ( ) [ ]

+

=h

hhlmf

525202'

22

+ 4h 4+

Pg.2$4 35 ( ) = xxfy 1 hi.r/ola euil%tera s:ntotas > C 9 @entro simetr:a > ( )00 > origen

( )

+= hh

hlmm 2

1

2

1

0 ( )

+

h22

1

4

1

( )2

12 =f angente > ( )= 2

4

1

2

1xy 1

4

1+= xy

Pg.2$4 36 ( ) = 2xyxf P!nto de tangencia > ( )=yx ( )2xx ( ) = 'yxf x2

Pendiente recta ( )10 ( )2xx =+0

12

x

x

x

x 12 + =+ x

x

x2

12

1=x

=1x P!nto tangencia" ( )11 Pendiente tangente" 2 ecta tangente" ( )= 121 xy 12 = xy

= 1x

P!nto tangencia"( )11

Pendiente tangente" 2ecta tangente" ( )+= 121 xy 12 = xy

Pg.2$4 37 += 3103' 2 xxy = 0'y =+ 03103 2 xx 3=x 31=x

=3x ( ) 113 =f 11=y = 31x ( )27

4131 =f

27

41=y

Pg.2$4 38 a) ( ) = 42xxfy xy 2'= =1x 3=y 2'=y 2=tgm 21=nom

g." ( )=+ 123 xy 52 = xy Ao." ( )=+ 12

13 xy

2

5

2

1= xy

) ( )

=5

2

x

xfy

( )2

5

2'

=x

y =1x 2

1=y

8

1' =y tgm 8=nom

g." ( )=+ 18

1

2

1xy

8

3

8

1 = xy Ao." ( )=+ 182

1xy

2

178 = xy

Pg.2$4 39 ?n ( )1 " ( ) < 0' xf ( ) edecrecientxf = ?n 1=x " ( ) =0' xf ( ) Mnimoxf = ?n ( )31 " ( ) > 0' xf ( ) crecientexf = ?n 3=x " ( ) =0' xf ( ) Mximoxf = ?n ( )+3 " ( ) < 0' xf ( ) edecrecientxf =

Pg.2$4 40 o/o# cada c!rva ( )xf 9 ( )xg *arece ser el traslado de la otra en direcci7n

(22


8/22


?n tal caso# se tiene" ( ) ( ) ( ) C#$T%TExgxfxh = 5!ego ( ) 0' =xh ( ) 029' =h

Pg.2$4 41 ( ) ( )= xfxg '' Para cada x #- 5as tangentes a amas son *aralelas- mas c!rvas son E*aralelasF

- rdenada en origen de( )xf

#0=y

- rdenada en origen de ( )xg # 2=y- ( ) ( ) 2+= xfxg

Pg.2$4 42 ( ) = 00f Gr6ica *asa *or ( )00 ( ) =10'f angente en origen con 1=m ( ) =03f Gr6ica *asa *or ( )03 ( ) = 13'f angente en ( )03 con 1=m

Pg.2$4 43 a) Falsa. 5:mite del en!nciado ( ) n!eri'a(leExf = 2=x ( ) 02' =f

) Falsa o er!a!era. Basta considerar( ) 02

1

=f con

0)2('1

=f

5a 6!nci7n ( ) ( ) 112 + xfxf c!m*le ( ) 02'2 =f *ero no c!m*le ( ) 022 =f c) Falsa o er!a!era. Basta

d) Falsa. 5:mite del en!nciado ( ) = !eri'a(lexf Continua en 2=x

Pg.2$$ 44 a) ( ) ( ) ( ) [ ] ++++h

xxhxhxhlm

1451450.

22

+

h

hxhh 4105 2

+ 4105 xh 410 x

) ( )( ) +

++h

xhxhlm 1

1

1

1

0.

22

( )( )[ ]+++

11

222

2

hxxh

hxh

( )22 1

2

+

x

x

Pg.2$$ 45 a) ( ) ( ) ( ) ( ) [ ] ++++=h

xxhxhxhlmxf

25250.'

