Post on 01-Jun-2018
8/9/2019 Repaso 2do Examen Parcial
1/15
Universidad Rafael Landivar
Facultad de Ingeniera
Matemtica I
Ingra. Adriana Lpez
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Determine whether the relation represents a unction. I it is a unction! state the "omain an" ran#e.
1)
!"# $
1 %"
1# #$
A) functi&n
d&main' ( # 1 1#*
range' (!" $ %" #$*
+) functi&n
d&main'(!" $ %" #$*
range' ( # 1 1#*
,) n&t a functi&n
1)
!) (-1.!! %.1!) -1.!!! -%.1) -$
") -1./% -1)*
A) functi&n
d&main' (%.1! -%.1 " -1*
range' (1.!! 1.!!!$
1./%*
+) functi&n
d&main' (1.!! 1.!!!$
1./%*
range' (%.1! -%.1 " -1*
,) n&t a functi&n
!)
$in" the value or the unction.
) Find f-!) 0en f-2) = 2!+/2.
A) +) 1" ,) ! 1" 3) !
)
) Find f-2 +1) 0en f-2) =2!-
2 +.
A)2!+!2 -!
2 ++)
2!+!2 -!
2 -!,)
2!+!2 +
2 +3)
2!-!
2 +
)
$in" the "omain o the unction.
$)2
2 -1
A) (242 >1* +) (242 1*,) all real num5ers 3) (242 1*
$)
/) g-2) =2
2!-#
A) (242 >#* +) all real num5ers
,) (242 -% %* 3) (242 "*
/)
1
8/9/2019 Repaso 2do Examen Parcial
2/15
The #raph o a unction is #iven. Use the #raph to answer the question.
%) F&r 0ic &f te f&ll&0ing values &f 2 d&es f-2) =-6"7
1""
-1"" 1""
-1""
A) " +) " ,) /" 3) -6"
%)
Use the #raph to in" the intervals on which it is increasin#! "ecreasin#! or constant.
6)
A) 3ecreasing &n -- -1) and -1 )8 increasing &n --! 1)
+) 3ecreasing &n -- -!) and -! )8 increasing &n --1 1)
,) 3ecreasing &n -- -!) and -! )8 increasing &n --1 1)8 c&nstant &n --! -1) and -1 !)
3) Increasing &n -- -!) and -! )8 decreasing &n --1 1)8 c&nstant &n --! -1) and -1 !)
6)
!
8/9/2019 Repaso 2do Examen Parcial
3/15
The #raph o a unction is #iven. Use the #raph to answer the question.
#) Find te num5ers if an9 at 0ic f as a l&cal ma2imum. :at are te l&cal ma2ima7
x-
-
2
2
y2
-2
x-
-
2
2
y2
-2
A) f as a l&cal ma2imum at -;8 te l&cal ma2imum is 1
+) f as a l&cal ma2imum at 2 ="8 te l&cal ma2imum is -1
,) f as a l&cal ma2imum at 2 =-; and ;8 te l&cal ma2imum is 1
3) f as n& l&cal ma2imum
#)
1")
x-10 10
y
10
-10
(-8, 5)
(-5, 0)
(0, 0)
(4, 0)
(5, -2.5)
(-9.5, 0)
(-2.5, -3.3)
(2.2, 3.9)
x-10 10
y
10
-10
(-8, 5)
(-5, 0)
(0, 0)
(4, 0)
(5, -2.5)
(-9.5, 0)
(-2.5, -3.3)
(2.2, 3.9)
Find te num5ers if an9 at 0ic f as a l&cal minimum. :at are te l&cal ma2ima7
A) f as a l&cal ma2imum at 2 =-. and -!.$8 te l&cal ma2imum at -. is -!.$8 te l&cal
ma2imum at -!.$ is $
+) f as a l&cal ma2imum at 2 =-!.$ and $8 te l&cal ma2imum at -!.$ is -.8 te l&cal
ma2imum at $ is -!.$
,) f as a l&cal minimum at 2 =-. and -!.$8 te l&cal minimum at -. is -!.$8 te l&cal
minimum at -!.$ is $
3) f as a l&cal minimum at 2 =-!.$ and $8 te l&cal minimum at -!.$ is -.8 te l&cal minimum
at $ is -!.$
1")
$in" the avera#e rate o chan#e or the unction between the #iven values.
