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Prévia do material em texto

Ministério
da Educação
LEIA COM ATENÇÃO AS INSTRUÇÕES ABAIXO.
ϭ͘��sĞƌŝĮƋƵĞ� ƐĞ͕� ĂůĠŵ� ĚĞƐƚĞ� ĐĂĚĞƌŶŽ͕� ǀŽĐġ� ƌĞĐĞďĞƵ� Ž� �ĂĚĞƌŶŽ� ĚĞ� ZĞƐƉŽƐƚĂƐ͕� ĚĞƐƟŶĂĚŽ� ă� ƚƌĂŶƐĐƌŝĕĆŽ� ĚĂƐ�
ƌĞƐƉŽƐƚĂƐ� ĚĂƐ� ƋƵĞƐƚƁĞƐ� ĚĞ� ŵƷůƟƉůĂ� ĞƐĐŽůŚĂ� ;ŽďũĞƟǀĂƐͿ͕� ĚĂƐ� ƋƵĞƐƚƁĞƐ� ĚŝƐĐƵƌƐŝǀĂƐ� Ğ� ĚŽ� ƋƵĞƐƟŽŶĄƌŝŽ� ĚĞ�
ƉĞƌĐĞƉĕĆŽ�ĚĂ�ƉƌŽǀĂ͘
Ϯ͘��ŽŶĮƌĂ�ƐĞ�ĞƐƚĞ�ĐĂĚĞƌŶŽ�ĐŽŶƚĠŵ�ĂƐ�ƋƵĞƐƚƁĞƐ�ĚŝƐĐƵƌƐŝǀĂƐ�Ğ�ĚĞ�ŵƷůƟƉůĂ�ĞƐĐŽůŚĂ�;ŽďũĞƟǀĂƐͿ͕�ĚĞ�ĨŽƌŵĂĕĆŽ�
ŐĞƌĂů�Ğ�ĚŽ�ĐŽŵƉŽŶĞŶƚĞ�ĞƐƉĞĐşĮĐŽ�ĚĂ�ĄƌĞĂ͕�Ğ�ĂƐ�ƌĞůĂƟǀĂƐ�ă�ƐƵĂ�ƉĞƌĐĞƉĕĆŽ�ĚĂ�ƉƌŽǀĂ͘��Ɛ�ƋƵĞƐƚƁĞƐ�ĞƐƚĆŽ�
ĂƐƐŝŵ�ĚŝƐƚƌŝďƵşĚĂƐ͗
Partes
Número das 
questões
Peso das 
questões no 
componente
Peso dos 
componentes no 
cálculo da nota
&ŽƌŵĂĕĆŽ�'ĞƌĂůͬ�ŝƐĐƵƌƐŝǀĂƐ �ϭ�Ğ��Ϯ 40%
25%
&ŽƌŵĂĕĆŽ�'ĞƌĂůͬKďũĞƟǀĂƐ ϭ�Ă�ϴ 60%
�ŽŵƉŽŶĞŶƚĞ��ƐƉĞĐşĮĐŽͬ�ŝƐĐƵƌƐŝǀĂƐ �ϯ�Ă��ϱ 15%
75%
�ŽŵƉŽŶĞŶƚĞ��ƐƉĞĐşĮĐŽͬKďũĞƟǀĂƐ ϵ�Ă�ϯϱ ϴϱй
YƵĞƐƟŽŶĄƌŝŽ�ĚĞ�WĞƌĐĞƉĕĆŽ�ĚĂ�WƌŽǀĂ� ϭ�Ă�ϵ - -
ϯ͘��sĞƌŝĮƋƵĞ�ƐĞ�Ă�ƉƌŽǀĂ�ĞƐƚĄ�ĐŽŵƉůĞƚĂ�Ğ�ƐĞ�Ž�ƐĞƵ�ŶŽŵĞ�ĞƐƚĄ�ĐŽƌƌĞƚŽ�ŶŽ��ĂĚĞƌŶŽ�ĚĞ�ZĞƐƉŽƐƚĂƐ͘��ĂƐŽ�ĐŽŶƚƌĄƌŝŽ͕�
ĂǀŝƐĞ�ŝŵĞĚŝĂƚĂŵĞŶƚĞ�Ƶŵ�ĚŽƐ�ƌĞƐƉŽŶƐĄǀĞŝƐ�ƉĞůĂ�ĂƉůŝĐĂĕĆŽ�ĚĂ�ƉƌŽǀĂ͘�sŽĐġ�ĚĞǀĞ�ĂƐƐŝŶĂƌ�Ž��ĂĚĞƌŶŽ�ĚĞ�ZĞƐƉŽƐƚĂƐ�
ŶŽ�ĞƐƉĂĕŽ�ƉƌſƉƌŝŽ͕�ĐŽŵ�ĐĂŶĞƚĂ�ĞƐĨĞƌŽŐƌĄĮĐĂ�ĚĞ�ƟŶƚĂ�ƉƌĞƚĂ͘
ϰ͘��KďƐĞƌǀĞ�ĂƐ�ŝŶƐƚƌƵĕƁĞƐ�ƐŽďƌĞ�Ă�ŵĂƌĐĂĕĆŽ�ĚĂƐ�ƌĞƐƉŽƐƚĂƐ�ĚĂƐ�ƋƵĞƐƚƁĞƐ�ĚĞ�ŵƷůƟƉůĂ�ĞƐĐŽůŚĂ�;ĂƉĞŶĂƐ�ƵŵĂ�ƌĞƐƉŽƐƚĂ�
ƉŽƌ�ƋƵĞƐƚĆŽͿ͕�ĞdžƉƌĞƐƐĂƐ�ŶŽ��ĂĚĞƌŶŽ�ĚĞ�ZĞƐƉŽƐƚĂƐ͘
ϱ͘��hƐĞ�ĐĂŶĞƚĂ�ĞƐĨĞƌŽŐƌĄĮĐĂ�ĚĞ�ƟŶƚĂ�ƉƌĞƚĂ͕�ƚĂŶƚŽ�ƉĂƌĂ�ŵĂƌĐĂƌ�ĂƐ�ƌĞƐƉŽƐƚĂƐ�ĚĂƐ�ƋƵĞƐƚƁĞƐ�ŽďũĞƟǀĂƐ�ƋƵĂŶƚŽ�ƉĂƌĂ�
ĞƐĐƌĞǀĞƌ�ĂƐ�ƌĞƐƉŽƐƚĂƐ�ĚĂƐ�ƋƵĞƐƚƁĞƐ�ĚŝƐĐƵƌƐŝǀĂƐ͘
ϲ͘�ZĞƐƉŽŶĚĂ� ĐĂĚĂ� ƋƵĞƐƚĆŽ� ĚŝƐĐƵƌƐŝǀĂ� Ğŵ͕� ŶŽ�ŵĄdžŝŵŽ͕� ϭϱ� ůŝŶŚĂƐ͘� YƵĂůƋƵĞƌ� ƚĞdžƚŽ� ƋƵĞ� ƵůƚƌĂƉĂƐƐĂƌ� Ž� ĞƐƉĂĕŽ�
ĚĞƐƟŶĂĚŽ�ă�ƌĞƐƉŽƐƚĂ�ƐĞƌĄ�ĚĞƐĐŽŶƐŝĚĞƌĂĚŽ͘
ϳ͘�EĆŽ�ƵƐĞ�ĐĂůĐƵůĂĚŽƌĂ͖�ŶĆŽ�ƐĞ�ĐŽŵƵŶŝƋƵĞ�ĐŽŵ�ŽƐ�ĚĞŵĂŝƐ�ĞƐƚƵĚĂŶƚĞƐ�ŶĞŵ�ƚƌŽƋƵĞ�ŵĂƚĞƌŝĂů�ĐŽŵ�ĞůĞƐ͖�
ŶĆŽ�ĐŽŶƐƵůƚĞ�ŵĂƚĞƌŝĂů�ďŝďůŝŽŐƌĄĮĐŽ͕�ĐĂĚĞƌŶŽƐ�ŽƵ�ĂŶŽƚĂĕƁĞƐ�ĚĞ�ƋƵĂůƋƵĞƌ�ĞƐƉĠĐŝĞ͘
ϴ͘�sŽĐġ�ƚĞƌĄ�ƋƵĂƚƌŽ�ŚŽƌĂƐ�ƉĂƌĂ�ƌĞƐƉŽŶĚĞƌ�ăƐ�ƋƵĞƐƚƁĞƐ�ĚĞ�ŵƷůƟƉůĂ�ĞƐĐŽůŚĂ�Ğ�ĚŝƐĐƵƌƐŝǀĂƐ�Ğ�ĂŽ�ƋƵĞƐƟŽŶĄƌŝŽ�ĚĞ�
ƉĞƌĐĞƉĕĆŽ�ĚĂ�ƉƌŽǀĂ͘
ϵ͘�YƵĂŶĚŽ�ƚĞƌŵŝŶĂƌ͕ �ĞŶƚƌĞŐƵĞ�ĂŽ��ƉůŝĐĂĚŽƌ�ŽƵ�&ŝƐĐĂů�Ž�ƐĞƵ��ĂĚĞƌŶŽ�ĚĞ�ZĞƐƉŽƐƚĂƐ͘
10. Atenção!�sŽĐġ�ĚĞǀĞƌĄ�ƉĞƌŵĂŶĞĐĞƌ͕ �ŶŽ�ŵşŶŝŵŽ͕�ƉŽƌ�ƵŵĂ�ŚŽƌĂ͕�ŶĂ�ƐĂůĂ�ĚĞ�ĂƉůŝĐĂĕĆŽ�ĚĂƐ�ƉƌŽǀĂƐ�Ğ�Ɛſ�ƉŽĚĞƌĄ�
ůĞǀĂƌ�ĞƐƚĞ��ĂĚĞƌŶŽ�ĚĞ�WƌŽǀĂ�ĂƉſƐ�ĚĞĐŽƌƌŝĚĂƐ�ƚƌġƐ�ŚŽƌĂƐ�ĚŽ�ŝŶşĐŝŽ�ĚŽ��džĂŵĞ͘
33Novembro/2014
Matemática
BACHARELADO
*R3320141*
2
D�d�D�d/��
FORMAÇÃO GERAL
QUESTÃO DISCURSIVA 1
KƐ�ĚĞƐĂĨŝŽƐ�ĚĂ�ŵŽďŝůŝĚĂĚĞ�ƵƌďĂŶĂ�ĂƐƐŽĐŝĂŵͲƐĞ�ă�ŶĞĐĞƐƐŝĚĂĚĞ�ĚĞ�ĚĞƐĞŶǀŽůǀŝŵĞŶƚŽ�ƵƌďĂŶŽ�ƐƵƐƚĞŶƚĄǀĞů͘�
��KEh�ĚĞĮŶĞ�ĞƐƐĞ�ĚĞƐĞŶǀŽůǀŝŵĞŶƚŽ�ĐŽŵŽ�ĂƋƵĞůĞ�ƋƵĞ�ĂƐƐĞŐƵƌĂ�ƋƵĂůŝĚĂĚĞ�ĚĞ�ǀŝĚĂ͕�ŝŶĐůƵşĚŽƐ�ŽƐ�ĐŽŵƉŽŶĞŶƚĞƐ�
ĞĐŽůſŐŝĐŽƐ͕�ĐƵůƚƵƌĂŝƐ͕�ƉŽůşƟĐŽƐ͕�ŝŶƐƟƚƵĐŝŽŶĂŝƐ͕�ƐŽĐŝĂŝƐ�Ğ�ĞĐŽŶƀŵŝĐŽƐ�ƋƵĞ�ŶĆŽ�ĐŽŵƉƌŽŵĞƚĂŵ�Ă�ƋƵĂůŝĚĂĚĞ�ĚĞ�
ǀŝĚĂ�ĚĂƐ�ĨƵƚƵƌĂƐ�ŐĞƌĂĕƁĞƐ͘
K�ĞƐƉĂĕŽ�ƵƌďĂŶŽ�ďƌĂƐŝůĞŝƌŽ�Ġ�ŵĂƌĐĂĚŽ�ƉŽƌ�ŝŶƷŵĞƌŽƐ�ƉƌŽďůĞŵĂƐ�ĐŽƟĚŝĂŶŽƐ�Ğ�ƉŽƌ�ǀĄƌŝĂƐ�ĐŽŶƚƌĂĚŝĕƁĞƐ͘�hŵĂ�ĚĂƐ�
ŐƌĂŶĚĞƐ�ƋƵĞƐƚƁĞƐ�Ğŵ�ĚĞďĂƚĞ�Ěŝnj�ƌĞƐƉĞŝƚŽ�ă�ŵŽďŝůŝĚĂĚĞ�ƵƌďĂŶĂ͕�ƵŵĂ�ǀĞnj�ƋƵĞ�Ž�ŵŽŵĞŶƚŽ�Ġ�ĚĞ�ŵŽƚŽƌŝnjĂĕĆŽ�
ĚŽƐ�ĚĞƐůŽĐĂŵĞŶƚŽƐ�ĚĂ�ƉŽƉƵůĂĕĆŽ͕�ƉŽƌ�ŵĞŝŽ�ĚĞ�ƚƌĂŶƐƉŽƌƚĞ�ĐŽůĞƟǀŽ�Ğ�ŝŶĚŝǀŝĚƵĂů͘
�ŽŶƐŝĚĞƌĞ�ŽƐ�ĚĂĚŽƐ�ĚŽ�ƐĞŐƵŝŶƚĞ�ƋƵĂĚƌŽ͘
Mobilidade urbana em cidade com mais de 500 mil habitantes
Modalidade Tipologia Porcentagem (%)
EĆŽ�ŵŽƚŽƌŝnjĂĚŽ ��ƉĠ ϭϱ͕ϵ�ŝĐŝĐůĞƚĂ� Ϯ͕ϳ
DŽƚŽƌŝnjĂĚŽ�ĐŽůĞƟǀŽ
NŶŝďƵƐ�ŵƵŶŝĐŝƉĂů ϮϮ͕Ϯ
NŶŝďƵƐ�ŵĞƚƌŽƉŽůŝƚĂŶŽ ϰ͕ϱ
DĞƚƌŽĨĞƌƌŽǀŝĄƌŝŽ Ϯϱ͕ϭ
DŽƚŽƌŝnjĂĚŽ�ŝŶĚŝǀŝĚƵĂů �ƵƚŽŵſǀĞů Ϯϳ͕ϱDŽƚŽĐŝĐůĞƚĂ Ϯ͕ϭ
dĞŶĚŽ� Ğŵ� ǀŝƐƚĂ� Ž� ƚĞdžƚŽ� Ğ� Ž� ƋƵĂĚƌŽ� ĚĞ� ŵŽďŝůŝĚĂĚĞ� ƵƌďĂŶĂ� ĂƉƌĞƐĞŶƚĂĚŽƐ͕� ƌĞĚŝũĂ� Ƶŵ� ƚĞdžƚŽ� ĚŝƐƐĞƌƚĂƟǀŽ͕�
ĐŽŶƚĞŵƉůĂŶĚŽ�ŽƐ�ƐĞŐƵŝŶƚĞƐ�ĂƐƉĞĐƚŽƐ͗
ĂͿ� ĐŽŶƐĞƋƵġŶĐŝĂƐ͕�ƉĂƌĂ�Ž�ĚĞƐĞŶǀŽůǀŝŵĞŶƚŽ�ƐƵƐƚĞŶƚĄǀĞů͕�ĚŽ�ƵƐŽ�ŵĂŝƐ�ĨƌĞƋƵĞŶƚĞ�ĚŽ�ƚƌĂŶƐƉŽƌƚĞ�ŵŽƚŽƌŝnjĂĚŽ͖�
;ǀĂůŽƌ͗�ϱ͕Ϭ�ƉŽŶƚŽƐͿ
ďͿ� ĚƵĂƐ�ĂĕƁĞƐ�ĚĞ�ŝŶƚĞƌǀĞŶĕĆŽ�ƋƵĞ�ĐŽŶƚƌŝďƵĂŵ�ƉĂƌĂ�Ă�ĐŽŶƐŽůŝĚĂĕĆŽ�ĚĞ�ƉŽůşƟĐĂ�ƉƷďůŝĐĂ�ĚĞ�ŝŶĐƌĞŵĞŶƚŽ�ĂŽ�ƵƐŽ�
ĚĞ�ďŝĐŝĐůĞƚĂ�ŶĂ�ĐŝĚĂĚĞ�ŵĞŶĐŝŽŶĂĚĂ͕�ĂƐƐĞŐƵƌĂŶĚŽͲƐĞ�Ž�ĚĞƐĞŶǀŽůǀŝŵĞŶƚŽ�ƐƵƐƚĞŶƚĄǀĞů͘�;ǀĂůŽƌ͗�ϱ͕Ϭ�ƉŽŶƚŽƐͿ
RASCUNHO
1
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4
5
6
7
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9
10
11
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ϭϯ
14
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*R3320142*
ϯ
D�d�D�d/��
QUESTÃO DISCURSIVA 2
dƌġƐ�ũŽǀĞŶƐ�ĚĞ�ϭϵ�ĂŶŽƐ�ĚĞ�ŝĚĂĚĞ͕�ŵŽƌĂĚŽƌĞƐ�ĚĞ�ƌƵĂ͕�ĨŽƌĂŵ�ƉƌĞƐŽƐ�Ğŵ�ŇĂŐƌĂŶƚĞ͕�ŶĞƐƚĂ�ƋƵĂƌƚĂͲĨĞŝƌĂ͕�ƉŽƌ�ƚĞƌĞŵ�
ĂƚĞĂĚŽ�ĨŽŐŽ�Ğŵ�Ƶŵ�ũŽǀĞŵ�ĚĞ�ϭϳ�ĂŶŽƐ͕�ŐƵĂƌĚĂĚŽƌ�ĚĞ�ĐĂƌƌŽƐ͘�K�ŵŽƟǀŽ͕�ƐĞŐƵŶĚŽ�Ă�ϭϰ͘a��W͕ �ĨŽŝ�ƵŵĂ�͚͛ďƌŝŐĂ�ƉŽƌ�
ƉŽŶƚŽ͛͛͘ �hŵ�ŵŽƚŽƌŝƐƚĂ�ĚĞƵ�͚͛Ƶŵ�ƚƌŽĐĂĚŽ͛͛�ĂŽ�ŵĞŶŽƌ͕ �Ž�ƋƵĞ�ŝƌƌŝƚŽƵ�ŽƐ�ƚƌġƐ�ŵŽƌĂĚŽƌĞƐ�ĚĞ�ƌƵĂ͕�ƋƵĞ�ƚĂŵďĠŵ�
ŐƵĂƌĚĂǀĂŵ�ĐĂƌƌŽƐ�ŶŽ� ůŽĐĂů͘�K�ŵĞŶŽƌ�ĨŽŝ� ůĞǀĂĚŽ�ĂŽ�,ŽƐƉŝƚĂů�ĚĂƐ��ůşŶŝĐĂƐ� ;,�Ϳ�ƉŽƌ�WDƐ�ƋƵĞ�ƉĂƐƐĂǀĂŵ�ƉĞůŽ�
ůŽĐĂů͘�^ĞŐƵŶĚŽ�Ž�,�͕�ĞůĞ�ƚĞǀĞ�ƋƵĞŝŵĂĚƵƌĂƐ�ůĞǀĞƐ�ŶŽ�ŽŵďƌŽ�ĞƐƋƵĞƌĚŽ͕�ĨŽŝ�ŵĞĚŝĐĂĚŽ�Ğ͕�Ğŵ�ƐĞŐƵŝĚĂ͕�ůŝďĞƌĂĚŽ͘�
KƐ�ŝŶĚŝĐŝĂĚŽƐ�ƉŽĚĞŵ�ƉĞŐĂƌ�ĚĞ�ϭϮ�Ă�ϯϬ�ĂŶŽƐ�ĚĞ�ƉƌŝƐĆŽ͕�ƐĞ�ĮĐĂƌ�ĐŽŵƉƌŽǀĂĚŽ�ƋƵĞ�Ă�ŝŶƚĞŶĕĆŽ�ĞƌĂ�ŵĂƚĂƌ�Ž�ŵĞŶŽƌ͘ �
�ĂƐŽ�ĐŽŶƚƌĄƌŝŽ͕�ĐŽŶĨŽƌŵĞ�Ă�ϭϰ͘a��W͕ �ŽƐ�ƚƌġƐ�ƉŽĚĞƌĆŽ�ƉĞŐĂƌ�ĚĞ�Ƶŵ�Ă�ƚƌġƐ�ĂŶŽƐ�ĚĞ�ĐĂĚĞŝĂ͘
�ŝƐƉŽŶşǀĞů�Ğŵ͗фŚƩƉ͗ͬͬǁǁǁϭ͘ĨŽůŚĂ͘ƵŽů͘ĐŽŵ͘ďƌх͘��ĐĞƐƐŽ�Ğŵ͗�Ϯϴ�ũƵů͘�ϮϬϭϯ�;ĂĚĂƉƚĂĚŽͿ͘
��ƉĂƌƟƌ�ĚĂ�ƐŝƚƵĂĕĆŽ�ŶĂƌƌĂĚĂ͕�ĞůĂďŽƌĞ�Ƶŵ�ƚĞdžƚŽ�ĚŝƐƐĞƌƚĂƟǀŽ�ƐŽďƌĞ�ǀŝŽůġŶĐŝĂ�ƵƌďĂŶĂ͕�ĂƉƌĞƐĞŶƚĂŶĚŽ͗