22

( )

++

h

hxh 52

52 +x

( ) ( ) ( ) ( )+

=

h

xfhxfhlmxf

''0"

( )[ ] [ ]

+++

h

xhx 5252 h

h2 2

) ( ) ( ) ( ) [ ] +++= hxhxhlmxf 330'

33

++ 22 33 hxhx

23x

( ) ( ) ( )+

=

h

hxhlmxf

23

0"

+ hx 36 x6

Pg.2$$ 46 ( ) ( )

=+

hhlmxf

xhx33

0'

h

hx 13

3 ( ) ( )0'fxf ( ) ( ) ( )0'' fxfxf =

5a constante !e roorcionali!a!es la derivada de la 6!nci7n en 0=x

'(22


9/22


=x -& -1 -0#$ 0 0#$ 1 1#$ 2 ( )=xf 0#04 0#11 0#$' 1 1#& & $#20 #00

( ) =xf' 0#04 0#12 0#%& 0#0 1#0 &0 $#1 #'

= 1,0h 1,0

13 1,0

1612,1 = 1,0h

1,0

13 1,0

0404,1

( ) ( ) + 20404,11612,10'f 22016,2 1008,1 1,1

( ) ( ) ( ) '0' aaa xx = =2a ( ) ( ) 69,022 ' = xx x2< =3a ( ) ( ) 1,133 ' = xx ( )x3> 32

23

4

3

ecta tangente"

=

4

31

2

3xy

4

3+= xy

(22


10/22


Pg.2$$ 52 =0x ( ) =

=2

2xf 1 ( )10 ( )

( )

+

22

4'

xxf

( ) =

+ 220

41

angente a ( )xf " ( )=+ 011 xy 1= xy =0x ( ) =+= 100xg 1 ( )10 ( ) + 12' xxg =+ 102 1

angente a ( )xg " ( )=+ 011 xy 1= xy

5as tangentes coinciden H mas curas son tangentesen( )10

Pg.2$$ 53 ( ) = 11

xx

xf == 22' 1

xxf

=+=

3

3" 22

xxf

==

4

4'" 3232

xxf

55 432432

xxf i' =+= ( ) ( ) = +14321 nnn x nf

( ) ( ) 1!1 += nnn xnf

Pg.2$$ 54 =x -1#$ -0#$0 -0#2$ 0 0#2$ 0#$0 0#$ 1 1#2$ 1#$0 1#$ 2 2#2$ ( )=xf ' '' $$ 4 4#$ %#1& #2& ' '#14 #&' $#42 2 -

( )=xg 14 #'' #$$ 10 10#$ 12#1& 1& 14 14#14 1&' 11#42 ' 2#'&

( ) ( ) ++= 489 2'' xxxgxf 9

434 =x

( ) ( ) += 818"" xxgxf 9

4=x

M:nimo relativo" =9434x 7286,0

M+imo relativo" +

=9

434x 1730,1

De las 6!nciones# *ara toda x " ( ) ( ) 6= xfxg 5a di6erencia de valores ta!lados# lo con6irma 5a c!rva ( )xg se otiene *or traslaci7n de ( )xf % !nidades hacia arria

Pg.2$$ 55 a) 5os valores de x 9 ( )xf son idnticos a los del 54

5os de ( )xg son simtricos de ( )xf res*ecto C ?sto es consec!encia de

) 5os valores de x 9 ( )xf son idnticos a los del 54 5os de ( )xg son los de ( )xf # m!lti*licados *or 2 ?sto es consec!encia de

10(22


11/22


Por otro lado# se veri6ica

88

f

fgfg

f

g ( )

2

4

614108

16

46823

6)

2

'''

5 f

fhfhg ( )

24

6341

5

10

16

222

8

112

8

5

Pg.2$$ 57 a) ( ) ( )+ 592533 223 xxx ( ) ( ) 53531233 + xxxx

) ( ) ( ) xxxx 4524 4325 ( )( )4524 333 xxx

c) ( ) ( )++ xxxx 2432 3324

( )324

3

3

48

+

+

xx

xx

d) ( ) ( ) 1634 52 xxx ( )523424

xx

x

+

e) xx

x

532

56

2

6)

4

3

12

4

x

x

4

3

1

2

x

x

Pg.2$% 58 a) ( ) ( )( )

+

+

2

2

1

3111

1

33

x

xx

x

x

( )

( ) 4

2

1

312

+x

x

11(22


12/22


)

( )

+

+

2

4

2

12

121

x

x

x

xxx

( )

+

+12

2

4

2

xx

xxx

12

23 +

+

xx

x

c)

2

2

231

32

2

x

xxx

x

( )