11) f-2) =2!+128 fr&m t& /
A)11
+) 11 ,) !1 3) %
11)
8/9/2019 Repaso 2do Examen Parcial
4/15
1!) f-2) =
2 -!8 fr&m t& %
A) % +)1
,) -
1"3) !
1!)
%e#in b& #raphin# the stan"ar" qua"ratic unction '()=(*. Then use transormations o this #raph to #raph the #iven
unction.
1) -2) =--2 +%)!+!
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
A)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
+)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
,)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
3)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
1)
8/9/2019 Repaso 2do Examen Parcial
5/15
%e#in b& #raphin# the stan"ar" absolute value unction '()= ( . Then use transormations o this #raph to #raph the
#iven unction.
1) g-2) =1
2 -/ +
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
A)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
+)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
,)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
3)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
1)
$
8/9/2019 Repaso 2do Examen Parcial
6/15
Use the #raph o the unction ! plotte" with a soli" line! to s+etch the #raph o the #iven unction #
1$) g-2) =-f-2 -!) -!
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
9 =f-2)
A)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
+)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
,)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
3)
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
x-6 -5 -4 -3 -2 -1 1 2 3 4 5 6
y6
5
4
3
2
1
-1
-2
-3
-4
-5
-6
1$)
/
8/9/2019 Repaso 2do Examen Parcial
7/15
%e#in b& #raphin# the stan"ar" qua"ratic unction '()=(*. Then use transormations o this #raph to #raph the #iven
unction.
1/) g-2) =-1
!-2 +%)!-!
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
6
4
2
-2
-4
-6
-8
-10
A)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
+)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
,)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
3)
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8 10
y10
8
64
2
-2
-4
-6
-8
-10
1/)
,iven unctions an" #! perorm the in"icate" operations.
1%) f-2) =2!-%2 g-2) =2!-$2 -1
Find fg
.
A)2
2 +1+)
2 -%
-$,)
2!-%2
2!-$2 -13)
-2
1
1%)
%
8/9/2019 Repaso 2do Examen Parcial
8/15
16) f-2) = 2 +! g-2) = !$2 -1/
Find fg.
A) -2 +!)-$2 -) +) - 2 +!)- !$2 -1/)
,) -$2 -)- 2 +!) 3) -2 +!)-!$2 -1/)
16)
$in" the "omain o the in"icate" combine" unction.
1#) Find te d&main &ff
g
-2) 0en f-2) =2!-62 and g-2) =2!-/2 -.
A) 3&main' - -! -!
+) 3&main' - -! -! +! -!
,) 3&main' --)
3) 3&main' - -! -! +! +!
1#)
$in" the "omain o the composite unction #.
!") f-2) =/
2 +% g-2) =2 +!
A) --) +) -- -%) &r --% )
,) -- -%) &r --% -!) &r --! ) 3) -- -#) &r --# )
!")
!1) f-2) =
2 +# g-2) =
#
2
A) -- -1) &r --1 ") &r -" ) +) -- -#) &r --# ") &r -" )
,) --) 3) -- -#) &r --# -1) &r --1 ") &r -" )
!1)
!!) f-2) =2 +#8 g-2) = 2
A) --) +)
8/9/2019 Repaso 2do Examen Parcial
9/15
8/9/2019 Repaso 2do Examen Parcial
10/15
A)
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
+)
x-10 -5 5 10
y
10
5
-5
-10
x-10 -5 5 10
y
10
5
-5
-10
,)
x-10 10
y
10
-10
x-10 10
y
10
-10
3)
x-10 10
y
10
-10
x-10 10
y
10
-10
,raph as a soli" line an" --as a "ashe" line in the same rectan#ular coor"inate space. Use interval notation to #ive the
"omain an" ran#e o an" --.
!6) f-2) =-2 -)! 2
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
!6)
1"
8/9/2019 Repaso 2do Examen Parcial
11/15
A)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
d&main =-- )8 range =-" )
-1d&main =-" )8 range =-- )
+)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
>as n& inverse
fd&main =-- )8 range =-")
,)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
d&main =-- )8 range =-" )-1d&main =-" )8 range =-- )
3)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
fd&main =- )8 range =-" )f-1d&main =-" )8 range =- )
!#) f-2) = 2 -$
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
!#)
11
8/9/2019 Repaso 2do Examen Parcial
12/15
A)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
d&main =-" )8 range =-$ )
-1d&main =-$ )8 range =-" )
+)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
fd&main =-" )8 range =-$ )
f-1d&main =--$ )8 range =-" )
,)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
d&main =-" )8 range =-$ )-1>as n& inverse.
