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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
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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
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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
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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
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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
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( ) ( ) ( ) ( ) ( ) ( )
++
=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$4 40 o/o# cada c!rva ( )xf 9 ( )xg *arece ser el traslado de la otra en direcci7n
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?n tal caso# se tiene" ( ) ( ) ( ) C#$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$$ 46 ( ) ( )
=+
hhlmxf
xhx33
0'
h
hx 13
3 ( ) ( )0'fxf ( ) ( ) ( )0'' fxfxf =
5a constante !e roorcionali!a!es la derivada de la 6!nci7n en 0=x
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=x -& -1 -0#$ 0 0#$ 1 1#$ 2 ( )=xf 0#04 0#11 0#$' 1 1#& & $#20 #00
( ) =xf' 0#04 0#12 0#%& 0#0 1#0 &0 $#1 #'
= 1,0h 1,0
13 1,0
1612,1 = 1,0h
1,0
13 1,0
0404,1
( ) ( ) + 20404,11612,10'f 22016,2 1008,1 1,1
( ) ( ) ( ) '0' aaa xx = =2a ( ) ( ) 69,022 ' = xx x2< =3a ( ) ( ) 1,133 ' = xx ( )x3> 32
23
4
3
ecta tangente"
=
4
31
2
3xy
4
3+= xy
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Pg.2$$ 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$$ 55 a) 5os valores de x 9 ( )xf son idnticos a los del 54
5os de ( )xg son simtricos de ( )xf res*ecto C ?sto es consec!encia de
) 5os valores de x 9 ( )xf son idnticos a los del 54 5os de ( )xg son los de ( )xf # m!lti*licados *or 2 ?sto es consec!encia de
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Por otro lado# se veri6ica
88
f
fgfg
f
g ( )
2
4
614108
16
46823
6)
2
'''
5 f
fhfhg ( )
24
6341
5
10
16
222
8
112
8
5
Pg.2$$ 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$% 58 a) ( ) ( )( )
+
+
2
2
1
3111
1
33
x
xx
x
x
( )
( ) 4
2
1
312
+x
x
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)
( )
+
+
2
4
2
12
121
x
x
x
xxx
( )
+
+12
2
4
2
xx
xxx
12
23 +
+
xx
x
c)
2
2
231
32
2
x
xxx
x
( )
3.2
32
22
22
xx
xx
3
3
22
xx
d)
+12
211
2
2
x
xxx
1
12
2
2
x
x
e) ( ) +
+12
22424
2
3
x
xx ( )
1428
2
3
++
x
xx
6) ( )( ) ( )
+ 13
2
132 xx
xxx ( ) ( )+ 323 2 xxx ( ) ( )133 xx
g) ( )
+
+
4
221
2
1
x
xxxxx
( ) ( )
++4
22
2
1421
xx
xxxx
xx
x
+
2
2
23
h) ( )
( )
2
1
2
111
x
xxx (
( )
2
12
12
x
xx ( )212
2
x
x
Pg.2$% 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
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=x -1 0 1 & 4 $ %( )= xpy 1' 12 ' % ' 12 1'
) ( ) xxxry 66 2'' ( ) 612" = xxr ( ) =0' xr 0=x
( )
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[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!
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1 Bachillerato - Matemticas - Unidad 10 - Derivadas
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$% 69 a) Falsa. ?n 3=x se tiene ( ) ( )xgxf >
;i ( ) =decrecexf ( ) 0'
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( )=#MsTVM. ( ) ( )
=M#
Ms#s=
310
1920 7
114,0 ( )mlg //
) 15,00008,0)(' += tts ( ) =0' ts =0008,0
15,0t 5,187t ( )das
M+ima salinidad entre d:as 69 7de /!lio
Pg.2$ 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
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1 Bachillerato - Matemticas - Unidad 10 - Derivadas
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$' 84 %ltura h =Radio*ases r minimi,ar" ( ) ( )=+= 22221 rrhC ( )222 rrh+
= hrVolumen 2333 2333
rh
= 24666
rr
C += ( ) rr
rC 8666
2
' +=
( ) =0' rC =+ 08666
2 r
r 3
4
333
=r 98,2 ( )cm 3 2 3334
=h 93,11 ( )cm
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&'ni$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
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( )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
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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
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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?"
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( ) %x1 ( ) *x1 ' ( ) Cx1 "