3.2

32

22

22

xx

xx

3

3

22

xx

d)

+12

211

2

2

x

xxx

1

12

2

2

x

x

e) ( ) +

+12

22424

2

3

x

xx ( )

1428

2

3

++

x

xx

6) ( )( ) ( )

+ 13

2

132 xx

xxx ( ) ( )+ 323 2 xxx ( ) ( )133 xx

g) ( )

+

+

4

221

2

1

x

xxxxx

( ) ( )

++4

22

2

1421

xx

xxxx

xx

x

+

2

2

23

h) ( )

( )

2

1

2

111

x

xxx (

( )

2

12

12

x

xx ( )212

2

x

x

Pg.2$% 59 =x -1#$ -1 -0#$ 0 0#$ 1 1#$ 2 2#2$ =f 1'#$& ' '' 4 %#1& ' #&' 2 - =x -%#$ -% -$#$ -$ -4#$ -4 -$ -& -2#$ =g 1'#$& ' '' 4 %#1& ' #&' 2 - 5as dos c!rvas ( )xf 9 ( )xg son ig!ales Ja9 des*la,amiento hori,ontal entre amas 5as tangentes en ( )5+xf 9 ( )xg son *aralelas

Pg.2$% 60 5a c!rva ( )xf es idntica a la ( )xf del 59 =x -0#$ -0#2$ 0 0#2$ 0#$ 1 1#$

=g ' '' 4 %#1& ' 2 -&2 Por com*resi7n# se otiene ( )xg a *artir de ( )xf angente en ( )xf 2 ms inclinada

( ) 389 2' ++= xxxf ( ) 63272 2' += xxxg

Por tanto# es ( ) ( )xfxg 22 ''

=

Pg.2$% 61 a) 'y ( ) 52' xxp

( ) ( )positi'oxp 2" = ( ) =0' xp 25=x ( )mnimo

?n ( )25 ( ) edecrecientxp = ?n ( )+25 ( ) crecientexp =

tro modo" ( ) par(olaxp = Krtice )(mnimo 25=x

ert!ra > sentido Decrece en ( )25 9 @rece en ( )+25

12(22


13/22


=x -1 0 1 & 4 $ %( )= xpy 1' 12 ' % ' 12 1'

) ( ) xxxry 66 2'' ( ) 612" = xxr ( ) =0' xr 0=x

( )


14/22


[II][III] = 069 (a 23= a( [III] ( ) =+ 023412 caa 06 =+ ca ac 6= [I] ( ) ( ) =++ 02622348 aaa 0210 =+a 51=a

( )= 2351( 103=( ( )= 516c 56=c

5a 6!nci7n es# *!es# ( ) ( ) ( ) ( ) 15610351 23 ++ xxxxf

Pg.2$% 63 ( ) c(xaxxf ++ 2 ( ) (axxf +2'

( )Pf ( ) 201P ( ) ( ) ++= c(a 1120 2 20=+ c(a [I] ( )Qf ( )123Q ( ) ( ) ++= c(a 3312 2 1239 =++ c(a [II] ( ) = 03'f ( ) =+ 032 (a 06 =+ (a [III] [II] [I] =+ 848 (a =+ 22 (a ( )12 += a( [III] ( ) =+ 0126 aa = 24a 21=a 3=(

[I] ( ) =+ 20321 c 233=c 5a 6!nci7n es# *!es# ( ) ( ) ( ) ( )233321 2 ++ xxxf

Pg.2$% 64 ( ) >0' xf 3x ( ) crecexf = ( ) ( )+ 43 ( ) decrecexf =' ( )43 ( ) > 43f Por e/em*lo" ( ) 53 =f ( ) > 24f Por e/em*lo" ( ) 14 =f

Pg.2$% 65 ( ) +

+

1

1

2

233

x

xxxxf

( )

( )

( ) 1111332

22

22224'

++

+++

= xx

xxxxxxf

( ) = 0' xf ( )( 011332 2222 =+++ xxxx =0x ( ) 00 =f EM:nimoF relativo 8*!nto de in6le+i7n)