ĂͿ��ĂŶĄůŝƐĞ�ĚĞ�ĚƵĂƐ�ĐĂƵƐĂƐ�ĚŽ�ƟƉŽ�ĚĞ�ǀŝŽůġŶĐŝĂ�ĚĞƐĐƌŝƚĂ�ŶŽ�ƚĞdžƚŽ͖�;ǀĂůŽƌ͗�ϳ͕Ϭ�ƉŽŶƚŽƐͿ
ďͿ��ĚŽŝƐ�ĨĂƚŽƌĞƐ�ƋƵĞ�ĐŽŶƚƌŝďƵŝƌŝĂŵ�ƉĂƌĂ�ƐĞ�ĞǀŝƚĂƌ�Ž�ĨĂƚŽ�ĚĞƐĐƌŝƚŽ�ŶĂ�ŶŽơĐŝĂ͘�;ǀĂůŽƌ͗�ϯ͕Ϭ�ƉŽŶƚŽƐͿ
RASCUNHO
1
2
ϯ
4
5
6
7
ϴ
9
10
11
12
ϭϯ
14
15
*R3320143*
4
D�d�D�d/��
QUESTÃO 01
K�ƚƌĞĐŚŽ�ĚĂ�ŵƷƐŝĐĂ�͞ EŽƐ��ĂŝůĞƐ�ĚĂ�sŝĚĂ͕͟ �ĚĞ�DŝůƚŽŶ�EĂƐĐŝŵĞŶƚŽ͕�͞ ƚŽĚŽ�ĂƌƟƐƚĂ�ƚĞŵ�ĚĞ�ŝƌ�ĂŽŶĚĞ�Ž�ƉŽǀŽ�ĞƐƚĄ͕͟ �Ġ�ĂŶƟŐŽ͕�
Ğ�Ă�ŵƷƐŝĐĂ͕�ĚĞ�ƚĆŽ�ƚŽĐĂĚĂ͕�ĂĐĂďŽƵ�ƉŽƌ�ƐĞ�ƚŽƌŶĂƌ�Ƶŵ�ĞƐƚĞƌĞſƟƉŽ�ĚĞ�ƚŽĐĂĚŽƌĞƐ�ĚĞ�ǀŝŽůƁĞƐ�Ğ�ĚĞ�ƌŽĚĂƐ�ĚĞ�ĂŵŝŐŽƐ�Ğŵ�
sŝƐĐŽŶĚĞ�ĚĞ�DĂƵĄ͕�ŶŽƐ�ĂŶŽƐ�ϭϵϳϬ͘��ŵ�ƚĞŵƉŽƐ�ĚŝŐŝƚĂŝƐ͕�ƉŽƌĠŵ͕�ĞůĂ�ĮĐŽƵ�ŵĂŝƐ�ĂƚƵĂů�ĚŽ�ƋƵĞ�ŶƵŶĐĂ͘���ĨĄĐŝů�ĞŶƚĞŶĚĞƌ�
Ž�ƉŽƌƋƵġ͗�ĂŶƟŐĂŵĞŶƚĞ͕�ƋƵĂŶĚŽ�Ă�ŝŶĨŽƌŵĂĕĆŽ�ƐĞ�ĐŽŶĐĞŶƚƌĂǀĂ�Ğŵ�ĐĞŶƚƌŽƐ�ĚĞ�ĞdžƉŽƐŝĕĆŽ͕�ǀĞşĐƵůŽƐ�ĚĞ�ĐŽŵƵŶŝĐĂĕĆŽ͕�
ĞĚŝƚŽƌĂƐ͕�ŵƵƐĞƵƐ�Ğ�ŐƌĂǀĂĚŽƌĂƐ͕�ĞƌĂ�ƉƌĞĐŝƐŽ�ƉĂƐƐĂƌ�ƉŽƌ�ƵŵĂ�ƐĠƌŝĞ�ĚĞ�ĐƵƌĂĚŽƌĞƐ͕�ƉĂƌĂ�ŐĂƌĂŶƟƌ�Ă�ƉƵďůŝĐĂĕĆŽ�ĚĞ�Ƶŵ�
ĂƌƟŐŽ�ŽƵ�ůŝǀƌŽ͕�Ă�ŐƌĂǀĂĕĆŽ�ĚĞ�Ƶŵ�ĚŝƐĐŽ�ŽƵ�Ă�ƉƌŽĚƵĕĆŽ�ĚĞ�ƵŵĂ�ĞdžƉŽƐŝĕĆŽ͘�K�ŵĞƐŵŽ�ĨƵŶŝů͕�ƋƵĞ�ƉŽĚĞƌŝĂ�ƐĞƌ�ŝŶũƵƐƚŽ�Ğ�
ĚĞŝdžĂƌ�ŐƌĂŶĚĞƐ�ƚĂůĞŶƚŽƐ�ĚĞ�ĨŽƌĂ͕�ƐŝŵƉůĞƐŵĞŶƚĞ�ƉŽƌƋƵĞ�ŶĆŽ�ƟŶŚĂŵ�ĂĐĞƐƐŽ�ăƐ�ĨĞƌƌĂŵĞŶƚĂƐ͕�ăƐ�ƉĞƐƐŽĂƐ�ŽƵ�ăƐ�ĨŽŶƚĞƐ�
ĚĞ�ŝŶĨŽƌŵĂĕĆŽ͕�ƚĂŵďĠŵ�ƐĞƌǀŝĂ�ĐŽŵŽ�ĮůƚƌŽ�ĚĞ�ƋƵĂůŝĚĂĚĞ͘�dŽĐĂƌ�ǀŝŽůĆŽ�ŽƵ�ĞŶĐĞŶĂƌ�ƵŵĂ�ƉĞĕĂ�ĚĞ�ƚĞĂƚƌŽ�Ğŵ�Ƶŵ�ŐƌĂŶĚĞ�
ĂƵĚŝƚſƌŝŽ�ĐŽƐƚƵŵĂǀĂ�ƚĞƌ�Ƶŵ�ƉĞƐŽ�ŵƵŝƚŽ�ŵĂŝŽƌ�ĚŽ�ƋƵĞ� ĨĂnjġͲůŽ�Ğŵ�Ƶŵ�ďĂƌ͕ �Ƶŵ�ĐĞŶƚƌŽ�ĐƵůƚƵƌĂů�ŽƵ�ƵŵĂ�ĐĂůĕĂĚĂ͘�
EĂƐ�ƌĂƌĂƐ�ŽĐĂƐŝƁĞƐ�Ğŵ�ƋƵĞ�ĞƐƐĞ�ǀĂůŽƌ�ƐĞ�ŝŶǀĞƌƟĂ͕�ĞƌĂ�ũƵƐƚĂŵĞŶƚĞ�ƉŽƌƋƵĞ͕�ƉĂƌĂ�ƵƐŽ�ĚŽ�ĞƐƉĂĕŽ�͞ĂůƚĞƌŶĂƟǀŽ͕͟ �ŚĂǀŝĂ�
ŵĞĐĂŶŝƐŵŽƐ�ĚĞ�ƐĞůĞĕĆŽ�ƚĆŽ�ŽƵ�ŵĂŝƐ�ƌşŐŝĚŽƐ�ƋƵĞ�ŽƐ�ĚŽ�ĞƐƉĂĕŽ�ŽĮĐŝĂů͘
Z��&�,Z�Z͕�>͘�dŽĚŽ�ĂƌƟƐƚĂ�ƚĞŵ�ĚĞ�ŝƌ�ĂŽŶĚĞ�Ž�ƉŽǀŽ�ĞƐƚĄ͘�ŝƐƉŽŶşǀĞů�Ğŵ͗�фŚƩƉ͗ͬͬŶŽǀŽ͘ŝƚĂƵĐƵůƚƵƌĂů͘ŽƌŐ͘ďƌх͘��ĐĞƐƐŽ�Ğŵ͗�Ϯϵ�ũƵů͘�ϮϬϭϰ�;ĂĚĂƉƚĂĚŽͿ͘
��ƉĂƌƟƌ�ĚŽ�ƚĞdžƚŽ�ĂĐŝŵĂ͕�ĂǀĂůŝĞ�ĂƐ�ĂƐƐĞƌĕƁĞƐ�Ă�ƐĞŐƵŝƌ�Ğ�Ă�ƌĞůĂĕĆŽ�ƉƌŽƉŽƐƚĂ�ĞŶƚƌĞ�ĞůĂƐ͘
/͘� K�ƉƌŽĐĞƐƐŽ�ĚĞ�ĞǀŽůƵĕĆŽ�ƚĞĐŶŽůſŐŝĐĂ�ĚĂ�ĂƚƵĂůŝĚĂĚĞ�ĚĞŵŽĐƌĂƟnjĂ�Ă�ƉƌŽĚƵĕĆŽ�Ğ�Ă�ĚŝǀƵůŐĂĕĆŽ�ĚĞ�ŽďƌĂƐ�
ĂƌơƐƟĐĂƐ͕�ƌĞĚƵnjŝŶĚŽ�Ă�ŝŵƉŽƌƚąŶĐŝĂ�ƋƵĞ�ŽƐ�ĐĞŶƚƌŽƐ�ĚĞ�ĞdžƉŽƐŝĕĆŽ�ƟŶŚĂŵ�ŶŽƐ�ĂŶŽƐ�ϭϵϳϬ͘
PORQUE
//͘� �Ɛ�ŶŽǀĂƐ�ƚĞĐŶŽůŽŐŝĂƐ�ƉŽƐƐŝďŝůŝƚĂŵ�ƋƵĞ�ĂƌƟƐƚĂƐ�ƐĞũĂŵ�ŝŶĚĞƉĞŶĚĞŶƚĞƐ͕�ŵŽŶƚĞŵ�ƐĞƵƐ�ƉƌſƉƌŝŽƐ�ĂŵďŝĞŶƚĞƐ�
ĚĞ�ƉƌŽĚƵĕĆŽ�Ğ�ĚŝƐƉŽŶŝďŝůŝnjĞŵ�ƐĞƵƐ�ƚƌĂďĂůŚŽƐ͕�ĚĞ�ĨŽƌŵĂ�ƐŝŵƉůĞƐ͕�ƉĂƌĂ�Ƶŵ�ŐƌĂŶĚĞ�ŶƷŵĞƌŽ�ĚĞ�ƉĞƐƐŽĂƐ͘
��ƌĞƐƉĞŝƚŽ�ĚĞƐƐĂƐ�ĂƐƐĞƌĕƁĞƐ͕�ĂƐƐŝŶĂůĞ�Ă�ŽƉĕĆŽ�ĐŽƌƌĞƚĂ͘
A �Ɛ�ĂƐƐĞƌĕƁĞƐ�/�Ğ�//�ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ�ǀĞƌĚĂĚĞŝƌĂƐ͕�Ğ�Ă�//�Ġ�ƵŵĂ�ũƵƐƟĮĐĂƟǀĂ�ĐŽƌƌĞƚĂ�ĚĂ�/͘
B �Ɛ�ĂƐƐĞƌĕƁĞƐ�/�Ğ�//�ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ�ǀĞƌĚĂĚĞŝƌĂƐ͕�ŵĂƐ�Ă�//�ŶĆŽ�Ġ�ƵŵĂ�ũƵƐƟĮĐĂƟǀĂ�ĐŽƌƌĞƚĂ�ĚĂ�/͘
C ��ĂƐƐĞƌĕĆŽ�/�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ǀĞƌĚĂĚĞŝƌĂ͕�Ğ�Ă�//�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ĨĂůƐĂ͘
D ��ĂƐƐĞƌĕĆŽ�/�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ĨĂůƐĂ͕�Ğ�Ă�//�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ǀĞƌĚĂĚĞŝƌĂ͘
E �Ɛ�ĂƐƐĞƌĕƁĞƐ�/�Ğ�//�ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ�ĨĂůƐĂƐ͘
QUESTÃO 02
�Žŵ�Ă�ŐůŽďĂůŝnjĂĕĆŽ�ĚĂ�ĞĐŽŶŽŵŝĂ�ƐŽĐŝĂů�ƉŽƌ�ŵĞŝŽ�ĚĂƐ�ŽƌŐĂŶŝnjĂĕƁĞƐ�ŶĆŽ�ŐŽǀĞƌŶĂŵĞŶƚĂŝƐ͕�ƐƵƌŐŝƵ�ƵŵĂ�ĚŝƐĐƵƐƐĆŽ�
ĚŽ�ĐŽŶĐĞŝƚŽ�ĚĞ�ĞŵƉƌĞƐĂ͕�ĚĞ�ƐƵĂ�ĨŽƌŵĂ�ĚĞ�ĐŽŶĐĞƉĕĆŽ�ũƵŶƚŽ�ăƐ�ŽƌŐĂŶŝnjĂĕƁĞƐ�ďƌĂƐŝůĞŝƌĂƐ�Ğ�ĚĞ�ƐƵĂƐ�ƉƌĄƟĐĂƐ͘�
�ĂĚĂ�ǀĞnj�ŵĂŝƐ͕�Ġ�ŶĞĐĞƐƐĄƌŝŽ�ĐŽŵďŝŶĂƌ�ĂƐ�ƉŽůşƟĐĂƐ�ƉƷďůŝĐĂƐ�ƋƵĞ�ƉƌŝŽƌŝnjĂŵ�ŵŽĚĞƌŶŝĚĂĚĞ�Ğ�ĐŽŵƉĞƟǀŝĚĂĚĞ�
ĐŽŵ�Ž�ĞƐĨŽƌĕŽ�ĚĞ�ŝŶĐŽƌƉŽƌĂĕĆŽ�ĚŽƐ�ƐĞƚŽƌĞƐ�ĂƚƌĂƐĂĚŽƐ͕�ŵĂŝƐ�ŝŶƚĞŶƐŝǀŽƐ�ĚĞ�ŵĆŽ�ĚĞ�ŽďƌĂ͘�
�ŝƐƉŽŶşǀĞů�Ğŵ͗�фŚƩƉ͗ͬͬƵŶƉĂŶϭ͘ƵŶ͘ŽƌŐх͘��ĐĞƐƐŽ�Ğŵ͗�ϰ�ĂŐŽ͘�ϮϬϭϰ�;ĂĚĂƉƚĂĚŽͿ͘
��ƌĞƐƉĞŝƚŽ�ĚĞƐƐĂ�ƚĞŵĄƟĐĂ͕�ĂǀĂůŝĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�Ă�ƐĞŐƵŝƌ͘
/͘� K� ƚĞƌĐĞŝƌŽ� ƐĞƚŽƌ�Ġ� ƵŵĂ�ŵŝƐƚƵƌĂ� ĚŽƐ� ĚŽŝƐ� ƐĞƚŽƌĞƐ� ĞĐŽŶƀŵŝĐŽƐ� ĐůĄƐƐŝĐŽƐ� ĚĂ� ƐŽĐŝĞĚĂĚĞ͗� Ž� ƉƷďůŝĐŽ͕�
ƌĞƉƌĞƐĞŶƚĂĚŽ�ƉĞůŽ��ƐƚĂĚŽ͕�Ğ�Ž�ƉƌŝǀĂĚŽ͕�ƌĞƉƌĞƐĞŶƚĂĚŽ�ƉĞůŽ�ĞŵƉƌĞƐĂƌŝĂĚŽ�Ğŵ�ŐĞƌĂů͘
//͘� ��Ž�ƚĞƌĐĞŝƌŽ�ƐĞƚŽƌ�ƋƵĞ�ǀŝĂďŝůŝnjĂ�Ž�ĂĐĞƐƐŽ�ĚĂ�ƐŽĐŝĞĚĂĚĞ�ă�ĞĚƵĐĂĕĆŽ�Ğ�ĂŽ�ĚĞƐĞŶǀŽůǀŝŵĞŶƚŽ�ĚĞ�ƚĠĐŶŝĐĂƐ�
ŝŶĚƵƐƚƌŝĂŝƐ͕�ĞĐŽŶƀŵŝĐĂƐ͕�ĮŶĂŶĐĞŝƌĂƐ͕�ƉŽůşƟĐĂƐ�Ğ�ĂŵďŝĞŶƚĂŝƐ͘
///͘� ��ƌĞƐƉŽŶƐĂďŝůŝĚĂĚĞ�ƐŽĐŝĂů�ƚĞŵ�ƌĞƐƵůƚĂĚŽ�ŶĂ�ĂůƚĞƌĂĕĆŽ�ĚŽ�ƉĞƌĮů�ĐŽƌƉŽƌĂƟǀŽ�Ğ�ĞƐƚƌĂƚĠŐŝĐŽ�ĚĂƐ�ĞŵƉƌĞƐĂƐ͕�
ƋƵĞ�ƚġŵ�ƌĞĨŽƌŵƵůĂĚŽ�Ă�ĐƵůƚƵƌĂ�Ğ�Ă�ĮůŽƐŽĮĂ�ƋƵĞ�ŽƌŝĞŶƚĂŵ�ĂƐ�ĂĕƁĞƐ�ŝŶƐƟƚƵĐŝŽŶĂŝƐ͘
�ƐƚĄ�ĐŽƌƌĞƚŽ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /͕�ĂƉĞŶĂƐ͘
B //͕�ĂƉĞŶĂƐ͘
C /�Ğ�///͕�ĂƉĞŶĂƐ͘
D //�Ğ�///͕�ĂƉĞŶĂƐ͘
E /͕�//�Ğ�///͘
*R3320144*
5
D�d�D�d/��
QUESTÃO 03
WeŐĂĚĂ�ĞĐŽůſŐŝĐĂ�Ġ�Ƶŵ� ŝŶĚŝĐĂĚŽƌ�ƋƵĞ�ĞƐƟŵĂ�Ă�ĚĞŵĂŶĚĂ�ŽƵ�Ă� ĞdžŝŐġŶĐŝĂ�ŚƵŵĂŶĂ� ƐŽďƌĞ�Ž�ŵĞŝŽ� ĂŵďŝĞŶƚĞ͕�
ĐŽŶƐŝĚĞƌĂŶĚŽͲƐĞ�Ž�ŶşǀĞů�ĚĞ�ĂƟǀŝĚĂĚĞ�ƉĂƌĂ�ĂƚĞŶĚĞƌ�ĂŽ�ƉĂĚƌĆŽ�ĚĞ�ĐŽŶƐƵŵŽ�ĂƚƵĂů�;ĐŽŵ�Ă�ƚĞĐŶŽůŽŐŝĂ�ĂƚƵĂůͿ͘��͕�ĚĞ�
ĐĞƌƚĂ�ĨŽƌŵĂ͕�ƵŵĂ�ŵĂŶĞŝƌĂ�ĚĞ�ŵĞĚŝƌ�Ž�ŇƵdžŽ�ĚĞ�ĂƟǀŽƐ�ĂŵďŝĞŶƚĂŝƐ�ĚĞ�ƋƵĞ�ŶĞĐĞƐƐŝƚĂŵŽƐ�ƉĂƌĂ�ƐƵƐƚĞŶƚĂƌ�ŶŽƐƐŽ�ƉĂĚƌĆŽ�
ĚĞ�ĐŽŶƐƵŵŽ͘��ƐƐĞ�ŝŶĚŝĐĂĚŽƌ�Ġ�ŵĞĚŝĚŽ�Ğŵ�ŚĞĐƚĂƌĞ�ŐůŽďĂů͕�ŵĞĚŝĚĂ�ĚĞ�ĄƌĞĂ�ĞƋƵŝǀĂůĞŶƚĞ�Ă�ϭϬ�ϬϬϬ�ŵϸ͘�EĂ�ŵĞĚŝĚĂ�
ŚĞĐƚĂƌĞ�ŐůŽďĂů͕�ƐĆŽ�ĐŽŶƐŝĚĞƌĂĚĂƐ�ĂƉĞŶĂƐ�ĂƐ�ĄƌĞĂƐ�ƉƌŽĚƵƟǀĂƐ�ĚŽ�ƉůĂŶĞƚĂ͘���ďŝŽĐĂƉĂĐŝĚĂĚĞ�ĚŽ�ƉůĂŶĞƚĂ͕�ŝŶĚŝĐĂĚŽƌ�
ƋƵĞ�ƌĞŇĞƚĞ�Ă�ƌĞŐĞŶĞƌĂĕĆŽ�;ŶĂƚƵƌĂůͿ�ĚŽ�ŵĞŝŽ�ĂŵďŝĞŶƚĞ͕�Ġ�ŵĞĚŝĚĂ�ƚĂŵďĠŵ�Ğŵ�ŚĞĐƚĂƌĞ�ŐůŽďĂů͘�hŵĂ�ƌĂnjĆŽ�ĞŶƚƌĞ�
ƉĞŐĂĚĂ�ĞĐŽůſŐŝĐĂ�Ğ�ďŝŽĐĂƉĂĐŝĚĂĚĞ�ĚŽ�ƉůĂŶĞƚĂ�ŝŐƵĂů�Ă�ϭ�ŝŶĚŝĐĂ�ƋƵĞ�Ă�ĞdžŝŐġŶĐŝĂ�ŚƵŵĂŶĂ�ƐŽďƌĞ�ŽƐ�ƌĞĐƵƌƐŽƐ�ĚŽ�
ŵĞŝŽ�ĂŵďŝĞŶƚĞ�Ġ�ƌĞƉŽƐƚĂ�ŶĂ�ƐƵĂ�ƚŽƚĂůŝĚĂĚĞ�ƉĞůŽ�ƉůĂŶĞƚĂ͕�ĚĞǀŝĚŽ�ă�ĐĂƉĂĐŝĚĂĚĞ�ŶĂƚƵƌĂů�ĚĞ�ƌĞŐĞŶĞƌĂĕĆŽ͘�^Ğ�ĨŽƌ�
ŵĂŝŽƌ�ƋƵĞ�ϭ͕�Ă�ƌĂnjĆŽ�ŝŶĚŝĐĂ�ƋƵĞ�Ă�ĚĞŵĂŶĚĂ�ŚƵŵĂŶĂ�Ġ�ƐƵƉĞƌŝŽƌ�ă�ĐĂƉĂĐŝĚĂĚĞ�ĚŽ�ƉůĂŶĞƚĂ�ĚĞ�ƐĞ�ƌĞĐƵƉĞƌĂƌ�Ğ͕�ƐĞ�ĨŽƌ�
ŵĞŶŽƌ�ƋƵĞ�ϭ͕�ŝŶĚŝĐĂ�ƋƵĞ�Ž�ƉůĂŶĞƚĂ�ƐĞ�ƌĞĐƵƉĞƌĂ�ŵĂŝƐ�ƌĂƉŝĚĂŵĞŶƚĞ͘
Razão entre a pegada ecológica
e a biocapacidade do planeta1.60
1.50
1.40
1.30
1.20
1.10
1.00
0.90
0.80
0.70
0.60
19
61
19
65
19
70
19
75
19
80
19
85
19
90
19
95
20
00
20
05
20
08
0.74
0.85
1.00
1.08
1.16 1.14
1.25 1.27
1.30
1.45
1.52
�ŝƐƉŽŶşǀĞů�Ğŵ͗фŚƩƉ͗ͬͬĨŝŶĂŶĐĂƐĨĂĐĞŝƐ͘ǁŽƌĚƉƌĞƐƐ͘ĐŽŵх͘��ĐĞƐƐŽ�Ğŵ͗�ϭϬ�ĂŐŽ͘�ϮϬϭϰ͘
K�ĂƵŵĞŶƚŽ�ĚĂ�ƌĂnjĆŽ�ĞŶƚƌĞ�ƉĞŐĂĚĂ�ĞĐŽůſŐŝĐĂ�Ğ�ďŝŽĐĂƉĂĐŝĚĂĚĞ�ƌĞƉƌĞƐĞŶƚĂĚŽ�ŶŽ�ŐƌĄĨŝĐŽ�ĞǀŝĚĞŶĐŝĂ
A ƌĞĚƵĕĆŽ�ĚĂƐ�ĄƌĞĂƐ�ĚĞ�ƉůĂŶƚŝŽ�ĚŽ�ƉůĂŶĞƚĂ�ƉĂƌĂ�ǀĂůŽƌĞƐ�ŝŶĨĞƌŝŽƌĞƐ�Ă�ϭϬ�ϬϬϬ�ŵ2�ĚĞǀŝĚŽ�ĂŽ�ƉĂĚƌĆŽ�ĂƚƵĂů�
ĚĞ�ĐŽŶƐƵŵŽ�ĚĞ�ƉƌŽĚƵƚŽƐ�ĂŐƌşĐŽůĂƐ͘
B ĂƵŵĞŶƚŽ�ŐƌĂĚƵĂů�ĚĂ�ĐĂƉĂĐŝĚĂĚĞ�ŶĂƚƵƌĂů�ĚĞ�ƌĞŐĞŶĞƌĂĕĆŽ�ĚŽ�ƉůĂŶĞƚĂ�Ğŵ�ƌĞůĂĕĆŽ�ăƐ�ĞdžŝŐġŶĐŝĂƐ�ŚƵŵĂŶĂƐ͘
C ƌĞƉŽƐŝĕĆŽ�ĚŽƐ�ƌĞĐƵƌƐŽƐ�ŶĂƚƵƌĂŝƐ�ƉĞůŽ�ƉůĂŶĞƚĂ�Ğŵ�ƐƵĂ�ƚŽƚĂůŝĚĂĚĞ�ĨƌĞŶƚĞ�ăƐ�ĞdžŝŐġŶĐŝĂƐ�ŚƵŵĂŶĂƐ͘
D ŝŶĐĂƉĂĐŝĚĂĚĞ�ĚĞ�ƌĞŐĞŶĞƌĂĕĆŽ�ŶĂƚƵƌĂů�ĚŽ�ƉůĂŶĞƚĂ�ĂŽ�ůŽŶŐŽ�ĚŽ�ƉĞƌşŽĚŽ�ϭϵϲϭͲϮϬϬϴ͘
E ƚĞŶĚġŶĐŝĂ�Ă�ĚĞƐĞƋƵŝůşďƌŝŽ�ŐƌĂĚƵĂů�Ğ�ĐŽŶƚşŶƵŽ�ĚĂ�ƐƵƐƚĞŶƚĂďŝůŝĚĂĚĞ�ĚŽ�ƉůĂŶĞƚĂ͘
*R3320145*
6
D�d�D�d/��
QUESTÃO 04
/ŵƉŽƌƚĂŶƚĞ�website� ĚĞ� ƌĞůĂĐŝŽŶĂŵĞŶƚŽ� ĐĂŵŝŶŚĂ� ƉĂƌĂ�
ϳϬϬ�ŵŝůŚƁĞƐ�ĚĞ�ƵƐƵĄƌŝŽƐ͘�KƵƚƌŽ�ĐŽŶŚĞĐŝĚŽ�ƐĞƌǀŝĚŽƌ�ĚĞ�
microblogging�ĂĐƵŵƵůĂ�ϭϰϬ�ŵŝůŚƁĞƐ�ĚĞ�ŵĞŶƐĂŐĞŶƐ�ĂŽ�
ĚŝĂ͘���ĐŽŵŽ�ƐĞ�ϳϱй�ĚĂ�ƉŽƉƵůĂĕĆŽ�ďƌĂƐŝůĞŝƌĂ�ƉŽƐƚĂƐƐĞ�Ƶŵ�
ĐŽŵĞŶƚĄƌŝŽ�Ă�ĐĂĚĂ�Ϯϰ�ŚŽƌĂƐ͘��Žŵ�ĂƐ�ƌĞĚĞƐ�ƐŽĐŝĂŝƐ�ĐĂĚĂ�
ǀĞnj�ŵĂŝƐ�ƉƌĞƐĞŶƚĞƐ�ŶŽ�ĚŝĂ�Ă�ĚŝĂ�ĚĂƐ�ƉĞƐƐŽĂƐ͕�Ġ�ŝŶĞǀŝƚĄǀĞů�
ƋƵĞ� ŵƵŝƚĂ� ŐĞŶƚĞ� ĞŶĐŽŶƚƌĞ� ŶĞůĂƐ� ƵŵĂ� ŵĂŶĞŝƌĂ� ĨĄĐŝů͕�
ƌĄƉŝĚĂ�Ğ�ĂďƌĂŶŐĞŶƚĞ�ĚĞ�ƐĞ�ŵĂŶŝĨĞƐƚĂƌ͘
hŵĂ�ƌĞĚĞ�ƐŽĐŝĂů�ĚĞ�ƌĞĐƌƵƚĂŵĞŶƚŽ�ƌĞǀĞůŽƵ�ƋƵĞ�ϵϮй�
ĚĂƐ� ĞŵƉƌĞƐĂƐ� ĂŵĞƌŝĐĂŶĂƐ� ũĄ� ƵƐĂƌĂŵ� ŽƵ� ƉůĂŶĞũĂŵ�
ƵƐĂƌ� ĂƐ� ƌĞĚĞƐ� ƐŽĐŝĂŝƐ� ŶŽ� ƉƌŽĐĞƐƐŽ� ĚĞ� ĐŽŶƚƌĂƚĂĕĆŽ͘�
�ĞƐƚĂƐ͕� ϲϬй� ĂƐƐƵŵĞŵ� ƋƵĞ� ďŝƐďŝůŚŽƚĂŵ� Ă� ǀŝĚĂ� ĚŽƐ�
ĐĂŶĚŝĚĂƚŽƐ�Ğŵ�websites�ĚĞ�ƌĞĚĞ�ƐŽĐŝĂů͘
ZĞĂůŝnjĂĚĂ� ƉŽƌ� ƵŵĂ� ĂŐġŶĐŝĂ� ĚĞ� ƌĞĐƌƵƚĂŵĞŶƚŽ͕� ƵŵĂ�
ƉĞƐƋƵŝƐĂ� ĐŽŵ� Ϯ� ϱϬϬ� ĞdžĞĐƵƟǀŽƐ� ďƌĂƐŝůĞŝƌŽƐ� ŵŽƐƚƌŽƵ�
ƋƵĞ�ϰϰй�ĚĞƐĐůĂƐƐŝĮĐĂƌŝĂŵ͕�ŶŽ�ƉƌŽĐĞƐƐŽ�ĚĞ�ƐĞůĞĕĆŽ͕�Ƶŵ�
ĐĂŶĚŝĚĂƚŽ�ƉŽƌ�ƐĞƵ�ĐŽŵƉŽƌƚĂŵĞŶƚŽ�Ğŵ�ƵŵĂ�ƌĞĚĞ�ƐŽĐŝĂů͘
DƵŝƚĂƐ� ƉĞƐƐŽĂƐ� ũĄ� ĞŶĨƌĞŶƚĂƌĂŵ� ƉƌŽďůĞŵĂƐ� ƉŽƌ�
ĐĂƵƐĂ� ĚĞ� ŝŶĨŽƌŵĂĕƁĞƐ� online͕� ƚĂŶƚŽ� ŶŽ� ĐĂŵƉŽ�
ƉĞƐƐŽĂů� ƋƵĂŶƚŽ� ŶŽ� ƉƌŽĮƐƐŝŽŶĂů͘� �ůŐƵŵĂƐ� ĞŵƉƌĞƐĂƐ�
Ğ� ŝŶƐƟƚƵŝĕƁĞƐ͕� ŝŶĐůƵƐŝǀĞ͕� ũĄ� ĂĚŽƚĂƌĂŵ� ĐĂƌƟůŚĂƐ� ĚĞ�
ĐŽŶĚƵƚĂ�Ğŵ�ƌĞĚĞƐ�ƐŽĐŝĂŝƐ͘
WK>KE/͕�'͘�K�ůĂĚŽ�ƉĞƌŝŐŽƐŽ�ĚĂƐ�ƌĞĚĞƐ�ƐŽĐŝĂŝƐ͘�ZĞǀŝƐƚĂ�/E&K͕�
Ɖ͘�ϳϬ�Ͳ�ϳϱ͕�ũƵůŚŽ�ϮϬϭϭ�;ĂĚĂƉƚĂĚŽͿ͘
�Ğ�ĂĐŽƌĚŽ�ĐŽŵ�Ž�ƚĞdžƚŽ͕
A ŵĂŝƐ�ĚĂ�ŵĞƚĂĚĞ�ĚĂƐ�ĞŵƉƌĞƐĂƐ�ĂŵĞƌŝĐĂŶĂƐ�ĞǀŝƚĂ�
ĂĐĞƐƐĂƌ�websites ĚĞ�ƌĞĚĞƐ�ƐŽĐŝĂŝƐ�ĚĞ�ĐĂŶĚŝĚĂƚŽƐ�
Ă�ĞŵƉƌĞŐŽ͘
B ĞŵƉƌĞƐĂƐ� Ğ� ŝŶƐƟƚƵŝĕƁĞƐ� ĞƐƚĆŽ� ĂƚĞŶƚĂƐ� ĂŽ�
ĐŽŵƉŽƌƚĂŵĞŶƚŽ�ĚĞ� ƐĞƵƐ� ĨƵŶĐŝŽŶĄƌŝŽƐ� Ğŵ�websites
ĚĞ�ƌĞĚĞƐ�ƐŽĐŝĂŝƐ͘
C Ă� ĐŽŵƉůĞdžŝĚĂĚĞ� ĚŽƐ� ƉƌŽĐĞĚŝŵĞŶƚŽƐ� ĚĞ� ƌĂƐƚƌĞŝŽ� Ğ�
ŵŽŶŝƚŽƌĂŵĞŶƚŽ� ĚĞ� ƵŵĂ� ƌĞĚĞ� ƐŽĐŝĂů� ŝŵƉĞĚĞ� ƋƵĞ�
ĂƐ� ĞŵƉƌĞƐĂƐ� ƚĞŶŚĂŵ� ĂĐĞƐƐŽ� ĂŽ� ƉĞƌĮů� ĚĞ� ƐĞƵƐ�
ĨƵŶĐŝŽŶĄƌŝŽƐ͘
D ĂƐ� ĐĂƌƟůŚĂƐ� ĚĞ� ĐŽŶĚƵƚĂ� ĂĚŽƚĂĚĂƐ� ŶĂƐ� ĞŵƉƌĞƐĂƐ�
ƉƌŽşďĞŵ�Ž�ƵƐŽ�ĚĞ�ƌĞĚĞƐ�ƐŽĐŝĂŝƐ�ƉĞůŽƐ�ĨƵŶĐŝŽŶĄƌŝŽƐ �͕Ğŵ�
ǀĞnj�ĚĞ�ƌĞĐŽŵĞŶĚĂƌ�ŵƵĚĂŶĕĂƐ�ĚĞ�ĐŽŵƉŽƌƚĂŵĞŶƚŽ͘
E Ă� ŵĂŝŽƌŝĂ� ĚŽƐ� ĞdžĞĐƵƟǀŽƐ� ďƌĂƐŝůĞŝƌŽƐ� ƵƟůŝnjĂƌŝĂ�
ŝŶĨŽƌŵĂĕƁĞƐ� ŽďƟĚĂƐ� Ğŵ� websites ĚĞ� ƌĞĚĞƐ�
ƐŽĐŝĂŝƐ͕� ƉĂƌĂ� ĚĞƐĐůĂƐƐŝĮĐĂƌ� Ƶŵ� ĐĂŶĚŝĚĂƚŽ� Ğŵ�
ƉƌŽĐĞƐƐŽ�ĚĞ�ƐĞůĞĕĆŽ͘
QUESTÃO 05
hŵĂ� ŝĚĞŝĂ� Ğ� Ƶŵ� ĂƉĂƌĞůŚŽ� ƐŝŵƉůĞƐ� ĚĞǀĞŵ͕� Ğŵ� ďƌĞǀĞ͕�
ĂũƵĚĂƌ�Ă�ƐĂůǀĂƌ�ǀŝĚĂƐ�ĚĞ�ƌĞĐĠŵͲŶĂƐĐŝĚŽƐ͘�/ĚĞĂůŝnjĂĚŽ�ƉĞůŽ�
ŵĞĐąŶŝĐŽ�ĂƌŐĞŶƟŶŽ�:ŽƌŐĞ�KĚſŶ͕�Ž�ĚŝƐƉŽƐŝƟǀŽ�ƋƵĞ�ůĞǀĂ�ƐĞƵ�
ƐŽďƌĞŶŽŵĞ�ĚĞƐĞŶƚĂůĂ�Ƶŵ�ďĞďġ�ƉƌĞƐŽ�ŶŽ�ĐĂŶĂů�ǀĂŐŝŶĂů�ͷ
Ğ͕�ƉŽƌ�ŵĂŝƐ�ŝŶƵƐŝƚĂĚŽ�ƋƵĞ�ƉĂƌĞĕĂ͕�ĨŽŝ�ĐƌŝĂĚŽ�ĐŽŵ�ďĂƐĞ�Ğŵ�
ƚĠĐŶŝĐĂ�ƵƐĂĚĂ�ƉĂƌĂ�ƌĞŵŽǀĞƌ�ƌŽůŚĂƐ�ĚĞ�ĚĞŶƚƌŽ�ĚĞ�ŐĂƌƌĂĨĂƐ͘�
K�ĂƉĂƌĞůŚŽ�ĐŽŶƐŝƐƚĞ�Ğŵ�ƵŵĂ�ďŽůƐĂ�ƉůĄƐƟĐĂ�ŝŶƐĞƌŝĚĂ�Ğŵ�
ƵŵĂ�ƉƌŽƚĞĕĆŽ�ĨĞŝƚĂ�ĚŽ�ŵĞƐŵŽ�ŵĂƚĞƌŝĂů�Ğ�ƋƵĞ�ĞŶǀŽůǀĞ�Ă�
ĐĂďĞĕĂ� ĚĂ� ĐƌŝĂŶĕĂ͘� �ƐƚĂŶĚŽ�Ž� ĚŝƐƉŽƐŝƟǀŽ� ĚĞǀŝĚĂŵĞŶƚĞ�
ƉŽƐŝĐŝŽŶĂĚŽ͕�Ă�ďŽůƐĂ�Ġ� ŝŶŇĂĚĂ�ƉĂƌĂ�ĂĚĞƌŝƌ�ă�ĐĂďĞĕĂ�ĚŽ�
ďĞďġ�Ğ�ƐĞƌ�ƉƵdžĂĚĂ�ĂŽƐ�ƉŽƵĐŽƐ͕�ĚĞ�ĨŽƌŵĂ�Ă�ŶĆŽ�ŵĂĐŚƵĐĄͲ
ůŽ͘�K�ŵĠƚŽĚŽ�ĚĞ�KĚſŶ�ĚĞǀĞ�ƐƵďƐƟƚƵŝƌ�ŽƵƚƌŽƐ�ũĄ�ĂƌĐĂŝĐŽƐ͕�
ĐŽŵŽ�Ž�ĚĞ�ĨſƌĐĞƉƐ�Ğ�Ž�ĚĞ�ƚƵďŽƐ�ĚĞ�ƐƵĐĕĆŽ͕�ŽƐ�ƋƵĂŝƐ͕�ƐĞ�
ƵƐĂĚŽƐ� ƉŽƌ� ŵĆŽƐ� ŵĂůƚƌĞŝŶĂĚĂƐ͕� ƉŽĚĞŵ� ĐŽŵƉƌŽŵĞƚĞƌ�
Ă�ǀŝĚĂ�ĚŽ�ďĞďġ͕�Ž�ƋƵĞ͕�ƐĞŐƵŶĚŽ�ĞƐƉĞĐŝĂůŝƐƚĂƐ͕�ŶĆŽ�ĚĞǀĞ�
ĂĐŽŶƚĞĐĞƌ�ĐŽŵ�Ž�ŶŽǀŽ�ĞƋƵŝƉĂŵĞŶƚŽ͘
^ĞŐƵŶĚŽ�Ž�dŚĞ�EĞǁ�zŽƌŬ�dŝŵĞƐ͕�Ă�ŝĚĞŝĂ�ƌĞĐĞďĞƵ�ĂƉŽŝŽ�
ĚĂ�KƌŐĂŶŝnjĂĕĆŽ�DƵŶĚŝĂů�ĚĞ�^ĂƷĚĞ�;KD^Ϳ�Ğ� ũĄ� ĨŽŝ�ĂƚĠ�
ůŝĐĞŶĐŝĂĚĂ� ƉŽƌ� ƵŵĂ� ĞŵƉƌĞƐĂ� ŶŽƌƚĞͲĂŵĞƌŝĐĂŶĂ� ĚĞ�
ƚĞĐŶŽůŽŐŝĂ�ŵĠĚŝĐĂ͘�EĆŽ�ƐĞ�ƐĂďĞ�ƋƵĂŶĚŽ�Ž�ĞƋƵŝƉĂŵĞŶƚŽ�
ĐŽŵĞĕĂƌĄ�Ă�ƐĞƌ�ƉƌŽĚƵnjŝĚŽ�ŶĞŵ�Ž�ƉƌĞĕŽ�Ă�ƐĞƌ�ĐŽďƌĂĚŽ͕�
ŵĂƐ�ƉƌĞƐƵŵĞͲƐĞ�ƋƵĞ�ĞůĞ�ŶĆŽ�ƉĂƐƐĂƌĄ�ĚĞ�ϱϬ�ĚſůĂƌĞƐ͕�
ĐŽŵ�ƌĞĚƵĕĆŽ�ĚŽ�ƉƌĞĕŽ�Ğŵ�ƉĂşƐĞƐ�ŵĂŝƐ�ƉŽďƌĞƐ͘
'h^D�K͕�'͘��ƉĂƌĞůŚŽ�ĚĞǀĞ�ĨĂĐŝůŝƚĂƌ�ƉĂƌƚŽƐ�Ğŵ�ƐŝƚƵĂĕƁĞƐ�ĚĞ�
emergência͘��ŝƐƉŽŶşǀĞů�Ğŵ͗�фŚƩƉ͗ͬͬĞdžĂŵĞ͘Ăďƌŝů͘ĐŽŵ͘ďƌх͘�
�ĐĞƐƐŽ�Ğŵ͗�ϭϴ�ŶŽǀ͘ �ϮϬϭϯ�;ĂĚĂƉƚĂĚŽͿ͘
�Žŵ�ƌĞůĂĕĆŽ�ĂŽ�ƚĞdžƚŽ�ĂĐŝŵĂ͕�ĂǀĂůŝĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�Ă�ƐĞŐƵŝƌ͘
/͘� �� ƵƟůŝnjĂĕĆŽ� ĚŽ� ŵĠƚŽĚŽ� ĚĞ� KĚſŶ� ƉŽĚĞƌĄ�
ƌĞĚƵnjŝƌ�Ă� ƚĂdžĂ�ĚĞ�ŵŽƌƚĂůŝĚĂĚĞ�ĚĞ�ĐƌŝĂŶĕĂƐ�ĂŽ�
ŶĂƐĐĞƌ͕ �ŵĞƐŵŽ�Ğŵ�ƉĂşƐĞƐ�ƉŽďƌĞƐ͘
//͘� WŽƌ� ƐĞƌ� ƵŵĂ� ǀĂƌŝĂŶƚĞ� ĚŽƐ� ƚƵďŽƐ� ĚĞ� ƐƵĐĕĆŽ͕� Ž�
ĂƉĂƌĞůŚŽ�ĚĞƐĞŶǀŽůǀŝĚŽ�ƉŽƌ�KĚſŶ�Ġ�ƌĞƐƵůƚĂĚŽ�ĚĞ�
ĂƉĞƌĨĞŝĕŽĂŵĞŶƚŽ�ĚĞ�ĞƋƵŝƉĂŵĞŶƚŽƐ�ĚĞ�ƉĂƌƚŽ͘
///͘� WŽƌ�ƐĞƵ�ƵƐŽ�ƐŝŵƉůĞƐ͕�Ž�ĚŝƐƉŽƐŝƟǀŽ�ĚĞ�MĚŽŶ�ƚĞŵ�
ŐƌĂŶĚĞ�ƉŽƚĞŶĐŝĂů�ĚĞ�ƐĞƌ�ƵƐĂĚŽ�Ğŵ�ƉĂşƐĞƐ�ŽŶĚĞ�
Ž�ƉĂƌƚŽ�Ġ�ƵƐƵĂůŵĞŶƚĞ�ƌĞĂůŝnjĂĚŽ�ƉŽƌ�ƉĂƌƚĞŝƌĂƐ͘
/s͘ � �� ƉŽƐƐŝďŝůŝĚĂĚĞ� ĚĞ͕� Ğŵ� ƉĂşƐĞƐ� ŵĂŝƐ� ƉŽďƌĞƐ͕�
ƌĞĚƵnjŝƌͲƐĞ� Ž� ƉƌĞĕŽ� ĚŽ� ĂƉĂƌĞůŚŽ� ŝĚĞĂůŝnjĂĚŽ�
ƉŽƌ� KĚſŶ� ĞǀŝĚĞŶĐŝĂ� ƉƌĞŽĐƵƉĂĕĆŽ� ĐŽŵ� Ă�
ƌĞƐƉŽŶƐĂďŝůŝĚĂĚĞ�ƐŽĐŝĂů͘
��ĐŽƌƌĞƚŽ�ĂƉĞŶĂƐ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /�Ğ�//͘
B /�Ğ�/s͘
C //�Ğ�///͘
D /͕�///�Ğ�/s͘
E //͕�///�Ğ�/s͘
*R3320146*
7
D�d�D�d/��
QUESTÃO 06
�Ɛ�ŵƵůŚĞƌĞƐ�ĨƌĞƋƵĞŶƚĂŵ�ŵĂŝƐ�ŽƐ�ďĂŶĐŽƐ�ĞƐĐŽůĂƌĞƐ�ƋƵĞ�ŽƐ�ŚŽŵĞŶƐ͕�ĚŝǀŝĚĞŵ�ƐĞƵ�ƚĞŵƉŽ�ĞŶƚƌĞ�Ž�ƚƌĂďĂůŚŽ�Ğ�ŽƐ�
ĐƵŝĚĂĚŽƐ�ĐŽŵ�Ă�ĐĂƐĂ͕�ŐĞƌĂŵ�ƌĞŶĚĂ�ĨĂŵŝůŝĂƌ͕ �ƉŽƌĠŵ�ĐŽŶƟŶƵĂŵ�ŐĂŶŚĂŶĚŽ�ŵĞŶŽƐ�Ğ�ƚƌĂďĂůŚĂŶĚŽ�ŵĂŝƐ�ƋƵĞ�ŽƐ�
ŚŽŵĞŶƐ͘
�Ɛ�ƉŽůşƟĐĂƐ�ĚĞ�ďĞŶĞİĐŝŽƐ�ŝŵƉůĞŵĞŶƚĂĚĂƐ�ƉŽƌ�ĞŵƉƌĞƐĂƐ�ƉƌĞŽĐƵƉĂĚĂƐ�Ğŵ�ĨĂĐŝůŝƚĂƌ�Ă�ǀŝĚĂ�ĚĂƐ�ĨƵŶĐŝŽŶĄƌŝĂƐ�ƋƵĞ�
ƚġŵ�ĐƌŝĂŶĕĂ�ƉĞƋƵĞŶĂ�Ğŵ�ĐĂƐĂ�ũĄ�ĞƐƚĆŽ�ĐŚĞŐĂŶĚŽ�ĂŽ��ƌĂƐŝů͘��ĐŽƌĚŽƐ�ĚĞ�ŚŽƌĄƌŝŽƐ�ŇĞdžşǀĞŝƐ͕�ƉƌŽŐƌĂŵĂƐ�ĐŽŵŽ�
ĂƵdžşůŝŽͲĐƌĞĐŚĞ͕�ĂƵdžşůŝŽͲďĂďĄ�Ğ�ĂƵdžşůŝŽͲĂŵĂŵĞŶƚĂĕĆŽ�ƐĆŽ�ĂůŐƵŶƐ�ĚŽƐ�ďĞŶĞİĐŝŽƐ�ŽĨĞƌĞĐŝĚŽƐ͘
�ŝƐƉŽŶşǀĞů�Ğŵ͗фŚƩƉ͗ͬͬǁǁǁϭ͘ĨŽůŚĂ͘ƵŽů͘ĐŽŵ͘ďƌх͘��ĐĞƐƐŽ�Ğŵ͗�ϯϬ�ũƵů͘�ϮϬϭϯ�;ĂĚĂƉƚĂĚŽͿ͘
30,3 36,5
10,1 44,4
31,7 34
15,5 41,4
JORNADA MÉDIA TOTAL DE TRABALHO POR
SEMANA NO BRASIL - (EM HORAS)
mãe com filhos
pai com filhos
mulher chefe de casal
com filhos
homem chefe de casal
com filhos
mulher cônjuge em
casal com filhos
tempo médio por semana dedicado aos afazeres domésticos
0 10 20 30 40 50 60 70tempo médio por semana dedicado ao trabalho remunerado
25,9 36,8
�ŝƐƉŽŶşǀĞů�Ğŵ͗�фŚƩƉ͗ͬͬŝƉĞĂ͘ŐŽǀ͘ ďƌх͘��ĐĞƐƐŽ�Ğŵ͗�ϯϬ�ũƵů͘�ϮϬϭϯ͘�
�ŽŶƐŝĚĞƌĂŶĚŽ�Ž�ƚĞdžƚŽ�Ğ�Ž�ŐƌĄĮĐŽ͕�ĂǀĂůŝĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�Ă�ƐĞŐƵŝƌ͘
/͘� K�ƐŽŵĂƚſƌŝŽ�ĚŽ�ƚĞŵƉŽ�ĚĞĚŝĐĂĚŽ�ƉĞůĂƐ�ŵƵůŚĞƌĞƐ�ĂŽƐ�ĂĨĂnjĞƌĞƐ�ĚŽŵĠƐƟĐŽƐ�Ğ�ĂŽ�ƚƌĂďĂůŚŽ�ƌĞŵƵŶĞƌĂĚŽ�
Ġ�ƐƵƉĞƌŝŽƌ�ĂŽ�ĚĞĚŝĐĂĚŽ�ƉĞůŽƐ�ŚŽŵĞŶƐ͕�ŝŶĚĞƉĞŶĚĞŶƚĞŵĞŶƚĞ�ĚŽ�ĨŽƌŵĂƚŽ�ĚĂ�ĨĂŵşůŝĂ͘
//͘� K�ĨƌĂŐŵĞŶƚŽ�ĚĞ�ƚĞdžƚŽ�Ğ�ŽƐ�ĚĂĚŽƐ�ĚŽ�ŐƌĄĮĐŽ�ĂƉŽŶƚĂŵ�ƉĂƌĂ�Ă�ŶĞĐĞƐƐŝĚĂĚĞ�ĚĞ�ĐƌŝĂĕĆŽ�ĚĞ�ƉŽůşƟĐĂƐ�ƋƵĞ�
ƉƌŽŵŽǀĂŵ�Ă�ŝŐƵĂůĚĂĚĞ�ĞŶƚƌĞ�ŽƐ�ŐġŶĞƌŽƐ�ŶŽ�ƋƵĞ�ĐŽŶĐĞƌŶĞ͕�ƉŽƌ�ĞdžĞŵƉůŽ͕�Ă�ƚĞŵƉŽ�ŵĠĚŝŽ�ĚĞĚŝĐĂĚŽ�
ĂŽ�ƚƌĂďĂůŚŽ�Ğ�ƌĞŵƵŶĞƌĂĕĆŽ�ƌĞĐĞďŝĚĂ͘�
///͘� EŽ� ĨƌĂŐŵĞŶƚŽ� ĚĞ� ƌĞƉŽƌƚĂŐĞŵ� ĂƉƌĞƐĞŶƚĂĚŽ͕� ƌĞƐƐĂůƚĂͲƐĞ� Ă� ĚŝĨĞƌĞŶĕĂ� ĞŶƚƌĞ� Ž� ƚĞŵƉŽ� ĚĞĚŝĐĂĚŽ� ƉŽƌ�
ŵƵůŚĞƌĞƐ�Ğ�ŚŽŵĞŶƐ�ĂŽ�ƚƌĂďĂůŚŽ�ƌĞŵƵŶĞƌĂĚŽ͕�ƐĞŵ�ĂůƵƐĆŽ�ĂŽƐ�ĂĨĂnjĞƌĞƐ�ĚŽŵĠƐƟĐŽƐ͘�
��ĐŽƌƌĞƚŽ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /͕�ĂƉĞŶĂƐ͘
B ///͕�ĂƉĞŶĂƐ͘
C /�Ğ�//͕�ĂƉĞŶĂƐ͘
D //�Ğ�///͕�ĂƉĞŶĂƐ͘
E /͕�//�Ğ�///͘
*R3320147*
ϴ
D�d�D�d/��
QUESTÃO 07
K� ƋƵĂĚƌŽ� Ă� ƐĞŐƵŝƌ� ĂƉƌĞƐĞŶƚĂ� Ă� ƉƌŽƉŽƌĕĆŽ� ;йͿ�
ĚĞ� ƚƌĂďĂůŚĂĚŽƌĞƐ� ƉŽƌ� ĨĂŝdžĂ� ĚĞ� ƚĞŵƉŽ� ŐĂƐƚŽ� ŶŽ�
ĚĞƐůŽĐĂŵĞŶƚŽ� ĐĂƐĂͲƚƌĂďĂůŚŽ͕� ŶŽ� �ƌĂƐŝů� Ğ� Ğŵ� ƚƌġƐ�
ĐŝĚĂĚĞƐ�ďƌĂƐŝůĞŝƌĂƐ͘
Tempo de 
deslocamento Brasil
Rio de 
Janeiro
São
Paulo �ƵƌŝƟďĂ�
Até cinco minutos ϭϮ͕ϳϬ ϱ͕ϴϬ ϱ͕ϭϬ ϳ͕ϴϬ
De seis minutos 
até meia hora ϱϮ͕ϮϬ ϯϮ͕ϭϬ ϯϭ͕ϲϬ ϰϱ͕ϴϬ
Mais de meia hora 
até uma hora Ϯϯ͕ϲϬ ϯϯ͕ϱϬ ϯϰ͕ϲϬ ϯϮ͕ϰϬ
Mais de uma hora 
até duas horas ϵ͕ϴϬ Ϯϯ͕ϮϬ Ϯϯ͕ϯϬ ϭϮ͕ϵϬ
Mais de duas 
horas ϭ͕ϴϬ ϱ͕ϱϬ ϱ͕ϯϬ ϭ͕ϮϬ
��E^K�ϮϬϭϬͬ/�'��;ĂĚĂƉƚĂĚŽͿ͘�
�Žŵ� ďĂƐĞ� ŶŽƐ� ĚĂĚŽƐ� ĂƉƌĞƐĞŶƚĂĚŽƐ� Ğ� ĐŽŶƐŝĚĞƌĂŶĚŽ� Ă�
ĚŝƐƚƌŝďƵŝĕĆŽ� ĚĂ� ƉŽƉƵůĂĕĆŽ� ƚƌĂďĂůŚĂĚŽƌĂ� ŶĂƐ� ĐŝĚĂĚĞƐ� Ğ�
ĂƐ�ƉŽůşƟĐĂƐ�ƉƷďůŝĐĂƐ�ĚŝƌĞĐŝŽŶĂĚĂƐ�ă�ŵŽďŝůŝĚĂĚĞ�ƵƌďĂŶĂ͕�
ĂǀĂůŝĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�Ă�ƐĞŐƵŝƌ͘
/͘� ��ĚŝƐƚƌŝďƵŝĕĆŽ�ĚĂƐ�ƉĞƐƐŽĂƐ�ƉŽƌ� ĨĂŝdžĂ�ĚĞ� ƚĞŵƉŽ�
ĚĞ� ĚĞƐůŽĐĂŵĞŶƚŽ� ĐĂƐĂͲƚƌĂďĂůŚŽ� ŶĂ� ƌĞŐŝĆŽ�
ŵĞƚƌŽƉŽůŝƚĂŶĂ�ĚŽ�ZŝŽ�ĚĞ�:ĂŶĞŝƌŽ�Ġ�ƉƌſdžŝŵĂ�ă�ƋƵĞ�
ƐĞ�ǀĞƌŝĮĐĂ�Ğŵ�^ĆŽ�WĂƵůŽ͕�ŵĂƐ�ŶĆŽ�Ğŵ��ƵƌŝƟďĂ�Ğ�
ŶĂ�ŵĠĚŝĂ�ďƌĂƐŝůĞŝƌĂ͘
//͘� EĂƐ�ŵĞƚƌſƉŽůĞƐ͕�Ğŵ�ŐĞƌĂů͕�Ă�ŵĂŝŽƌŝĂ�ĚŽƐ�ƉŽƐƚŽƐ�
ĚĞ� ƚƌĂďĂůŚŽ� ĞƐƚĄ� ůŽĐĂůŝnjĂĚĂ� ŶĂƐ� ĄƌĞĂƐ� ƵƌďĂŶĂƐ�
ĐĞŶƚƌĂŝƐ͕�Ğ�ĂƐ�ƌĞƐŝĚġŶĐŝĂƐ�ĚĂ�ƉŽƉƵůĂĕĆŽ�ĚĞ�ďĂŝdžĂ�
ƌĞŶĚĂ�ĞƐƚĆŽ� ĐŽŶĐĞŶƚƌĂĚĂƐ� Ğŵ�ĄƌĞĂƐ� ŝƌƌĞŐƵůĂƌĞƐ�
ŽƵ�ŶĂ�ƉĞƌŝĨĞƌŝĂ͕�Ž�ƋƵĞ�ĂƵŵĞŶƚĂ�Ž�ƚĞŵƉŽ�ŐĂƐƚŽ�ƉŽƌ�
ĞƐƚĂ�ƉŽƉƵůĂĕĆŽ�ŶŽ�ĚĞƐůŽĐĂŵĞŶƚŽ�ĐĂƐĂͲƚƌĂďĂůŚŽ�Ğ�
Ž�ĐƵƐƚŽ�ĚŽ�ƚƌĂŶƐƉŽƌƚĞ͘
///͘� �Ɛ� ƉŽůşƟĐĂƐ� ƉƷďůŝĐĂƐ� ƌĞĨĞƌĞŶƚĞƐ� Ă� ƚƌĂŶƐƉŽƌƚĞƐ�
ƵƌďĂŶŽƐ͕� ĐŽŵŽ͕� ƉŽƌ� ĞdžĞŵƉůŽ͕� �ŝůŚĞƚĞ� jŶŝĐŽ�
Ğ� sĞşĐƵůŽ� >ĞǀĞ� ƐŽďƌĞ� dƌŝůŚŽƐ� ;s>dͿ͕� ĂŽ� ƐĞƌĞŵ�
ŝŵƉůĞŵĞŶƚĂĚĂƐ͕� ĐŽŶƚƌŝďƵĞŵ� ƉĂƌĂ� ƌĞĚƵĕĆŽ� ĚŽ�
ƚĞŵƉŽ�ŐĂƐƚŽ�ŶŽ�ĚĞƐůŽĐĂŵĞŶƚŽ� ĐĂƐĂͲƚƌĂďĂůŚŽ�Ğ�
ĚŽ�ĐƵƐƚŽ�ĚŽ�ƚƌĂŶƐƉŽƌƚĞ͘
��ĐŽƌƌĞƚŽ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /͕�ĂƉĞŶĂƐ͘
B ///͕�ĂƉĞŶĂƐ͘
C /�Ğ�//͕�ĂƉĞŶĂƐ͘
D //�Ğ�///͕�ĂƉĞŶĂƐ͘
E /͕�//�Ğ�///͘
QUESTÃO 08
�ŽŶƐƚĂŶƚĞƐ� ƚƌĂŶƐĨŽƌŵĂĕƁĞƐ� ŽĐŽƌƌĞƌĂŵ� ŶŽƐ� ŵĞŝŽƐ�
ƌƵƌĂů�Ğ�ƵƌďĂŶŽ͕�Ă�ƉĂƌƟƌ�ĚŽ�ƐĠĐƵůŽ�yy͘��Žŵ�Ž�ĂĚǀĞŶƚŽ�
ĚĂ�ŝŶĚƵƐƚƌŝĂůŝnjĂĕĆŽ͕�ŚŽƵǀĞ�ŵƵĚĂŶĕĂƐ�ŝŵƉŽƌƚĂŶƚĞƐ�ŶŽ�
ŵŽĚŽ�ĚĞ�ǀŝĚĂ�ĚĂƐ�ƉĞƐƐŽĂƐ͕�Ğŵ�ƐĞƵƐ�ƉĂĚƌƁĞƐ�ĐƵůƚƵƌĂŝƐ͕�
ǀĂůŽƌĞƐ� Ğ� ƚƌĂĚŝĕƁĞƐ͘�K� ĐŽŶũƵŶƚŽ�ĚĞ� ĂĐŽŶƚĞĐŝŵĞŶƚŽƐ�
ƉƌŽǀŽĐŽƵ͕� ƚĂŶƚŽ� ŶĂ� njŽŶĂ� ƵƌďĂŶĂ� ƋƵĂŶƚŽ� ŶĂ� ƌƵƌĂů͕�
ƉƌŽďůĞŵĂƐ�ĐŽŵŽ�ĞdžƉůŽƐĆŽ�ĚĞŵŽŐƌĄĮĐĂ͕�ƉƌĞũƵşnjŽ�ŶĂƐ�
ĂƟǀŝĚĂĚĞƐ�ĂŐƌşĐŽůĂƐ�Ğ�ǀŝŽůġŶĐŝĂ͘
/ŶŝĐŝĂƌĂŵͲƐĞ� ŝŶƷŵĞƌĂƐ� ƚƌĂŶƐĨŽƌŵĂĕƁĞƐ� ŶĂ� ŶĂƚƵƌĞnjĂ͕�
ĐƌŝĂŶĚŽͲƐĞ�ƚĠĐŶŝĐĂƐ�ƉĂƌĂ�ŽďũĞƚŽƐ�ĂƚĠ�ĞŶƚĆŽ�ƐĞŵ�ƵƟůŝĚĂĚĞ�
ƉĂƌĂ� Ž� ŚŽŵĞŵ͘� /ƐƐŽ� Ɛſ� ĨŽŝ� ƉŽƐƐşǀĞů� Ğŵ� ĚĞĐŽƌƌġŶĐŝĂ�
ĚŽƐ� ƌĞĐƵƌƐŽƐ� ŶĂƚƵƌĂŝƐ� ĞdžŝƐƚĞŶƚĞƐ͕� ƋƵĞ� ƉƌŽƉŝĐŝĂƌĂŵ�
ĞƐƚƌƵƚƵƌĂ�ĚĞ�ĐƌĞƐĐŝŵĞŶƚŽ�Ğ�ďƵƐĐĂ�ĚĞ�ƉƌŽƐƉĞƌŝĚĂĚĞ͕�Ž�
ƋƵĞ�ĨĂnj�ĚĂ�ĞdžƉĞƌŝŵĞŶƚĂĕĆŽ�Ƶŵ�ŵĠƚŽĚŽ�ĚĞ�ƚƌĂŶƐĨŽƌŵĂƌ�
ŽƐ�ƌĞĐƵƌƐŽƐ�Ğŵ�ďĞŶĞİĐŝŽ�ƉƌſƉƌŝŽ͘
^�EdK^͕�D͘�DĞƚĂŵŽƌĨŽƐĞƐ�ĚŽ�ĞƐƉĂĕŽ�ŚĂďŝƚĂĚŽ.
