8/9/2019 Repaso 2do Examen Parcial

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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

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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)

!

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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)

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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)

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%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)

$

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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$)

/

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%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

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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) --) +)

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8/9/2019 Repaso 2do Examen Parcial

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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"

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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

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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!

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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

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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

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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

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