( ) =+++ 011332 222 xxx ( ) =+ 065 22 xx 0=x imaginariox == 56 ( ) ( )xfxf = L!nci7n IMP ;imtrica res*ecto origen ( )00 =x -$ -4 -& -2 -1 0 1 2 & 4 $ ( )= xfy 100#4 4'#4' 1'#4% 4#42 0#2 0 -0#2 -4#42 -1'#4% -4'#4' -100#4 &%xi$oasol!to en { }55 " 5=x ( ) 49,100xf &'ni$oasol!to en { }55 " 5=x ( ) 49,100xf

Pg.2$% 66 Dimensiones" yxx Kol!men" yxV = 2 rea total" 19242 =+ xyx

x

xy

4

192 2= =

= x

xV

4

192 2

448

3xx

4

348

2' xV =

=0'V = 04

348

2x

8=x 8=x 4=y

Dimensiones de la ca/a" ( )cm488

Pg.2$% 67 P!nto !scado" ( )yxM

2

30P Distancia MP" ( )

2

2

2

30

+= yx!

14(22


15/22


2

22

2

3

+= xx! 2

22

2

'

2

32

22

322

+

+

=

xx

xxx

! =0'! ( ) 014 2 =xx

oValex =0 ( )2301 =x 1=x 1=y ( )11M

Pg.2$% 68 @oordenadas al origen de la recta" ( )0a ( )(0 ecta" 1=+(

y

a

x

( ) 41% =+ 141

(a

4=(

(a = a($

2

1

42

1 2

=

(

($

=0'$ ( )

( ) =

0

4

42

2

12

2

(

((( = 082 (( ( )oVale( 0= 8=(

=48

8a 2=a 1

82=+

yx 84 += xy

Pg.2$% 69 a) Falsa. ?n 3=x se tiene ( ) ( )xgxf >

;i ( ) =decrecexf ( ) 0'


16/22


( )=#MsTVM. ( ) ( )

=M#

Ms#s=

310

1920 7

114,0 ( )mlg //

) 15,00008,0)(' += tts ( ) =0' ts =0008,0

15,0t 5,187t ( )das

M+ima salinidad entre d:as 69 7de /!lio

Pg.2$ 73 a) 5a velocidad es la derivada del es*acio# res*ecto al tiem*o. Por tanto"( ) += 4812' tts ( ) ( ) += 480120's ( ) =0's 48 ( )sm

) ?l descenso em*ie,a c!ando la velocidad es cero 8m:nima). 5!ego"( ) =0' ts =+ 04812t = 1248t =t 4 ( )s

c) 5a alt!ra m+ima corres*onde a la velocidad n!la. ?n consec!encia"=4t += tts 486 2 ( ) ( )+= 44846 2s =s 96 ( )m

d) ras ascenso 9 descenso# el es*acio total es n!lo. De ah:

( ) == 521

522

2'

xxxf == 5321

tgm 1 == tgno mm

1

1

angente" ( )= 311 xy 2= xy @orte C" =0y 2=x ( )02=P Aormal" ( )= 311 xy 4+= xy @orte C" =0y 4=x ( )04=Q

rea PN" hPQ2

1 ( ) 1242

1 122

11 ( )2u

Pg.2$ 77 l*aseInicia* 0 t**t 20+= ial%lturaInic% =0 t%%t 30+=

rea ring!lo" tt%*$2

1 ( )+= '''

2

1%*%*$ ( )*%$ 32

2

1' +=

$TVI.

( )=+= 5342

2

1'$ ( )=+1582

1= 23

2

15,11 ( )scm2

Pg.2$ 78 =1x ( ) == 311f 1 ( ) = 2' 3xxf ( ) == 2' 131f 3 ( ) 3=ftgm

=1x ( ) ( ) h-g == 1111 ( ) = -xxg 2' ( ) = -g 21' ( ) -m gtg =21%(22


17/22


a) ( ) ( ) = gtgftg mm = -23 1=- ) ( ) ( ) = gtgftg mm 1 ( )= -213 37=-

Pg.2$' 79 =1x 112 ==y ( )11=% =3x 932 ==y ( )93=*

==

=2

8

13

19%*m 4 ( ) = xxy 2' ( )=xmtg x2 = x24 2=x 4=y

angente" ( )= 244 xy 44 = xy

Pg.2$' 80 a) = xy xxM x*ase = 9 x%ltura = ( ) xx$ = 9

( )x

xx$2

19

' += =0'$ =

02

39

x

x 3=x 33M

Dimensiones con %ream+ima" ( ) xx9 36 ) = xy ( )xx x*ase =9 x%ltura = xxP += 92

x

P 12' += =0'P =+ 0

12

x

4

1=x

2

1

4

1

Dimensiones con er'$etrom+imo"2

1

4

35

Pg.2$' 81 2' 1

xy = 2

1

xmtg

= P. g"