^ĆŽ�WĂƵůŽ͗�,ƵĐŝƚĞĐ͕�ϭϵϴϴ�;ĂĚĂƉƚĂĚŽͿ͘
��ƉĂƌƟƌ�ĚĂƐ�ŝĚĞŝĂƐ�ĞdžƉƌĞƐƐĂƐ�ŶŽ�ƚĞdžƚŽ�ĂĐŝŵĂ͕�ĐŽŶĐůƵŝͲƐĞ�
ƋƵĞ͕�ŶŽ��ƌĂƐŝů�ĚŽ�ƐĠĐƵůŽ�yy͕
A Ă� ŝŶĚƵƐƚƌŝĂůŝnjĂĕĆŽ� ŽĐŽƌƌĞƵ� ŝŶĚĞƉĞŶĚĞŶƚĞŵĞŶƚĞ�
ĚŽ�ġdžŽĚŽ�ƌƵƌĂů�Ğ�ĚŽƐ�ƌĞĐƵƌƐŽƐ�ŶĂƚƵƌĂŝƐ�ĚŝƐƉŽŶşǀĞŝƐ͘
B Ž�ġdžŽĚŽ�ƌƵƌĂů�ƉĂƌĂ�ĂƐ�ĐŝĚĂĚĞƐ�ŶĆŽ�ƉƌĞũƵĚŝĐŽƵ�ĂƐ�
ĂƚŝǀŝĚĂĚĞƐ�ĂŐƌşĐŽůĂƐ�ŶĞŵ�Ž�ŵĞŝŽ�ƌƵƌĂů�ƉŽƌƋƵĞ�
ŶŽǀĂƐ� ƚĞĐŶŽůŽŐŝĂƐ� ŚĂǀŝĂŵ� ƐŝĚŽ� ŝŶƚƌŽĚƵnjŝĚĂƐ�
ŶŽ�ĐĂŵƉŽ͘
C ŚŽŵĞŶƐ�Ğ�ŵƵůŚĞƌĞƐ�ĂĚǀŝŶĚŽƐ�ĚŽ�ĐĂŵƉŽ�ĚĞŝdžĂƌĂŵ�
ƐƵĂ� ĐƵůƚƵƌĂ� Ğ� ƐĞ� ĂĚĂƉƚĂƌĂŵ� Ă� ŽƵƚƌĂ͕� ĐŝƚĂĚŝŶĂ͕�
ƚŽƚĂůŵĞŶƚĞ�ĚŝĨĞƌĞŶƚĞ�Ğ�ŽƉŽƐƚĂ�ĂŽƐ�ƐĞƵƐ�ǀĂůŽƌĞƐ͘
D ƚĂŶƚŽ�Ž�ĞƐƉĂĕŽ�ƵƌďĂŶŽ�ƋƵĂŶƚŽ�Ž� ƌƵƌĂů� ƐŽĨƌĞƌĂŵ�
ƚƌĂŶƐĨŽƌŵĂĕƁĞƐ� ĚĞĐŽƌƌĞŶƚĞƐ� ĚĂ� ĂƉůŝĐĂĕĆŽ� ĚĞ�
ŶŽǀĂƐ� ƚĞĐŶŽůŽŐŝĂƐ� ăƐ� ĂƟǀŝĚĂĚĞƐ� ŝŶĚƵƐƚƌŝĂŝƐ� Ğ�
ĂŐƌşĐŽůĂƐ͘
E ŽƐ� ŵŝŐƌĂŶƚĞƐ� ĐŚĞŐĂƌĂŵ� ăƐ� ŐƌĂŶĚĞƐ� ĐŝĚĂĚĞƐ�
ƚƌĂnjĞŶĚŽ� ĐŽŶƐŝŐŽ� ǀĂůŽƌĞƐ� Ğ� ƚƌĂĚŝĕƁĞƐ͕� ƋƵĞ� ůŚĞƐ�
ƉŽƐƐŝďŝůŝƚĂƌĂŵ� ŵĂŶƚĞƌ� ŝŶƚĂĐƚĂ� ƐƵĂ� ĐƵůƚƵƌĂ͕� ƚĂů�
ĐŽŵŽ�ƐĞ�ŵĂŶŝĨĞƐƚĂǀĂ�ŶĂƐ�ƉĞƋƵĞŶĂƐ�ĐŝĚĂĚĞƐ�Ğ�ŶŽ�
ŵĞŝŽ�ƌƵƌĂů͘
*R3320148*
9
D�d�D�d/��
9
D�d�D�d/��
QUESTÃO DISCURSIVA 3
KƐ� ƉƌŝŶĐŝƉĂŝƐ� ĞĨĞŝƚŽƐ� ǀŝƐƵĂŝƐ� ĚĂ� ĐŽŵƉƵƚĂĕĆŽ� ŐƌĄĮĐĂ� ǀŝƐƚŽƐ� Ğŵ� ƵŵĂ� ƚĞůĂ� ƐĆŽ� ƌĞƐƵůƚĂĚŽƐ� ĚĞ� ĂƉůŝĐĂĕƁĞƐ� ĚĞ�
ƚƌĂŶƐĨŽƌŵĂĕƁĞƐ� ůŝŶĞĂƌĞƐ͘�dƌĂŶƐůĂĕĆŽ͕�ƌŽƚĂĕĆŽ͕�ƌĞĚŝŵĞŶƐŝŽŶĂŵĞŶƚŽ�Ğ�ĂůƚĞƌĂĕĆŽ�ĚĞ�ĐŽƌĞƐ�ƐĆŽ�ĂƉĞŶĂƐ�ĂůŐƵŶƐ�
ĞdžĞŵƉůŽƐ͘�
�ŽŶƐŝĚĞƌĞ� ƋƵĞ�ƵŵĂ� ƚĞůĂ� Ġ� ĐŽƌƚĂĚĂ�ƉŽƌ� ĚŽŝƐ� ĞŝdžŽƐ͕� x e y͕� ŽƌƚŽŐŽŶĂŝƐ� ĞŶƚƌĞ� Ɛŝ͕� ĨŽƌŵĂŶĚŽ�Ƶŵ� ƐŝƐƚĞŵĂ�ĚĞ�
ĐŽŽƌĚĞŶĂĚĂƐ� ĐŽŵ� ŽƌŝŐĞŵ� ŶŽ� ĐĞŶƚƌŽ� ĚĂ� ƚĞůĂ͘� ^ƵƉŽŶŚĂ� ƋƵĞ͕� ŶĞƐƐĂ� ƚĞůĂ� ƉůĂŶĂ͕� ĞdžŝƐƚĞ� Ă� ŝŵĂŐĞŵ� ĚĞ� ƵŵĂ�
ĞůŝƉƐĞ�ĐŽŵ�ĞŝdžŽ�ŵĂŝŽƌ�ĚĞ�ƚĂŵĂŶŚŽ�ϰ͕�ƉĂƌĂůĞůŽ�ĂŽ�ĞŝdžŽ�x͕�Ğ�ĐƵũŽƐ�ĨŽĐŽƐ�ƚġŵ�ĐŽŽƌĚĞŶĂĚĂƐ�(�1, 2) e (1, 2).
�ŽŶƐŝĚĞƌĞ�T Ƶŵ�ŽƉĞƌĂĚŽƌ�ůŝŶĞĂƌ�ĚĞĮŶŝĚŽ�Ğŵ�R2.
�Ğ� ĂĐŽƌĚŽ� ĐŽŵ� ĂƐ� ŝŶĨŽƌŵĂĕƁĞƐ� ĂĐŝŵĂ͕� ĨĂĕĂ� Ž� ƋƵĞ� ƐĞ� ƉĞĚĞ� ŶŽƐ� ŝƚĞŶƐ� Ă� ƐĞŐƵŝƌ͕ � ĂƉƌĞƐĞŶƚĂŶĚŽ� ŽƐ� ĐĄůĐƵůŽƐ�
ƵƟůŝnjĂĚŽƐ�ŶĂ�ƐƵĂ�ƌĞƐŽůƵĕĆŽ͘
ĂͿ�� DŽƐƚƌĞ�ƋƵĞ�Ž�ƉŽŶƚŽ�(0, 2 +
√
3)�ƉĞƌƚĞŶĐĞ�ă�ĞůŝƉƐĞ͘��;ǀĂůŽƌ͗�ϯ͕Ϭ�ƉŽŶƚŽƐͿ
ďͿ��^ƵƉŽŶŚĂ�ƋƵĞ͕�Ğŵ�ĐĂĚĂ�ƉŽŶƚŽ�ĚĂ�ƚĞůĂ͕�ƐĞũĂ�ĂƉůŝĐĂĚŽ�Ž�ŽƉĞƌĂĚŽƌ�ůŝŶĞĂƌ�T(x, y) = (x + y, �2x + 4y).
YƵĂŝƐ�ƐĞƌĆŽ�ĂƐ�ĐŽŽƌĚĞŶĂĚĂƐ�ĚŽƐ�ĨŽĐŽƐ�ĚĂ�ĞůŝƉƐĞ�ĂƉſƐ�Ă�ĂƉůŝĐĂĕĆŽ�ĚĞ�T ͍ �;ǀĂůŽƌ͗�ϯ͕Ϭ�ƉŽŶƚŽƐͿ
ĐͿ�� �ĂůĐƵůĞ�ŽƐ�ĂƵƚŽǀĂůŽƌĞƐ�ĚŽ�ŽƉĞƌĂĚŽƌ�ůŝŶĞĂƌ�T(x, y) = (x + y, �2x + 4y)͘�;ǀĂůŽƌ͗�ϰ͕Ϭ�ƉŽŶƚŽƐͿ
RASCUNHO
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COMPONENTE ESPECÍFICO
*R3320149*
10
D�d�D�d/��
QUESTÃO DISCURSIVA 4
�ƚƵĂůŵĞŶƚĞ͕�Ă�ŵĂŝŽƌŝĂ�ĚŽƐ�ĞĚŝƚŽƌĞƐ�ĚĞ�ƚĞdžƚŽ�ŽĨĞƌĞĐĞ�Ž�ƌĞĐƵƌƐŽ�ĚĞ�ĐŽƌƌĞĕĆŽ�ŽƌƚŽŐƌĄĮĐĂ͘��ƐƐĞ�ƌĞĐƵƌƐŽ�ĐŽŶƐŝƐƚĞ�
Ğŵ�ĚĞƐƚĂĐĂƌ͕ � ĞŶƚƌĞ� ĂƐ� ƉĂůĂǀƌĂƐ�ĚŝŐŝƚĂĚĂƐ͕� ĂƋƵĞůĂƐ� ĐŽŵ�ƉŽƐƐşǀĞŝƐ� ĞƌƌŽƐ�ĚĞ� ŐƌĂĮĂ͘� WŽƌ� ĞdžĞŵƉůŽ͕� ƋƵĂŶĚŽ� ƐĞ�
ĚŝŐŝƚĂ�Ă�ƉĂůĂǀƌĂ�͞ĐĂnjĂ͕͟ �Ž�ƌĞĐƵƌƐŽ�ĚĞ�ĐŽƌƌĞĕĆŽ�ĚĞƐƚĂĐĂ�ĞƐƐĂ�ƉĂůĂǀƌĂ͕�ƉŽŝƐ�Ă�ƉĂůĂǀƌĂ�͞ĐĂnjĂ͟�ŶĆŽ�ĞdžŝƐƚĞ�ŶĂ�ůşŶŐƵĂ�
ƉŽƌƚƵŐƵĞƐĂ͘�dĂŵďĠŵ�Ġ�ĐŽŵƵŵ�Ž�ƌĞĐƵƌƐŽ�ĚĞ�ĐŽƌƌĞĕĆŽ�ŽƌƚŽŐƌĄĮĐĂ�ƐƵŐĞƌŝƌ�ƵŵĂ�ŽƵƚƌĂ�ƉĂůĂǀƌĂ�ƉĂƌĂ�ƐƵďƐƟƚƵŝƌ�Ă�
ƉĂůĂǀƌĂ�ŝŶĐŽƌƌĞƚĂ͘
��ƐƵŐĞƐƚĆŽ�ĚĞ�ƋƵĂŝƐ�ƉĂůĂǀƌĂƐ�ƉŽĚĞŵ�ƐƵďƐƟƚƵŝƌ�Ă�ƉĂůĂǀƌĂ�ŝŶĐŽƌƌĞƚĂ�Ġ�ĨĞŝƚĂ�ĐŽŵ�ƵŵĂ�ŵĞĚŝĚĂ�ĚĂ�ĚŝƐƚąŶĐŝĂ�ĞŶƚƌĞ�
Ă�ƉĂůĂǀƌĂ�ŝŶĐŽƌƌĞƚĂ�Ğ�ĂƐ�ƉĂůĂǀƌĂƐ�ƋƵĞ�ĐŽŶƐƚĂŵ�ŶŽ�ĚŝĐŝŽŶĄƌŝŽ�ĚŽ�ĞĚŝƚŽƌ�ĚĞ�ƚĞdžƚŽ͘��džŝƐƚĞŵ�ĚŝǀĞƌƐĂƐ�ŵĂŶĞŝƌĂƐ�ĚĞ�
ŵĞĚŝƌ�Ă�ĚŝƐƚąŶĐŝĂ�ĞŶƚƌĞ�ĚƵĂƐ�ƉĂůĂǀƌĂƐ͘�hŵĂ�ĚĞůĂƐ�Ġ�Ă�ĚĞŶŽŵŝŶĂĚĂ��ŝƐƚąŶĐŝĂ�ĚĞ�,ĂŵŵŝŶŐ͕�ŶĂ�ƋƵĂů�Ă�ŵĞĚŝĚĂ�
ĚĂ�ĚŝƐƚąŶĐŝĂ�ĞŶƚƌĞ�ĚƵĂƐ�ƉĂůĂǀƌĂƐ�x e y͕�ĐŽŵ�Ă�ŵĞƐŵĂ�ƋƵĂŶƚŝĚĂĚĞ�ĚĞ�ůĞƚƌĂƐ͕�Ġ�ĨĞŝƚĂ�ĚĂ�ƐĞŐƵŝŶƚĞ�ĨŽƌŵĂ͗�
d(x, y) с�ŶƷŵĞƌŽ�ĚĞ�ůĞƚƌĂƐ�ƋƵĞ�ƐĆŽ�ĚŝĨĞƌĞŶƚĞƐ�Ğŵ�x e y͕�Ğŵ�ƐƵĂƐ�ƌĞƐƉĞĐƟǀĂƐ�ƉŽƐŝĕƁĞƐ͘�DĂŝƐ�ĨŽƌŵĂůŵĞŶƚĞ͕�
ƐĞ�x = x
1x2x3...xn e y = y1y2y3...yn�ƐĆŽ�ƉĂůĂǀƌĂƐ�Ğŵ�ƋƵĞ�xi e yi ƐĆŽ�ůĞƚƌĂƐ�ĚŽ�ĂůĨĂďĞƚŽ͕�ƉĂƌĂ�i = 1,2,3,...,n͕�ĞŶƚĆŽ�
d(x, y) = #({i : xi z yi, ĐŽŵ i = 1,2,3...,n})͕�Ğŵ�ƋƵĞ�#(A)�ƌĞƉƌĞƐĞŶƚĂ�Ž�ŶƷŵĞƌŽ�ĚĞ�ĞůĞŵĞŶƚŽƐ�ĚŽ�ĐŽŶũƵŶƚŽ�A.
WŽƌ�ĞdžĞŵƉůŽ͕�d(caza, casa) = #({3}) = 1͕�ũĄ�ƋƵĞ�ĞůĂƐ�ĚŝĨĞƌĞŵ�ĂƉĞŶĂƐ�ŶĂ�ƚĞƌĐĞŝƌĂ�ůĞƚƌĂ͘
��ƉĂƌƟƌ�ĚĞƐƐĂƐ�ŝŶĨŽƌŵĂĕƁĞƐ͕�ĨĂĕĂ�Ž�ƋƵĞ�ƐĞ�ƉĞĚĞ�ŶŽƐ�ŝƚĞŶƐ�Ă�ƐĞŐƵŝƌ͘
ĂͿ�� DŽƐƚƌĞ�ƋƵĞ�Ă��ŝƐƚąŶĐŝĂ�ĚĞ�,ĂŵŵŝŶŐ�Ġ�ƵŵĂ�ŵĠƚƌŝĐĂ�ŶŽ�ĐŽŶũƵŶƚŽ�ĚĂƐ�ƉĂůĂǀƌĂƐ�ĐŽŵ�ůĞƚƌĂƐ͘�;ǀĂůŽƌ͗�ϱ͕Ϭ�ƉŽŶƚŽƐͿ
ďͿ��DŽƐƚƌĞ�ƋƵĞ�Ž�ĐŽŶũƵŶƚŽ�ĚĂƐ�ƉĂůĂǀƌĂƐ�ĐŽŵ�ůĞƚƌĂƐ͕�ŵƵŶŝĚŽ�ĚĂ��ŝƐƚąŶĐŝĂ�ĚĞ�,ĂŵŵŝŶŐ͕�Ġ�Ƶŵ�ĞƐƉĂĕŽ�
ŵĠƚƌŝĐŽ�ĚŝƐĐƌĞƚŽ͘�;ǀĂůŽƌ͗�ϱ͕Ϭ�ƉŽŶƚŽƐͿ
RASCUNHO
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QUESTÃO DISCURSIVA 5
hŵĂ�ĞƋƵĂĕĆŽ�ĚŝŽĨĂŶƟŶĂ�ůŝŶĞĂƌ�ŶĂƐ�ŝŶĐſŐŶŝƚĂƐ�x e y�Ġ�ƵŵĂ�ĞƋƵĂĕĆŽ�ĚĂ�ĨŽƌŵĂ�ax + by = c͕�Ğŵ�ƋƵĞ�a, b e c
ƐĆŽ�ŝŶƚĞŝƌŽƐ͕�Ğ�ĂƐ�ƷŶŝĐĂƐ�ƐŽůƵĕƁĞƐ�(x0, y0)�ƋƵĞ�ŝŶƚĞƌĞƐƐĂŵ�ƐĆŽ�ĂƋƵĞůĂƐ�Ğŵ�ƋƵĞ�x0͕ y0 �Z .