3)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
fd&main =-" )8 range =--$ )f-1d&main =--$ )8 range =-" )
") f-2) =
2 +$
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
")
1!
8/9/2019 Repaso 2do Examen Parcial
13/15
A)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
d&main =-- )8 range =-" )
-1d&main =-" )8 range =-- )
+)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
fd&main =-- )8 range =--)
f-1d&main =-- )8 range =-- )
,)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
d&main =-- )8 range =-- )-1d&main =--)8 range =-- )
3)
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
x-10 -8 -6 -4 -2 2 4 6 8
y10
8
6
4
2
-2
-4
-6
-8
-10
fd&main =-- )8 range =--)f-1d&main =-- )8 range =-- )
1
8/9/2019 Repaso 2do Examen Parcial
14/15
Ans0er ?e9@estname' RBACD !3D EAM BAR,IAL
1) AD5Gective' -.1) 3etermine :eter a Relati&n Represents a Functi&n
!) +D5Gective' -.1) 3etermine :eter a Relati&n Represents a Functi&n
) AD5Gective' -.1) Find te Halue &f a Functi&n
) AD5Gective' -.1) Find te Halue &f a Functi&n
$) AD5Gective' -.1) Find te 3&main &f a Functi&n
/) ,D5Gective' -.1) Find te 3&main &f a Functi&n
%) +D5Gective' -.!) D5tain Inf&rmati&n fr&m &r a5&ut te rap &f a Functi&n
6) ,
D5Gective' -.) Use a rap t& 3etermine :ere a Functi&n Is Increasing 3ecreasing &r ,&nstant#) ,
D5Gective' -.) Use a rap t& L&cate L&cal Ma2ima and L&cal Minima
1") 3D5Gective' -.) Use a rap t& L&cate L&cal Ma2ima and L&cal Minima
11) +D5Gective' -.) Find te Average Rate &f ,ange &f a Functi&n
1!) ,D5Gective' -.) Find te Average Rate &f ,ange &f a Functi&n
1) 3D5Gective' -1./) rap Functi&ns Inv&lving a CeJuence &f @ransf&rmati&ns
1) ,D5Gective' -1./) rap Functi&ns Inv&lving a CeJuence &f @ransf&rmati&ns
1$) +D5Gective' -1./) rap Functi&ns Inv&lving a CeJuence &f @ransf&rmati&ns
1/) 3D5Gective' -1./) rap Functi&ns Inv&lving a CeJuence &f @ransf&rmati&ns
1%) ,D5Gective' -1.%) ,&m5ine Functi&ns Using te Alge5ra &f Functi&ns Cpecif9ing 3&mains
16) +D5Gective' -1.%) ,&m5ine Functi&ns Using te Alge5ra &f Functi&ns Cpecif9ing 3&mains
1#) 3D5Gective' -1.%) ,&m5ine Functi&ns Using te Alge5ra &f Functi&ns Cpecif9ing 3&mains
!") 3D5Gective' -1.%) 3etermine 3&mains f&r ,&mp&site Functi&ns
!1) AD5Gective' -1.%) 3etermine 3&mains f&r ,&mp&site Functi&ns
!!) ,D5Gective' -1.%) 3etermine 3&mains f&r ,&mp&site Functi&ns
!) +D5Gective' -1.6) Find te Inverse &f a Functi&n
1
8/9/2019 Repaso 2do Examen Parcial
15/15
Ans0er ?e9@estname' RBACD !3D EAM BAR,IAL
!) +D5Gective' -1.6) Find te Inverse &f a Functi&n
!$) AD5Gective' -1.6) Find te Inverse &f a Functi&n
!/) AD5Gective' -1.6) Use te rap &f a Dne-t&-Dne Functi&n t& rap Its Inverse Functi&n
!%) 3D5Gective' -1.6) Use te rap &f a Dne-t&-Dne Functi&n t& rap Its Inverse Functi&n
!6) 3D5Gective' -1.6) Find te Inverse &f a Functi&n and rap +&t Functi&ns &n te Came A2es
!#) AD5Gective' -1.6) Find te Inverse &f a Functi&n and rap +&t Functi&ns &n te Came A2es
") +D5Gective' -1.6) Find te Inverse &f a Functi&n and rap +&t Functi&ns &n te Came A2es
1$