aaT 1

g" ( )axaa

y =2

11

@orte C" =0y ax 2= ( )02 aM @orte " =0xa

y 2

=

a

20

P!nto $e!ioMA =

++

2

20

2

02 aa

aa

1 P!nto tangencia

Pg.2$' 82 'olumenmasa

d= menMnimoVoluidadMxima!ens ( ) 2' 002037,001700874,006426,0 TTTV += ( ) =0' TV Ao tiene sol!ciones reales ( ) 87,9990 =V ( ) 26,98730 V ?ntre 09 30 @# el olu$en $'ni$o8'#2%) se da a C30

Por tanto# entre 0 9 &0 @# la !ensi!a! $%xi$ase otiene a C30

Pg.2$ 83 ( ) 14

11

43

2

'

= n

nc ( ) =0' nc

=

3

214n ( )+ 0n

+=

3

214n 6188,8n A art:c!los a 6aricar" 619.8

Pg.2$' 84 %ltura h =Radio*ases r minimi,ar" ( ) ( )=+= 22221 rrhC ( )222 rrh+

= hrVolumen 2333 2333

rh

= 24666

rr

C += ( ) rr

rC 8666

2

' +=

( ) =0' rC =+ 08666

2 r

r 3

4

333

=r 98,2 ( )cm 3 2 3334

=h 93,11 ( )cm

1(22


18/22


&'ni$o# 9a

( ) + 813323

"

rrC

0"

>C *ara todo r

Dimensiones *ara m:nimo coste de 6aricaci7n" ( )cmr 98,2 ( )cmh 63,11=

Pg.2$' 85 5ongit!d list7n ( )x oc!*ada *or tela naranarectang!lar ( )xx 3 5ongit!d list7n ( )y oc!*ada *or tela er!ec!adrang!lar ( )yy 6=+ yx ;!*er6icie a minimi,ar" += 223 yx$ ( )22 63 xx$ +=

( ) ( )( )=+= 1626' xxx$ 128 x ( ) >= 08

"x$ M:nimo

( ) =0' x$ 2

3=x = ( )m5,1

2

9=y = ( )m5,4

ela naran/a > ( )m5,45,1 ela verde > ( )m5,45,4

Pg.2$' 86 x.n/mero x#tro/mero 20 minimi,ar" ( )xx$ += 2022

( ) 4022 += xxx$ ( ) 22' = xx$ ( ) >= 02" x$ M:nimo ( ) =0' x$ = 022x =x 1 =x20 19 ( ) += 120212Mnimo$ 39

Pg.2$' 87 (*ase 2 h%ltura 22 (hx&adoIgual += =+ 5022 x( 2522 =++ (h( ma+imi,ar" h($ =

( ) ( ) = 6255025 xxx$ ( )252

25015'

+

=x

xx$

( ) =0' x$ =+ 025015x 350=x 3502 =( ring!lo euil%tero

lt!ra =2

3

3

50h

3

325=h

Pg.2$' 88 Dimensiones *gina > yx 600= Parte im*resa > ( ) ( )44 = yx$

( ) ( )

= 46004x

xx$ ( )2

2' 24004

x

xx$ += ( ) =0' x$ 610=x 610=y

5a *arte im*resa es m+ima# c!ando la *gina es cua!ra!a

Pg.2$' 89 Per:metro" ++= rxr 226 ++

= xr2

23

rx2

23

+=

( ) ( ) 22 262

rrrr$

++= ( ) ( ) ++= rrr$ 226' ( ) ( )rr$ += 46'

( ) =0' r$ ( ) =+ 046 r +

=4

6r

+=

4

12x

Pg.2$' 90 lt!ra > *&> 1 =xPM xP% =1 22 M*PMPCP* +== ( )==++ x$PCP*P% ( ) ++ 22 2421 xx ( ) 421 2 ++ xxx$

1'(22


19/22


( )4

21

2

'