EĞƐƐĞ�ĐŽŶƚĞdžƚŽ͕�ĐŽŶƐŝĚĞƌĞ�ƋƵĞ�ŽƐ�ŝŶŐƌĞƐƐŽƐ�ĚĞ�Ƶŵ�ĐŝŶĞŵĂ�ĐƵƐƚĂŵ�ZΨ�ϵ͕ϬϬ�ƉĂƌĂ�ĞƐƚƵĚĂŶƚĞƐ�Ğ�ZΨ�ϭϱ͕ϬϬ�ƉĂƌĂ�Ž�
ƉƷďůŝĐŽ�ŐĞƌĂů͕�Ğ�ƋƵĞ͕�Ğŵ�ĐĞƌƚŽ�ĚŝĂ͕�ĚƵƌĂŶƚĞ�ĚĞƚĞƌŵŝŶĂĚŽ�ƉĞƌşŽĚŽ͕�Ă�ĂƌƌĞĐĂĚĂĕĆŽ�ŶĂƐ�ďŝůŚĞƚĞƌŝĂƐ�ĚĞƐƐĞ�ĐŝŶĞŵĂ�ĨŽŝ�ĚĞ�ZΨ�Ϯϰϲ͕ϬϬ͘
��ƉĂƌƟƌ�ĚĂƐ�ŝŶĨŽƌŵĂĕƁĞƐ�ĂĐŝŵĂ͕�ĨĂĕĂ�Ž�ƋƵĞ�ƐĞ�ƉĞĚĞ�ŶŽƐ�ŝƚĞŶƐ�Ă�ƐĞŐƵŝƌ͘
ĂͿ� KďƚĞŶŚĂ�ƵŵĂ�ĞƋƵĂĕĆŽ�ĚŝŽĨĂŶƟŶĂ� ůŝŶĞĂƌ�ƋƵĞ�ŵŽĚĞůĞ�Ă�ƐŝƚƵĂĕĆŽ�ĂĐŝŵĂ͕� ŝŶĚŝĐĂŶĚŽ�Ž�ƐŝŐŶŝĮĐĂĚŽ�ĚĂƐ�
ŝŶĐſŐŶŝƚĂƐ͘�;ǀĂůŽƌ͗�ϯ͕Ϭ�ƉŽŶƚŽƐͿ
ďͿ� YƵĂŶƚĂƐ�Ğ�ƋƵĂŝƐ�ƐĆŽ�ĂƐ�ƐŽůƵĕƁĞƐ�ĚŽ�ƉƌŽďůĞŵĂ�ĚĞƐĐƌŝƚŽ�ŶŽ�ŝƚĞŵ�;ĂͿ͍�;ǀĂůŽƌ͗�ϳ͕Ϭ�ƉŽŶƚŽƐͿ
RASCUNHO
1
2
ϯ
4
5
6
7
ϴ
9
10
11
12
ϭϯ
14
15
*R33201411*
12
D�d�D�d/��
QUESTÃO 09
K� ĐŽŶũƵŶƚŽ� M2;ZͿ� Ġ� ĨŽƌŵĂĚŽ� ƉĞůĂƐ� ŵĂƚƌŝnjĞƐ�
ƋƵĂĚƌĂĚĂƐ� ĚĞ�ŽƌĚĞŵ�Ϯ� ĐŽŵ�ĞŶƚƌĂĚĂƐ� ŝŶƚĞŝƌĂƐ͘� �ƐƐĞ�
ĐŽŶũƵŶƚŽ�Ġ�ĨĞĐŚĂĚŽ�ƐŽď�ĂƐ�ŽƉĞƌĂĕƁĞƐ�ƵƐƵĂŝƐ�ĚĞ�ƐŽŵĂ�
Ğ�ŵƵůƟƉůŝĐĂĕĆŽ�ĚĞ�ŵĂƚƌŝnjĞƐ͕�ƵŵĂ�ǀĞnj�ƋƵĞ�ĂƐ�ĞŶƚƌĂĚĂƐ�
ĚĂƐ�ŵĂƚƌŝnjĞƐ�ƌĞƐƵůƚĂŶƚĞƐ�ĚĂ�ƐŽŵĂ�Ğ�ĚĂ�ŵƵůƟƉůŝĐĂĕĆŽ�
ƐĆŽ�ŶƷŵĞƌŽƐ�ŝŶƚĞŝƌŽƐ͘
�Žŵ� ƌĞůĂĕĆŽ� ă� ĞƐƚƌƵƚƵƌĂ� ĂůŐĠďƌŝĐĂ� ĚĞƐƐĞ� ĐŽŶũƵŶƚŽ�
ĐŽŵ� ĂƐ� ŽƉĞƌĂĕƁĞƐ� ĚĞƐĐƌŝƚĂƐ͕� ĂǀĂůŝĞ� ĂƐ� ƐĞŐƵŝŶƚĞƐ�
ĂƐƐĞƌĕƁĞƐ�Ğ�Ă�ƌĞůĂĕĆŽ�ƉƌŽƉŽƐƚĂ�ĞŶƚƌĞ�ĞůĂƐ͘
/͘�K� ĐŽŶũƵŶƚŽ�M2;ZͿ͕�ŵƵŶŝĚŽ�ĚĂƐ�ŽƉĞƌĂĕƁĞƐ�ƵƐƵĂŝƐ�
ĚĞ�ƐŽŵĂ�Ğ�ŵƵůƟƉůŝĐĂĕĆŽ͕�ĨŽƌŵĂ�Ƶŵ�ĂŶĞů͘
PORQUE
//͘� K� ĐŽŶũƵŶƚŽ�M2;ZͿ͕� ŵƵŶŝĚŽ� ĚĂ� ŽƉĞƌĂĕĆŽ� ƵƐƵĂů�
ĚĞ� ƐŽŵĂ� ĚĞ� ŵĂƚƌŝnjĞƐ͕� ĨŽƌŵĂ� Ƶŵ� ŐƌƵƉŽ� Ğ� ĞdžŝƐƚĞ� Ž�
ĞůĞŵĞŶƚŽ� ƵŶŝĚĂĚĞ� ĚĂĚŽ� ƉĞůĂ�ŵĂƚƌŝnj� ŝĚĞŶƟĚĂĚĞ� ĚĞ�
ŽƌĚĞŵ�Ϯ͘
�� ƌĞƐƉĞŝƚŽ� ĚĞƐƐĂƐ� ĂƐƐĞƌĕƁĞƐ͕� ĂƐƐŝŶĂůĞ� Ă� ĂůƚĞƌŶĂƟǀĂ�
ĐŽƌƌĞƚĂ͘
A �Ɛ�ĂƐƐĞƌĕƁĞƐ�/�Ğ�//�ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ�ǀĞƌĚĂĚĞŝƌĂƐ͕�Ğ�
Ă�//�Ġ�ƵŵĂ�ũƵƐƟĮĐĂƟǀĂ�ĐŽƌƌĞƚĂ�ĚĂ�/͘
B �Ɛ�ĂƐƐĞƌĕƁĞƐ� /� Ğ� //� ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ� ǀĞƌĚĂĚĞŝƌĂƐ͕�
ŵĂƐ�Ă�//�ŶĆŽ�Ġ�ƵŵĂ�ũƵƐƟĮĐĂƟǀĂ�ĐŽƌƌĞƚĂ�ĚĂ�/͘
C ��ĂƐƐĞƌĕĆŽ�/�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ǀĞƌĚĂĚĞŝƌĂ͕�Ğ�Ă�//�Ġ�
ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ĨĂůƐĂ͘
D ��ĂƐƐĞƌĕĆŽ�/�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ĨĂůƐĂ͕�Ğ�Ă�//�Ġ�ƵŵĂ�
ƉƌŽƉŽƐŝĕĆŽ�ǀĞƌĚĂĚĞŝƌĂ͘
E �Ɛ�ĂƐƐĞƌĕƁĞƐ�/�Ğ�//�ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ�ĨĂůƐĂƐ͘
ÁREA LIVRE
QUESTÃO 10
EŽ�ĐŽŶƚĞdžƚŽ�ĚĞ�ŝŶǀĞƐƟŵĞŶƚŽ�Ğ�ĨŽƌŵĂĕĆŽ�ĚĞ�ĐĂƉŝƚĂů͕�
ƐĞ� M(t)� ƌĞƉƌĞƐĞŶƚĂ� Ž� ŵŽŶƚĂŶƚĞ� ĚŽ� ĐĂƉŝƚĂů� ĚĞ�
ƵŵĂ� ĞŵƉƌĞƐĂ� ĞdžŝƐƚĞŶƚĞ� Ğŵ� ĐĂĚĂ� ŝŶƐƚĂŶƚĞ� t e I(t)
ƌĞƉƌĞƐĞŶƚĂ� Ă� ƚĂdžĂ� ĚĞ� ŝŶǀĞƐƟŵĞŶƚŽ� ůşƋƵŝĚŽ� ƉŽƌ� Ƶŵ�
ƉĞƌşŽĚŽ�ĚĞ�ƚĞŵƉŽ͕�ĞŶƚĆŽ
M =
∫ b
a
I(t) dt
ĨŽƌŶĞĐĞ�Ž�ŵŽŶƚĂŶƚĞ�ĂĐƵŵƵůĂĚŽ�ŶŽ�ƉĞƌşŽĚŽ��a d t d b�
�ŝƐƉŽŶşǀĞů�Ğŵ͗�фŚƩƉ͗ͬͬǁǁǁ͘ŝŵĞ͘ƵĞƌũ͘ďƌх͘��ĐĞƐƐŽ�Ğŵ͗�ϯ�ĂŐŽ͘�ϮϬϭϰ�
;ĂĚĂƉƚĂĚŽͿ͘
�ŽŶƐŝĚĞƌĞ�ƋƵĞ�Ă� ĨƵŶĕĆŽ� I(t) = t ln(t)� ĚĞĮŶŝĚĂ�ƉĂƌĂ�
t t 1͕�ƌĞƉƌĞƐĞŶƚĂ�Ă�ƚĂdžĂ�ĚĞ�ŝŶǀĞƐƟŵĞŶƚŽ�ůşƋƵŝĚŽ͕�Ğŵ�
ŵŝůŚĂƌĞƐ�ĚĞ�ƌĞĂŝƐ͕�ĚĞ�ƵŵĂ�ĞŵƉƌĞƐĂ�ĚĞ�ĐŽƐŵĠƟĐŽƐ͘�
EĞƐƐĞ� ĐĂƐŽ͕� ƵƟůŝnjĂŶĚŽ� ln(3) ∼= 1, 1 ͕� Ž� ǀĂůŽƌ� ĚŽ�
ŵŽŶƚĂŶƚĞ�ĂĐƵŵƵůĂĚŽ�ŶŽ�ƉĞƌşŽĚŽ�1 d t d 3�Ġ�ŝŐƵĂů�Ă
A ZΨ�ϭ�ϭϬϬ͕ϬϬ͘
B ZΨ�Ϯ�ϭϬϬ͕ϬϬ͘
C ZΨ�Ϯ�ϵϱϬ͕ϬϬ͘
D ZΨ�ϯ�ϳϱϬ͕ϬϬ͘
E ZΨ�ϰ�ϵϱϬ͕ϬϬ͘
ÁREA LIVRE
*R33201412*
ϭϯ
D�d�D�d/��
QUESTÃO 11
�ŵ� ƵŵĂ� ůŽũĂ� ĚĞ� ŵĂƚĞƌŝĂů� ĞƐĐŽůĂƌ͕ � ĂƐ� ŵĞƌĐĂĚŽƌŝĂƐ�
ĐĂŶĞƚĂ͕� ůĄƉŝƐ� Ğ� ďŽƌƌĂĐŚĂ͕� ĚĞ� Ƶŵ� ƷŶŝĐŽ� ƟƉŽ� ĐĂĚĂ�
ƵŵĂ͕�ƐĆŽ�ǀĞŶĚŝĚĂƐ�ƉĂƌĂ�ƚƌġƐ�ĞƐƚƵĚĂŶƚĞƐ͘�K�ƉƌŝŵĞŝƌŽ�
ĐŽŵƉƌŽƵ� ƵŵĂ� ĐĂŶĞƚĂ͕� ƚƌġƐ� ůĄƉŝƐ� Ğ� ĚƵĂƐ� ďŽƌƌĂĐŚĂƐ�
ƉĂŐĂŶĚŽ�ZΨ�ϭϬ͕ϬϬ͖�Ž�ƐĞŐƵŶĚŽ�ĂĚƋƵŝƌŝƵ�ĚƵĂƐ�ĐĂŶĞƚĂƐ͕�
Ƶŵ�ůĄƉŝƐ�Ğ�ƵŵĂ�ďŽƌƌĂĐŚĂ�ƉĂŐĂŶĚŽ�ZΨ�ϵ͕ϬϬ͖�Ž�ƚĞƌĐĞŝƌŽ�
ĐŽŵƉƌŽƵ�ƚƌġƐ�ĐĂŶĞƚĂƐ͕�ƋƵĂƚƌŽ�ůĄƉŝƐ�Ğ�ƚƌġƐ�ďŽƌƌĂĐŚĂƐ�
ƉĂŐĂŶĚŽ�ZΨ�ϭϵ͕ϬϬ͘
KƐ� ĞƐƚƵĚĂŶƚĞƐ͕� ĂƉſƐ� ĂƐ� ĐŽŵƉƌĂƐ͕� ƐĞŵ� ǀĞƌŝĮĐĂƌĞŵ�
ŽƐ�ǀĂůŽƌĞƐ�ĚĞ�ĐĂĚĂ�ŵĞƌĐĂĚŽƌŝĂ͕�ƉƌŽĐƵƌĂƌĂŵ�ƌĞƐŽůǀĞƌ�
Ž�ƉƌŽďůĞŵĂ͗�͞��ƉĂƌƟƌ�ĚĂƐ�ĐŽŵƉƌĂƐ�ĞĨĞƚƵĂĚĂƐ�Ğ�ĚŽƐ�
ƌĞƐƉĞĐƟǀŽƐ� ǀĂůŽƌĞƐ� ƚŽƚĂŝƐ� ƉĂŐŽƐ� ƉŽƌ� ĞůĞƐ͕� ƋƵĂů� Ž�
ƉƌĞĕŽ�ĚĂ�ĐĂŶĞƚĂ͕�ĚŽ�ůĄƉŝƐ�Ğ�ĚĂ�ďŽƌƌĂĐŚĂ͍͘͟ �WĂƌĂ�ŝƐƐŽ͕�
ŵŽŶƚĂƌĂŵ� Ƶŵ� ƐŝƐƚĞŵĂ� ĚĞ� ĞƋƵĂĕƁĞƐ� ůŝŶĞĂƌĞƐ� ĐƵũĂƐ�
ŝŶĐſŐŶŝƚĂƐ�ƐĆŽ�ŽƐ�ƉƌĞĕŽƐ�ĚĂƐ�ŵĞƌĐĂĚŽƌŝĂƐ͘�
�ƐƐĞ�ƐŝƐƚĞŵĂ�ĚĞ�ĞƋƵĂĕƁĞƐ�Ġ�
A ƉŽƐƐşǀĞů�ĚĞƚĞƌŵŝŶĂĚŽ͕�ƐĞŶĚŽ�Ž�ƉƌĞĕŽ�ĚĂ�ďŽƌƌĂĐŚĂ�
ŵĂŝƐ�ĐĂƌŽ�ƋƵĞ�Ž�ĚŽ�ůĄƉŝƐ͘�
B ŝŵƉŽƐƐşǀĞů͕�ƉŽŝƐ�ƐĂďĞƌ�ŽƐ�ƚŽƚĂŝƐ�ĚĂƐ�ĐŽŵƉƌĂƐ�ŶĆŽ�
ŐĂƌĂŶƚĞ�Ă�ĞdžŝƐƚġŶĐŝĂ�ĚĞ�ƐŽůƵĕĆŽ͘�
C ƉŽƐƐşǀĞů� ĚĞƚĞƌŵŝŶĂĚŽ͕� ƉŽĚĞŶĚŽ� ĂĚŵŝƟƌ� ĐŽŵŽ�
ƐŽůƵĕĆŽ�Ž�ǀĂůŽƌ�ĚŽ�ƉƌĞĕŽ�ĚĂ�ĐĂŶĞƚĂ͕�ĚŽ�ůĄƉŝƐ�Ğ�ĚĂ�
ďŽƌƌĂĐŚĂ͘
D ƉŽƐƐşǀĞů� ŝŶĚĞƚĞƌŵŝŶĂĚŽ͕� ĚĞ� ĨŽƌŵĂ� ƋƵĞ� Ă� ƐŽŵĂ�
ĚŽƐ� ǀĂůŽƌĞƐ� ƉŽƐƐşǀĞŝƐ� ĚĂ� ĐĂŶĞƚĂ͕� ĚŽ� ůĄƉŝƐ� Ğ� ĚĂ�
ďŽƌƌĂĐŚĂ� Ġ� ŝŐƵĂů� Ă� ĐŝŶĐŽ� ǀĞnjĞƐ� Ž� ƉƌĞĕŽ� ĚŽ� ůĄƉŝƐ�
ƐƵďƚƌú̎�ĚĞ�ZΨ�ϵ͕ϬϬ͘�
E ƉŽƐƐşǀĞů�ŝŶĚĞƚĞƌŵŝŶĂĚŽ͕�ĚĞ�ĨŽƌŵĂ�ƋƵĞ�Ă�ƐŽŵĂ�ĚŽƐ�
ǀĂůŽƌĞƐ�ƉŽƐƐşǀĞŝƐ�ĚĂ�ĐĂŶĞƚĂ͕�ĚŽ�ůĄƉŝƐ�Ğ�ĚĂ�ďŽƌƌĂĐŚĂ�
Ġ�ŝŐƵĂů�Ă�ϭͬϱ�ĚĂ�ĂĚŝĕĆŽ�ĚŽ�ƉƌĞĕŽ�ĚĂ�ďŽƌƌĂĐŚĂ�ĐŽŵ�
ZΨ�Ϯϴ͕ϬϬ͘�
ÁREA LIVRE
QUESTÃO 12
�ĞƐĞũĂͲƐĞ�ƉŝŶƚĂƌ� Ă� ƐƵƉĞƌĨşĐŝĞ� ĞdžƚĞƌŶĂ�Ğ� ůĂƚĞƌĂů� ĚĞ�
Ƶŵ�ŵŽŶƵŵĞŶƚŽ�Ğŵ�ĨŽƌŵĂ�ĚĞ�Ƶŵ�ƉĂƌĂďŽůŽŝĚĞ͕�ƋƵĞ�
ƉŽĚĞ�ƐĞƌ�ĚĞƐĐƌŝƚĂ�ƉĞůĂ�ĞƋƵĂĕĆŽ�z = x 2 + y 2͕�ƐŝƚƵĂĚĂ�
ŶĂ� ƌĞŐŝĆŽ� ĚŽ� ĞƐƉĂĕŽ� ĚĞ� ĐŽŽƌĚĞŶĂĚĂƐ� ĐĂƌƚĞƐŝĂŶĂƐ�
(x, y, z)� ĚĂĚĂ� ƉĞůĂ� ĐŽŶĚŝĕĆŽ� z d 9͘� KƐ� ĞŝdžŽƐ�
ĐŽŽƌĚĞŶĂĚŽƐ� ĞƐƚĆŽ� ĚŝŵĞŶƐŝŽŶĂĚŽƐ� Ğŵ� ŵĞƚƌŽƐ�
Ğ�ŐĂƐƚĂͲƐĞ�Ƶŵ� ůŝƚƌŽ�Ğ�ŵĞŝŽ�ĚĞ� ƚŝŶƚĂ�Ă� ĐĂĚĂ�ŵĞƚƌŽ�
ƋƵĂĚƌĂĚŽ�ĚĞ�ĄƌĞĂ�ĚĂ�ƐƵƉĞƌĨşĐŝĞ�Ă�ƐĞƌ�ƉŝŶƚĂĚĂ͘
��ƋƵĂŶƟĚĂĚĞ�ĚĞ�ƟŶƚĂ͕�Ğŵ�ůŝƚƌŽƐ͕�ŶĞĐĞƐƐĄƌŝĂ�ƉĂƌĂ�ƐĞ�
ƉŝŶƚĂƌ�Ă�ƐƵƉĞƌİĐŝĞ�ůĂƚĞƌĂů�ĚŽ�ŵŽŶƵŵĞŶƚŽ�Ġ�ĚĂĚĂ�ƉĞůĂ�
ŝŶƚĞŐƌĂů�ĚƵƉůĂ
A 4
∫ 3
0
∫ √9−x2
0
(x2 + y2)dxdy
.
B 6
∫ 3
0
∫ √9−x2
0
(x2 + y2)dxdy
.
C 4
∫ π/2
0
∫ 3
0
√
1 + 4r2 rdrdθ
.
D 6
∫ π/2
0
∫ 3
0
√
1 + 4r2 rdrdθ
.
E 6
∫ π/2
0
∫ 3
−3
√
1 + 4r2 rdrdθ
.
ÁREA LIVRE
*R33201413*
14
D�d�D�d/��
QUESTÃO 13
DƵŝƚŽƐ�ĨĞŶƀŵĞŶŽƐ�ƉƌŽďĂďŝůşƐƟĐŽƐ�ƐĞŐƵĞŵ�ƵŵĂ�ůĞŝ�ĚĞ�ĚŝƐƚƌŝďƵŝĕĆŽ�ĚĞŶŽŵŝŶĂĚĂ�EŽƌŵĂů͕�ŶĂ�ƋƵĂů�ŽƐ�ǀĂůŽƌĞƐ�
ŵĂŝƐ�ĨƌĞƋƵĞŶƚĞƐ�ƐĞ�ĞŶĐŽŶƚƌĂŵ�ƉƌſdžŝŵŽƐ�ă�ŵĠĚŝĂ͘���ĐƵƌǀĂ�ƋƵĞ�ƌĞƉƌĞƐĞŶƚĂ�ĞƐƐĂ�ĚŝƐƚƌŝďƵŝĕĆŽ�ƚĞŵ�Ă�ĨŽƌŵĂ�ĚĞ�Ƶŵ�
ƐŝŶŽ͕�Ġ�ƐŝŵĠƚƌŝĐĂ�Ğŵ�ƚŽƌŶŽ�ĚĂ�ŵĠĚŝĂ�P͕�ƚĞŵ�ŶŽ�ĞŝdžŽ�ĚĂƐ�ĂďƐĐŝƐƐĂƐ�ƵŵĂ�ĂƐƐşŶƚŽƚĂ�ŚŽƌŝnjŽŶƚĂů�Ğ�Ġ�ĚĞƚĞƌŵŝŶĂĚĂ�
ƉĞůĂ�ƐĞŐƵŝŶƚĞ�ĨƵŶĕĆŽ�ĚĞ�ĚĞŶƐŝĚĂĚĞ͗
f(x) =
1√
2πσ2
e−
1
2(
x−μ
σ )
2
>ĞǀĂŶĚŽ�Ğŵ�ĐŽŶƐŝĚĞƌĂĕĆŽ�ƋƵĞ�ĐĂĚĂ�ĐƵƌǀĂ�ĚĞ�ĚŝƐƚƌŝďƵŝĕĆŽ�EŽƌŵĂů�Ġ�ĚĞƚĞƌŵŝŶĂĚĂ�ƉĞůĂ�ƐƵĂ�ŵĠĚŝĂ�P�Ğ�ƉĞůŽ�ƐĞƵ�
ĚĞƐǀŝŽͲƉĂĚƌĆŽ�V͕�'ĂƵƐƐ�ĚĞƐĞŶǀŽůǀĞƵ�ƵŵĂ�ĨŽƌŵĂ�ĚĞ�ƉĂĚƌŽŶŝnjĄͲůĂƐ�Ğŵ�ƵŵĂ�ƷŶŝĐĂ�EŽƌŵĂů͕�ĐĂƌĂĐƚĞƌŝnjĂĚĂ�ƉŽƌ�
ƚĞƌ�ŵĠĚŝĂ�Ϭ�Ğ�ĚĞƐǀŝŽͲƉĂĚƌĆŽ�ϭ͘��ƐƐŝŵ͕�Ă�EŽƌŵĂů�WĂĚƌĆŽ�Ġ�ĚĞƚĞƌŵŝŶĂĚĂ�ƉĞůĂ�ĨƵŶĕĆŽ�f(z) =
1√
2π
e−
z2
2 ͕
ŶĂ�ƋƵĂů�ĐĂĚĂ�Ƶŵ�ĚŽƐ�ǀĂůŽƌĞƐ�x�ĚĂ�ĨƵŶĕĆŽ�ĚĞ�ĚŝƐƚƌŝďƵŝĕĆŽ�EŽƌŵĂů�N;P͕�VͿĠ�ĐŽŶǀĞƌƟĚŽ�Ğŵ�ƵŵĂ�ŶŽǀĂ�ǀĂƌŝĄǀĞů�
ĂĚŝŵĞŶƐŝŽŶĂů͕�ĚĞƐŝŐŶĂĚĂ�ŐĞŶĞƌŝĐĂŵĞŶƚĞ�ƉŽƌ�z͕�Ă�ƋƵĂů�ƚĞŵ�ĚŝƐƚƌŝďƵŝĕĆŽ�EŽƌŵĂů�N(0, 1)͘���ĐŽŶǀĞƌƐĆŽ�ĚĞƐƐĂ�
ǀĂƌŝĄǀĞů�ƐĞ�ĚĄ�ƉŽƌ�ŵĞŝŽ�ĚĂ�ƐĞŐƵŝŶƚĞ�ĞdžƉƌĞƐƐĆŽ͗�z =
x− μ
σ
.
^ĂďĞͲƐĞ�ƋƵĞ�Ă�ĄƌĞĂ�ƐŽď�Ž�ŐƌĄĮĐŽ�ĚĂ�ĨƵŶĕĆŽ�ĚĞ�ĚĞŶƐŝĚĂĚĞ�ĚĞ�ƉƌŽďĂďŝůŝĚĂĚĞ�Ğŵ�ĚĞƚĞƌŵŝŶĂĚŽ�ŝŶƚĞƌǀĂůŽ�ĨŽƌŶĞĐĞ�
Ă�ƉƌŽďĂďŝůŝĚĂĚĞ�ĚĞ�ŽĐŽƌƌġŶĐŝĂ�ĚĞ�Ƶŵ�ǀĂůŽƌ�ĚĞŶƚƌŽ�ĚĞƐƐĞ�ŝŶƚĞƌǀĂůŽ͘��ƐƐŝŵ͕�ĐŽŶƐŝĚĞƌĂͲƐĞ�ƋƵĞ�Ă�ĄƌĞĂ�ĞŶƚƌĞ�Ă�
ĐƵƌǀĂ�EŽƌŵĂů�Ğ�Ă�ĂƐƐşŶƚŽƚĂ�ĚĞƚĞƌŵŝŶĂĚĂ�ƉĞůŽ�ĞŝdžŽ�ĚĂƐ�ĂďƐĐŝƐƐĂƐ�Ġ�ŝŐƵĂů�Ă�ϭ͘
�Ğ�ĂĐŽƌĚŽ�ĐŽŵ�ĚĂĚŽƐ�ŽďƟĚŽƐ�ŶŽ�ƉŽƌƚĂů�ĚŽ�/E�WͬD���ƌĞůĂƟǀŽƐ�ĂŽƐ�ϭϭ�ϯϬϯ�ĞƐƚƵĚĂŶƚĞƐ�ĚĞ�>ŝĐĞŶĐŝĂƚƵƌĂ�Ğŵ�
DĂƚĞŵĄƟĐĂ�ƋƵĞ�ƌĞĂůŝnjĂƌĂŵ�Ă�ƉƌŽǀĂ�ĚŽ��ŶĂĚĞ�Ğŵ�ϮϬϭϭ͕�Ă�ŵĠĚŝĂ�Ğ�Ž�ĚĞƐǀŝŽͲƉĂĚƌĆŽ�ĚŽ�ĚĞƐĞŵƉĞŶŚŽ�ŐĞƌĂů�
ĚĞƐƐĞƐ�ĞƐƚƵĚĂŶƚĞƐ�ĨŽƌĂŵ͕�ƌĞƐƉĞĐƟǀĂŵĞŶƚĞ͕�ŝŐƵĂŝƐ�Ă�ϯϮ͕ϰ�Ğ�ϭϭ͕ϲ�ƉŽŶƚŽƐ͘�
�ŽŶƐŝĚĞƌĂŶĚŽ� ƋƵĞ� Ă� ĚŝƐƚƌŝďƵŝĕĆŽ� ĚŽ� ĚĞƐĞŵƉĞŶŚŽ� ĚĞƐƐĞƐ� ĂůƵŶŽƐ� ŶŽ� �E���� ϮϬϭϭ� ƉŽĚĞ� ƐĞƌ� ĂƉƌŽdžŝŵĂĚĂ�
ƉĞůĂ� ĚŝƐƚƌŝďƵŝĕĆŽ� EŽƌŵĂů͕� ĂƐƐŝŶĂůĞ� Ă� ĂůƚĞƌŶĂƟǀĂ� ĐƵũĂ� ĞdžƉƌĞƐƐĆŽ� ĨŽƌŶĞĐĞ� Ž� ƉĞƌĐĞŶƚƵĂů� ĚĞ� ĞƐƚƵĚĂŶƚĞƐ� ĐŽŵ�
ĚĞƐĞŵƉĞŶŚŽ�ŝŶĨĞƌŝŽƌ�Ă�ϮϬ͕ϴ�ƉŽŶƚŽƐ�ŽƵ�ƐƵƉĞƌŝŽƌ�Ă�ϱϱ͕ϲ�ƉŽŶƚŽƐ͘
A
∫ 2
−1
f(z)dz
.
B
∫ 55,6
20,8
f(z)dz
.
C 1−
∫ 2
−1
f(z)dz
.
D
∫ 32,4
11,6
f(z)dz
.
E 1−
(∫ −1
−∞
f(z)dz +
∫ ∞
2
f(z)dz
)
.
*R33201414*
15
D�d�D�d/��
QUESTÃO 14
hŵ�ƉƌŽďůĞŵĂ�ŵƵŝƚŽ�ĐŽŵƵŵ�Ğŵ�ŐĞŽŵĞƚƌŝĂ�Ġ�Ž�ĚĂƐ�
ƚƌĂũĞƚſƌŝĂƐ� ŽƌƚŽŐŽŶĂŝƐ͕� Ž� ƋƵĞ� ĞƋƵŝǀĂůĞ� Ă� ĚŝnjĞƌ� ƋƵĞ͕�
ĚĂĚĂ�ƵŵĂ�ĐƵƌǀĂ�ĚĞ�ƵŵĂ�ĨĂŵşůŝĂ͕�ĞůĂ�ŝŶƚĞƌĐĞƉƚĂ�ƵŵĂ�
ĐƵƌǀĂ�ĚĂ�ŽƵƚƌĂ�ĨĂŵşůŝĂ�ĚĞ�ŵŽĚŽ�ƋƵĞ�ƐƵĂƐ�ƚĂŶŐĞŶƚĞƐ�
ƐĆŽ�ƉĞƌƉĞŶĚŝĐƵůĂƌĞƐ�ĞŶƚƌĞ�Ɛŝ͕�ŶŽ�ƉŽŶƚŽ�ĚĞ�ŝŶƚĞƌƐĞĕĆŽ͘�
�ƐƐĞ� ƉƌŽďůĞŵĂ� ƉŽĚĞ� ƐĞƌ� ĂďŽƌĚĂĚŽ͕� ƚĂŵďĠŵ͕� ƉĞůŽ�
ĐĄůĐƵůŽ�ĚŝĨĞƌĞŶĐŝĂů� Ğ� ŝŶƚĞŐƌĂů� Ğ͕� ĐŽŶƐĞƋƵĞŶƚĞŵĞŶƚĞ͕�
ƉĞůĂƐ�ĞƋƵĂĕƁĞƐ�ĚŝĨĞƌĞŶĐŝĂŝƐ�ŽƌĚŝŶĄƌŝĂƐ͘
�Žŵ� Ž� ĂƵdžşůŝŽ� ĚĞƐƐĂƐ� ŝŶĨŽƌŵĂĕƁĞƐ͕� ĐŽŶĐůƵŝͲƐĞ� ƋƵĞ͕�
ƉĂƌĂ� c e k� ŶƷŵĞƌŽƐ� ƌĞĂŝƐ� ŶĆŽ� ŶƵůŽƐ͕� ŶŽ� ƉůĂŶŽ� ĚĞ�
ĐŽŽƌĚĞŶĂĚĂƐ�ĐĂƌƚĞƐŝĂŶĂƐ�xOy͕�Ă�ĨĂŵşůŝĂ�ĚĞ�ƚƌĂũĞƚſƌŝĂƐ�
ŽƌƚŽŐŽŶĂŝƐ�ă�ĨĂŵşůŝĂ�ĚĞ�ŚŝƉĠƌďŽůĞƐ�xy = c�Ġ�ĚĂĚĂ�ƉŽƌ
A x � y 2 = k.
B x 2 + y = k.
C x 2 � y = k.
D x 2 + y 2 = k.
E x 2 � y 2 = k.
ÁREA LIVRE
QUESTÃO 15
hŵĂ�ĨƵŶĕĆŽ�ĚŝĨĞƌĞŶĐŝĄǀĞů͕�f͕�ĐƌĞƐĐĞŶƚĞ�Ă�ƉĂƌƚŝƌ�ĚĂ�
ŽƌŝŐĞŵ�Ğ�ƐŝƚƵĂĚĂ�ŶŽ�ƉƌŝŵĞŝƌŽ�ƋƵĂĚƌĂŶƚĞ�Ġ�ƚĂů�ƋƵĞ�
Ă�ĄƌĞĂ�ĚĂ�ƌĞŐŝĆŽ�ƐŽď�ƐĞƵ�ŐƌĄĨŝĐŽ�Ğ�ĂĐŝŵĂ�ĚŽ�ĞŝdžŽ�
ĚĂƐ�ĂďƐĐŝƐƐĂƐ͕�ĚĞ�0�ĂƚĠ�x͕�ǀĂůĞ�Ƶŵ�ƋƵŝŶƚŽ�ĚĂ�ĄƌĞĂ�
ĚŽ�ƚƌŝąŶŐƵůŽ�ĐŽŵ�ǀĠƌƚŝĐĞƐ�ŶŽƐ�ƉŽŶƚŽƐ�(0͕ 0)͕ (x͕y)
e (x͕ 0)͕�Ğŵ�ƋƵĞ�y = f(x).
��ĞƋƵĂĕĆŽ�ĚŝĨĞƌĞŶĐŝĂů�ƋƵĞ�ĚĞƐĐƌĞǀĞ�ĞƐƐĂ�ƐŝƚƵĂĕĆŽ�Ġ
A xy' � 9y = x.
B xy' � 9y = 0.
C x2y' � 9y = 0.
D y' � 9xy = 0.
E y' � 9x2y = 0.
ÁREA LIVRE
*R33201415*
16
D�d�D�d/��
QUESTÃO 16
�ŽŶƐŝĚĞƌĞ�ƵŵĂ�ĨƵŶĕĆŽ�f�͗ �Ro R�ĚŝĨĞƌĞŶĐŝĄǀĞů�Ğŵ�ƚŽĚŽ�
Ž�ƐĞƵ�ĚŽŵşŶŝŽ͕�ĐŽŵ�f '(x) d x , � x�R.
^Ğ�f(1) = 1͕�ĞŶƚĆŽ͕�ƉĞůŽ�dĞŽƌĞŵĂ�ĚŽ�sĂůŽƌ�DĠĚŝŽ͕�Ž�
ǀĂůŽƌ�ŵĄdžŝŵŽ�ĚĞ�f(3)�Ġ�ŝŐƵĂů�Ă
A ϯ͘
B 5.
C 7.
D 9.
E 11.
QUESTÃO 17
�ŽŶƐŝĚĞƌĞ (xn)͕� n ͕� ƵŵĂ� ƐĞƋƵġŶĐŝĂ� ĚĞ� ŶƷŵĞƌŽƐ�
ƌĞĂŝƐ�ƉŽƐŝƟǀŽƐ�ƚĂů�ƋƵĞ�xn+1 =
xn
n + 1
EĞƐƐĞ�ĐĂƐŽ͕� lim xn
no+ь
�Ġ�ŝŐƵĂů�Ă
A +ь.
B 0.
C x1.
D 1.
E e.
ÁREA LIVRE
QUESTÃO 18
�ŵ�ƵŵĂ�ĨĞƐƚĂ�ŝŶĨĂŶƟů͕�Ƶŵ�ŐƌƵƉŽ�ĚĞ�ϳ�ĐƌŝĂŶĕĂƐ�ͷ��ŶĂ͕�
�ĞĂƚƌŝnj͕� �ĂƌůŽƐ͕��Ăǀŝ͕� �ĚƵĂƌĚŽ͕� &ĞƌŶĂŶĚĂ� Ğ�'ĂďƌŝĞůĂ�ͷ�
ƌĞƵŶŝƵͲƐĞ�ƉƌſdžŝŵŽ�Ă�ƵŵĂ�ŵĞƐĂ�ƉĂƌĂ�ďƌŝŶĐĂƌ�ĚĞ�͞ ĞƐĐŽŶĚĞͲ
ĞƐĐŽŶĚĞ͕͟ �Ƶŵ�ũŽŐŽ�ŶŽ�ƋƵĂů�ƵŵĂ�ĐƌŝĂŶĕĂ�Ġ�ƐĞƉĂƌĂĚĂ�ĚŽƐ�
ĚĞŵĂŝƐ͕� ƋƵĞ� ƉƌŽĐƵƌĂŵ� ůŽĐĂŝƐ� ƉĂƌĂ� ƐĞ� ĞƐĐŽŶĚĞƌ͕ � ƐĞŵ�
ƋƵĞ�Ă�ĞƐĐŽůŚŝĚĂ�ĂƐ�ǀĞũĂ͕�ƉŽŝƐ�ĞƐƐĂ� ƚĞŶƚĂƌĄ�ĞŶĐŽŶƚƌĄͲůĂƐ�
ĂƉſƐ� ĂůŐƵŵ� ƚĞŵƉŽ� ĞƐƚĂďĞůĞĐŝĚŽ� ƉƌĞǀŝĂŵĞŶƚĞ͘� �ƐƐŝŵ͕�
ĞƌĂ�ŶĞĐĞƐƐĄƌŝŽ�ĞƐĐŽůŚĞƌ�ƋƵĂů�ĚĞůĂƐ�ƐĞƌŝĂ�ĂƋƵĞůĂ�ƋƵĞ�ŝƌŝĂ�
ƉƌŽĐƵƌĂƌ�ƚŽĚĂƐ�ĂƐ�ŽƵƚƌĂƐ͘
WĂƌĂ�ĞĨĞƚƵĂƌ�ĞƐƐĂ�ĞƐĐŽůŚĂ͕�ĂƐ�ĐƌŝĂŶĕĂƐ�ƐĞ�ĚŝƐƉƵƐĞƌĂŵ�
Ğŵ� Ƶŵ� ĐşƌĐƵůŽ� ŶĂ� ŵĞƐŵĂ� ŽƌĚĞŵ� ĚĞƐĐƌŝƚĂ�
ĂŶƚĞƌŝŽƌŵĞŶƚĞ� Ğ͕� ƐŝŵƵůƚĂŶĞĂŵĞŶƚĞ͕� ŵŽƐƚƌĂƌĂŵ�
Ƶŵ� ŶƷŵĞƌŽ� ĚĞ� ĚĞĚŽƐ� ĚĂƐ� ŵĆŽƐ͘� KƐ� ŶƷŵĞƌŽƐ� ĚĞ�
ĚĞĚŽƐ� ŵŽƐƚƌĂĚŽƐ� ĨŽƌĂŵ� ƐŽŵĂĚŽƐ͕� ƌĞƐƵůƚĂŶĚŽ� Ğŵ�
ƵŵĂ�ƋƵĂŶƟĚĂĚĞ�ƋƵĞ�ǀĂŵŽƐ�ĐŚĂŵĂƌ�ĚĞ�dKd�>͘��ŶĂ�
ĐŽŵĞĕŽƵ�Ă�ĐŽŶƚĂƌ�ĚĞ�ϭ�ĂƚĠ�Ž�dKd�>͕�Ğ͕�Ă�ĐĂĚĂ�ŶƷŵĞƌŽ�
ĚŝƚŽ͕�ĂƉŽŶƚĂǀĂ�ƉĂƌĂ�ƵŵĂ�ĐƌŝĂŶĕĂ�ĚĂ�ƐĞŐƵŝŶƚĞ�ĨŽƌŵĂ͗�
ϭ�Ͳ��ŶĂ͕�Ϯ�Ͳ��ĞĂƚƌŝnj͕�ϯ�Ͳ��ĂƌůŽƐ͕�ϰ�Ͳ��Ăǀŝ͕�Ğ�ĂƐƐŝŵ�ƉŽƌ�
ĚŝĂŶƚĞ͘�YƵĂŶĚŽ�ĐŚĞŐĂƐƐĞ�ĂŽ�ŶƷŵĞƌŽ�dKd�>͕�Ă�ĐƌŝĂŶĕĂ�
ĐŽƌƌĞƐƉŽŶĚĞŶƚĞ�Ă�ĞƐƐĞ�ŶƷŵĞƌŽ�ƐĞƌŝĂ�ĂƋƵĞůĂ�ƋƵĞ�ŝƌŝĂ�
ƉƌŽĐƵƌĂƌ�ĂƐ�ĚĞŵĂŝƐ͘
^Ğ�Ž�ŶƷŵĞƌŽ�dKd�>�Ġ�ŝŐƵĂů�Ă�ϲϰ͕�Ă�ĐƌŝĂŶĕĂ�ĚĞƐŝŐŶĂĚĂ�
ƉĂƌĂ�ƉƌŽĐƵƌĂƌ�ĂƐ�ĚĞŵĂŝƐ�Ġ
A �ŶĂ͘
B �ĞĂƚƌŝnj͘
C �ĂƌůŽƐ͘
D �Ăǀŝ͘
E �ĚƵĂƌĚŽ͘
ÁREA LIVRE
*R33201416*
17
D�d�D�d/��
QUESTÃO 19
WĂƌĂ� ƌĞĂůŝnjĂƌ� ƐĞƵ� ƚƌĂďĂůŚŽ� ĐŽƟĚŝĂŶŽ͕� Ƶŵ�ĞŶŐĞŶŚĞŝƌŽ�
Đŝǀŝů� ƉƌĞĐŝƐĂ� ŵŽĚĞůĂƌ� ŵĂƚĞŵĂƟĐĂŵĞŶƚĞ� ĂůŐƵŵĂƐ�
ƚĂƌĞĨĂƐ͘� �ŵ� ĚĞƚĞƌŵŝŶĂĚŽ� ƉƌŽũĞƚŽ͕� ƵŵĂ� ƐŝƚƵĂĕĆŽ�
ƉƌŽďůĞŵĂ͕�ĚĞƉŽŝƐ�ĚĞ�ŵŽĚĞůĂĚĂ͕�ƌĞĐĂŝƵ�Ğŵ�Ƶŵ�ƐŝƐƚĞŵĂ�
ĚĞ�ĞƋƵĂĕƁĞƐ�ůŝŶĞĂƌĞƐ�ĐŽŵ�m�ĞƋƵĂĕƁĞƐ�Ğ�n ŝŶĐſŐŶŝƚĂƐ͕�
ƉĂƌĂ�Ž�ƋƵĂů�Ă�ŵĂƚƌŝnj�ĚŽƐ�ĐŽĞĮĐŝĞŶƚĞƐ�ĨŽŝ�ĚĞŶŽŵŝŶĂĚĂ�M.