++=

x

xr$ ( ) =0' r$

3

4=x { }10

?l m:nimo estar en e+tremo" ( ) 50 =$ 9 ( ) 521 =$ &'ni$ocon 1=x %P

Pg.2$' 91 ecorrido *or calle > x ecorrido *or *ar ( )22 400100 x+

iem*o a minimi,ar" ( ) ( )3400100

5

22

xxxT ++=

( ) ( )( )

( )22'

4001006

14002

5

1

x

xxT

+

+= ( ) =0' xT

( ) ( )=+ xx 400104001006 22 ( ) ( ) 22222 40054001003 xx =+

( ) = 2222 40041003 x

=4

1003400 x = 75400 x 325=x

ecorrido *or *ar


20/22


Polinomio de6inido" ( )30 3=d ( ) =40'f 4=c ( ) = 01'f 0423 =++ (a ( ) =44'f 0848 =+ (a 94=a 38=( ( ) ( ) 343894 23 ++= xxxy P!ntos" =x -1 -0#$ -0#2$ 0 0#$ 1 2 & 4 $

=y -4 0#2' 1#'& & 4#& 4#' ' & 4#' 11#'

Pg.2$ 95 ( ) -xxxxf +++=2

1

2

1 23 ( ) =71f =+++ 7

2

1

2

11 - 5=-

5a 67rm!la es" ( ) 52

1

2

1 23 +++= xxxxf

Pg.2$ 96 ?s claro

dems# derivadas laterales idnticas en todos los *!ntos# es decir"( ) ( )= + 11 '' ff (=1

Pg.2$ 98 ?n ( )2 ( ) crecientexf =' ( ) 0' >xf ( )xf > creciente 9 c7ncava en sentido

?n = 2x ( ) ?2" realf =( )xf > *!nto de in6le+i7n

?n ( ) 12 ( ) edecrecientxf =' ( ) 0' creciente 9 c7ncava en sentido

?n =1x ( ) 01' =f ( )xf > *!nto m+imo

?n ( )+1 ( ) edecrecientxf =' ( ) 0' decreciente 9 c7ncava en sentido

Pg.2$ 99 ecta tangente" nmxy +=

P!ntos corte c!rva" xxxnmx =+ 24 2 ( ) 012

24 =+ nxmxx

P!ntos tangencia" ax= (x= ( ) ( ) ( )2224 12 (xaxnxmxx + ( ) nxmxx + 12 24 = ( ) ( ) ( ) 2222234 242 (ax(aa(x(a(ax(ax ++++++ 0=+(a 24 22 =++ (a(a ( ) m(aa( +=+ 12 n(a =22 n 1=m 1=a 1=( 1=n angente" 1= xy Pts.angencia" += 1a ( ) 21 =+f ( )21 = 1a ( ) 01 =f ( )01

Pg.2$ 100 ?n [ ](a " ( ) =deri'a(lexf ( ) continuaxf = ( ) 0' xf ( ) montonaxf = a) P!ede oc!rrir

c) Ao es *osile


21/22


Pg.2$ 101 Pro9ecci7n vertical escalera > y Pro9ecci7n hori,ontal escalera > x

( ) ( )= stsmx 5,0 tx = 5,0 ( )m ( )22 5,07 ty = 2'

25,049

25,0

t

ty

=

a) 2=x ( )m ==5,0

2t 4 ( )s =

=

2

'

425,049

425,0yVy

45

1 15,0

( )sm

) 4=x ( )m ==5,0

4t 8 ( )s =

=

2

'

825,049

825,0yVy

33

2 35,0

( )sm

c) 6=x ( )m ==5,0

6t 12 ( )s =

=

2

'

1225,049

625,0yVy

13

3 83,0

( )sm

Pg.2$ 102 ?n res!men las il!straciones corres*onden a ( )x1 # ( )x1' 9

( )x1" 5as c!rvas re*resentadas# en s!s corres*ondencias# de/an m!cho

In6le+i7n en ( )x1 M+imo o M:nimo en ( )x1'

@ero en ( )x1"

M+imo o M:nimo en ( )x1 @ero en ( )x1'

( )x1 > @reciente ( )x1' > Positiva

( )x1 > Decreciente ( )x1' > Aegativa

( )x1 > @7ncava sentido ( )x1"> Positiva

( )x1 > @7ncava sentido ( )x1"> Aegativa

5 ;5U@IA ?;# J# ?KID?A?"

21(22


22/22


( ) %x1 ( ) *x1 ' ( ) Cx1 "

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