�ƉſƐ� Ă� ŵŽĚĞůĂŐĞŵ͕� Ž� ĞŶŐĞŶŚĞŝƌŽ� ĚĞƐĐŽďƌŝƵ� ƋƵĞ� Ž�
ƉŽƐƚŽ�ĚĂ�ŵĂƚƌŝnj�ĂŵƉůŝĂĚĂ�ĚŽ�ƐŝƐƚĞŵĂ�(Pa)�ĞƌĂ�ŝŐƵĂů�ĂŽ�
ƉŽƐƚŽ�ĚĂ�ŵĂƚƌŝnj�ĚŽƐ�ĐŽĞĮĐŝĞŶƚĞƐ�(Pc)�Ğ�ƋƵĞ�ĂŵďŽƐ͕�
(Pa) e (Pc)͕� ƚġŵ� ǀĂůŽƌ� ĞƋƵŝǀĂůĞŶƚĞ� ĂŽ� ŶƷŵĞƌŽ� ĚĞ�
ŝŶĐſŐŶŝƚĂƐ�ĚŽ�ƐŝƐƚĞŵĂ͕�ŽƵ�ƐĞũĂ͕�Pa = Pc = n.
�ĚŵŝƟŶĚŽ�ƋƵĞ�Ž�ŵŽĚĞůŽ�ĐŽŶƐƚƌƵşĚŽ�ƉĞůŽ�ĞŶŐĞŶŚĞŝƌŽ�
ĞƐƚĄ�ŵĂƚĞŵĂƟĐĂŵĞŶƚĞ�ĐŽƌƌĞƚŽ͕�ĂǀĂůŝĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�
Ă�ƐĞŐƵŝƌ͘
/͘� ��ŵĂƚƌŝnj�M�Ġ�ƐŝŶŐƵůĂƌ͘
//͘� K� ƐŝƐƚĞŵĂ� ĚĞ� ĞƋƵĂĕƁĞƐ� ůŝŶĞĂƌĞƐ� ŵŽĚĞůĂĚŽ�
ĂĚŵŝƚĞ�ƵŵĂ�ƷŶŝĐĂ�ƐŽůƵĕĆŽ͘
///͘� ��ŝŵƉŽƐƐşǀĞů�ĞŶĐŽŶƚƌĂƌ�Ă�ƐŽůƵĕĆŽ�ĚŽ�ƉƌŽďůĞŵĂ�
ƵƟůŝnjĂŶĚŽ�Ž�ƐŝƐƚĞŵĂ�ĐŽŶĨŽƌŵĞ�ŵŽĚĞůĂĚŽ͘
/s͘ � K�ǀĂůŽƌ�ĚĞ�Pc�Ġ�ĐĂůĐƵůĂĚŽ�ŽďƚĞŶĚŽͲƐĞ�Ă�ŵĂŝŽƌ�
ŽƌĚĞŵ�ƉŽƐƐşǀĞů�ĚĂƐ�ƐƵďŵĂƚƌŝnjĞƐ�ƋƵĂĚƌĂĚĂƐ�ĚĞ�
M ƋƵĞ�ƚĞŶŚĂŵ�ĚĞƚĞƌŵŝŶĂŶƚĞƐ�ŶĆŽ�ŶƵůŽƐ͘
��ĐŽƌƌĞƚŽ�ĂƉĞŶĂƐ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /͘
B //͘
C /�Ğ�///͘
D //�Ğ�/s͘
E ///�Ğ�/s͘
ÁREA LIVRE
QUESTÃO 20
�ŽŶƐŝĚĞƌĞ�ƵŵĂ�ƉĂƌĄďŽůĂ�ĚĞ�ĨŽĐŽ�F�Ğ�ƌĞƚĂ�ĚŝƌĞƚƌŝnj�d.
�ĞŶŽƚĞ� ƉŽƌ�P Ƶŵ�ƉŽŶƚŽ� ƉĞƌƚĞŶĐĞŶƚĞ� ă� ƉĂƌĄďŽůĂ� Ğ�
ƉŽƌ�D�Ă�ƐƵĂ�ƉƌŽũĞĕĆŽ�ŽƌƚŽŐŽŶĂů�ŶĂ�ƌĞƚĂ�ĚŝƌĞƚƌŝnj�d.
ZĞƉƌĞƐĞŶƚĂĚŽ�ƉŽƌ�r�Ă�ƌĞƚĂ�ďŝƐƐĞƚƌŝnj�ĚŽ�ąŶŐƵůŽ�FP̂D͕
ĂǀĂůŝĞ� ĂƐ� ĂƐƐĞƌĕƁĞƐ� Ă� ƐĞŐƵŝƌ� Ğ� Ă� ƌĞůĂĕĆŽ� ƉƌŽƉŽƐƚĂ�
ĞŶƚƌĞ�ĞůĂƐ͘
/͘� ��ƌĞƚĂ�r�Ġ�ƚĂŶŐĞŶƚĞ�ă�ƉĂƌĄďŽůĂ�ŶŽ�ƉŽŶƚŽ�P.
PORQUE
//͘� WĂƌĂ� ƋƵĂůƋƵĞƌ� ƉŽŶƚŽ�Q� ƉĞƌƚĞŶĐĞŶƚĞ� ă� ƌĞƚĂ�
r͕ Q z P͕� Ă� ĚŝƐƚąŶĐŝĂ� ĚĞ�Q ĂŽ�ƉŽŶƚŽ�D� Ġ�
ŵĂŝŽƌ�ƋƵĞ�Ă�ĚŝƐƚąŶĐŝĂ�ĚĞ�Q ă�ƌĞƚĂ�d.
��ƌĞƐƉĞŝƚŽ�ĚĞƐƐĂƐ�ĂƐƐĞƌĕƁĞƐ͕�ĂƐƐŝŶĂůĞ�Ă�ŽƉĕĆŽ�ĐŽƌƌĞƚĂ͘
A �Ɛ�ĂƐƐĞƌĕƁĞƐ�/�Ğ�//�ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ�ǀĞƌĚĂĚĞŝƌĂƐ͕�Ğ�
Ă�//�Ġ�ƵŵĂ�ũƵƐƟĮĐĂƟǀĂ�ĐŽƌƌĞƚĂ�ĚĂ�/͘
B �Ɛ�ĂƐƐĞƌĕƁĞƐ� /� Ğ� //� ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ� ǀĞƌĚĂĚĞŝƌĂƐ͕�
ŵĂƐ�Ă�//�ŶĆŽ�Ġ�ƵŵĂ�ũƵƐƟĮĐĂƟǀĂ�ĐŽƌƌĞƚĂ�ĚĂ�/͘
C ��ĂƐƐĞƌĕĆŽ�/�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ǀĞƌĚĂĚĞŝƌĂ͕�Ğ�Ă�//�Ġ�
ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ĨĂůƐĂ͘
D ��ĂƐƐĞƌĕĆŽ�/�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ĨĂůƐĂ͕�Ğ�Ă�//�Ġ�ƵŵĂ�
ƉƌŽƉŽƐŝĕĆŽ�ǀĞƌĚĂĚĞŝƌĂ͘
E �Ɛ�ĂƐƐĞƌĕƁĞƐ�/�Ğ�//�ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ�ĨĂůƐĂƐ͘
ÁREA LIVRE
*R33201417*
ϭϴ
D�d�D�d/��
QUESTÃO 21
EŽ�ĞƐƚƵĚŽ�ĚĞ�ĨƵŶĕƁĞƐ�ĚĞ�ǀĂƌŝĄǀĞŝƐ�ƌĞĂŝƐ͕�ďƵƐĐĂŵͲƐĞ�
ŝŶĨŽƌŵĂĕƁĞƐ�ƐŽďƌĞ�ĐŽŶƟŶƵŝĚĂĚĞ͕�ĚŝĨĞƌĞŶĐŝĂďŝůŝĚĂĚĞ͕�
ĞŶƚƌĞ�ŽƵƚƌĂƐ͘��ŽŶƐŝĚĞƌĞ�ƵŵĂ�ĨƵŶĕĆŽ�ĚĞ�ĚƵĂƐ�ǀĂƌŝĄǀĞŝƐ�f�͗ �
R
2 o R͕�ĚĞĮŶŝĚĂ�ƉŽƌ
f(x, y) =
⎧⎪⎪⎨
⎪⎪⎩
x2y2
x4 + y4
, se (x, y) �= (0, 0)
0, se (x, y) = (0, 0).
��ƌĞƐƉĞŝƚŽ�ĚĞƐƐĂ�ĨƵŶĕĆŽ͕�ĂǀĂůŝĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�Ă�ƐĞŐƵŝƌ͘
/͘� �Ž�ůŽŶŐŽ�ĚĂƐ�ƌĞƚĂƐ�y = cx͕�Ž�ǀĂůŽƌ�ĚĂ�ĨƵŶĕĆŽ�
f�Ġ�ĐŽŶƐƚĂŶƚĞ͘
//͘� ��ĨƵŶĕĆŽ�f�Ġ�ĚĞƐĐŽŶơŶƵĂ�Ğŵ�(0, 0).
///͘� ��ĨƵŶĕĆŽ�f�ƐĂƟƐĨĂnj |f (x, y)| <
1
2
͕�ƋƵĂŝƐƋƵĞƌ�
ƋƵĞ�ƐĞũĂŵ�(x, y)  R2͕�ĐŽŵ�x z y.
��ĐŽƌƌĞƚŽ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A //͕�ĂƉĞŶĂƐ͘
B ///͕�ĂƉĞŶĂƐ͘
C /�Ğ�//͕�ĂƉĞŶĂƐ͘
D /�Ğ�///͕�ĂƉĞŶĂƐ͘
E /͕�//�Ğ�///͘
ÁREA LIVRE
QUESTÃO 22
hŵ�ĚŽƐ�ƉƌŽďůĞŵĂƐ�ŵĂŝƐ�ŝŵƉŽƌƚĂŶƚĞƐ�ĞƐƚƵĚĂĚŽƐ�ƉĞůŽ�
ĐĄůĐƵůŽ�ĚŝĨĞƌĞŶĐŝĂů�Ğ�ŝŶƚĞŐƌĂů�Ěŝnj�ƌĞƐƉĞŝƚŽ�ă�ŵĂdžŝŵŝnjĂĕĆŽ�
Ğ�ŵŝŶŝŵŝnjĂĕĆŽ�ĚĞ�ĨƵŶĕƁĞƐ͘�hŵ�ĚĞƐƐĞƐ�ƉƌŽďůĞŵĂƐ�ĞƐƚĄ�
ƌĞůĂĐŝŽŶĂĚŽ�ă�ĨƵŶĕĆŽ�ĐƷďŝĐĂ�ĚĞĮŶŝĚĂ�ƉŽƌ
f(x) = ax3 + bx2 + cx + d,
Ğŵ�ƋƵĞ�a͕ b͕ c e d�ƐĆŽ�ĐŽŶƐƚĂŶƚĞƐ�ƌĞĂŝƐ͕�ĐŽŵ�a z 0.
�ĐĞƌĐĂ�ĚĞƐƐĂ�ĐƷďŝĐĂ͕�ĂǀĂůŝĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�Ă�ƐĞŐƵŝƌ͘
/͘� �� ĨƵŶĕĆŽ� f� ƉŽƐƐƵŝ� ĂƉĞŶĂƐ� Ƶŵ� ƉŽŶƚŽ� ĚĞ�
ŝŶŇĞdžĆŽ͕� ŝŶĚĞƉĞŶĚĞŶƚĞŵĞŶƚĞ�ĚŽƐ�ǀĂůŽƌĞƐ�ĚĞ�
a͕ b͕ c e d.
//͘� ^Ğ�b2 � 3ac > 0͕�ĞŶƚĆŽ�f�ƉŽƐƐƵŝ�Ƶŵ�ƉŽŶƚŽ�ĚĞ�
ŵĄdžŝŵŽ�ůŽĐĂů�Ğ�Ƶŵ�ƉŽŶƚŽ�ĚĞ�ŵşŶŝŵŽ�ůŽĐĂů͘
///͘� ^Ğ� f� ƉŽƐƐƵŝ� Ƶŵ� ƉŽŶƚŽ� ĚĞ� ŵĄdžŝŵŽ� ůŽĐĂů� Ğ�
Ƶŵ� ƉŽŶƚŽ� ĚĞ� ŵşŶŝŵŽ� ůŽĐĂů͕� ĞŶƚĆŽ� Ă� ŵĠĚŝĂ�
ĂƌŝƚŵĠƟĐĂ� ĚĂƐ� ĂďƐĐŝƐƐĂƐ� ĚĞƐƐĞƐ� ĚŽŝƐ� ƉŽŶƚŽƐ�
ĞdžƚƌĞŵŽƐ�ĐŽƌƌĞƐƉŽŶĚĞ�ă�ĂďƐĐŝƐƐĂ�ĚŽ�ƉŽŶƚŽ�ĚĞ�
ŝŶŇĞdžĆŽ͘
��ĐŽƌƌĞƚŽ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /͕�ĂƉĞŶĂƐ͘
B //͕�ĂƉĞŶĂƐ͘
C /�Ğ�///͕�ĂƉĞŶĂƐ͘
D //�Ğ�///͕�ĂƉĞŶĂƐ͘
E /͕�//�Ğ�///͘
ÁREA LIVRE
*R33201418*
19
D�d�D�d/��
QUESTÃO 23
�ǀĂůŝĞ� ĂƐ� ƐĞŐƵŝŶƚĞƐ� ĂĮƌŵĂĕƁĞƐ� Ă� ƌĞƐƉĞŝƚŽ� ĚĂƐ�
ĞƐƚƌƵƚƵƌĂƐ�ĂůŐĠďƌŝĐĂƐ͘
/͘� ^Ğ� I � Ġ� Ƶŵ� ŝĚĞĂů� ĚĞ� Z ͕� ĞŶƚĆŽ��
I = mZ = {ma : a ∈ Z}͕� ƉĂƌĂ� ĂůŐƵŵ�
m ∈ Z .
//͘� K�ĐŽŶũƵŶƚŽ�ĚĂƐ�ĐůĂƐƐĞƐ�ĚĞ�ĞƋƵŝǀĂůġŶĐŝĂ�ŵſĚƵůŽ�
n͕�Zn = Z/nZ ͕�ƉĂƌĂ�n�ƉƌŝŵŽ͕�ŵƵŶŝĚŽ�ĚĂƐ�
ŽƉĞƌĂĕƁĞƐ�ƵƐƵĂŝƐ�ĚĞ�ĂĚŝĕĆŽ�Ğ�ŵƵůƟƉůŝĐĂĕĆŽ͕�
Ġ�Ƶŵ�ĐŽƌƉŽ͘
///͘� K� ĐŽŶũƵŶƚŽ� ĚŽƐ� ŶƷŵĞƌŽƐ� ĐŽŵƉůĞdžŽƐ� ĐŽŵ�
ƉĂƌƚĞ�ƌĞĂů�ŶƵůĂ͕�ŵƵŶŝĚŽ�ĚĂ�ŽƉĞƌĂĕĆŽ�ƵƐƵĂů�ĚĞ�
ĂĚŝĕĆŽ͕�ĨŽƌŵĂ�Ƶŵ�ŐƌƵƉŽ�ĂďĞůŝĂŶŽ͘
/s͘ � K�ĐŽŶũƵŶƚŽ�ĚĂƐ�ŵĂƚƌŝnjĞƐ�ƋƵĂĚƌĂĚĂƐ�ĚĞ�ŽƌĚĞŵ�
Ϯ�ĐŽŵ�ĞŶƚƌĂĚĂƐ�ƌĞĂŝƐ͕�ŵƵŶŝĚŽ�ĚĂƐ�ŽƉĞƌĂĕƁĞƐ�
ƵƐƵĂŝƐ� ĚĞ� ĂĚŝĕĆŽ� Ğ�ŵƵůƟƉůŝĐĂĕĆŽ͕� Ġ� Ƶŵ�ĂŶĞů�
ŶĆŽ�ĐŽŵƵƚĂƟǀŽ�ĐŽŵ�ƵŶŝĚĂĚĞ͘
��ĐŽƌƌĞƚŽ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ��Ğŵ
A /͕�ĂƉĞŶĂƐ͘
B //͕�ĂƉĞŶĂƐ͘
C /͕�///�Ğ�/s͕�ĂƉĞŶĂƐ͘
D //͕�///�Ğ�/s͕�ĂƉĞŶĂƐ͘
E /͕�//͕�///�Ğ�/s͘
ÁREA LIVRE
QUESTÃO 24
hŵ� ƉĞƐƋƵŝƐĂĚŽƌ� ŶĞĐĞƐƐŝƚĂ� ĚĂ� ƐŽůƵĕĆŽ� ĚĞ� ƵŵĂ�
ĞƋƵĂĕĆŽ�ĚŝĨĞƌĞŶĐŝĂů�ŽƌĚŝŶĄƌŝĂ�ƉĂƌĂ�ŝŵƉůĞŵĞŶƚĂƌ�Ğŵ�
ƐĞƵ�ĐſĚŝŐŽ�ĐŽŵƉƵƚĂĐŝŽŶĂů͘�ZĞƐŽůǀĞŶĚŽ�ƉŽƌ�ŵĞŝŽ�ĚĞ�
ƐĠƌŝĞ�ĚĞ�ƉŽƚġŶĐŝĂƐ͕�ĞůĞ�ĞŶĐŽŶƚƌĂ�Ă�ƐĞŐƵŝŶƚĞ�ƐŽůƵĕĆŽ͗
y =
∞∑
n=1
xn
n2
.
K� ƉĞƐƋƵŝƐĂĚŽƌ� ŶĞĐĞƐƐŝƚĂ� ƋƵĞ� Ă� ƐŽůƵĕĆŽ� ƐĞũĂ�
ĐŽŵƉƵƚĂĚĂ�ĐŽŵ�ƵŵĂ�ƉƌĞĐŝƐĆŽ�ĚĞ�0,01�ŶŽ�ŝŶƚĞƌǀĂůŽ�
�2 < x < 2͘� WĂƌĂ� ŝƐƐŽ͕� ĞůĞ� ĞƐĐƌĞǀĞ� ŽƐ� ƉƌŝŵĞŝƌŽƐ�
ϮϬϬ� ƚĞƌŵŽƐ� ĚĂ� ƐĠƌŝĞ� ŶŽ� ĐſĚŝŐŽ͘� EŽ� ĞŶƚĂŶƚŽ͕� ĂŽ�
ƌŽĚĂƌ�Ž�ƉƌŽŐƌĂŵĂ�ŶĞƐƐĞ� ŝŶƚĞƌǀĂůŽ͕�ĞůĞ�ƉĞƌĐĞďĞ�Ƶŵ�
ĐŽŵƉŽƌƚĂŵĞŶƚŽ�ĂŶƀŵĂůŽ͘
�ŽŶƐŝĚĞƌĂŶĚŽ�ƋƵĞ�Ž�ƉĞƐƋƵŝƐĂĚŽƌ�ŶĆŽ�ƚĞŶŚĂ�ĐŽŵĞƟĚŽ�
ĞƌƌŽ�ĂŽ�ŝŵƉůĞŵĞŶƚĂƌ�Ă�ƐĠƌŝĞ͕�ĂǀĂůŝĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�Ă�
ƐĞŐƵŝƌ͘
/͘� ��ƐĠƌŝĞ�ŶĆŽ�ƉŽĚĞ�ƌĞƉƌĞƐĞŶƚĂƌ�ƵŵĂ�ƐŽůƵĕĆŽ�
ŶŽ� ŝŶƚĞƌǀĂůŽ��2 < x < 2͕�ƉŽŝƐ�ĚŝǀĞƌŐĞ�ƉĂƌĂ�
| x | > 1.
//͘� ^ĆŽ�ŶĞĐĞƐƐĄƌŝŽƐ�ŵĂŝƐ�ƚĞƌŵŽƐ�ĚŽ�ƋƵĞ�ŽƐ�ƋƵĞ�Ž�
ƉĞƐƋƵŝƐĂĚŽƌ�ĞƐĐƌĞǀĞƵ�ƉĂƌĂ�ĂƟŶŐŝƌ�Ă�ƉƌĞĐŝƐĆŽ�
ƌĞƋƵĞƌŝĚĂ�ŶŽ�ŝŶƚĞƌǀĂůŽ��1 < x < 1.
///͘� �� ƉƌĞĐŝƐĆŽ� 0,01� ŶĆŽ� ƉŽĚĞ� ƐĞƌ� ĂƟŶŐŝĚĂ� ƉŽƌ�
ĞƐƐĂ�ƐĠƌŝĞ�ĚĞ�ƉŽƚġŶĐŝĂƐ͘
��ĐŽƌƌĞƚŽ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /͕�ĂƉĞŶĂƐ͘
B ///͕�ĂƉĞŶĂƐ͘
C /�Ğ�//͕�ĂƉĞŶĂƐ͘
D //�Ğ�///͕�ĂƉĞŶĂƐ͘
E /͕�//�Ğ�///͘
ÁREA LIVRE
*R33201419*
20
D�d�D�d/��
QUESTÃO 25
^Ğ� V� Ġ� Ƶŵ� ĞƐƉĂĕŽ� ǀĞƚŽƌŝĂů� ĐŽŵ� ƉƌŽĚƵƚŽ� ŝŶƚĞƌŶŽ�
ƐŽďƌĞ� Ž� ĐŽƌƉŽ� ĚŽƐ� ŶƷŵĞƌŽƐ� ĐŽŵƉůĞdžŽƐ͕� ĞŶƚĆŽ͕� Ă�
ĐĂĚĂ�ŽƉĞƌĂĚŽƌ� ůŝŶĞĂƌ�T : V o�V͕�ĞƐƚĄ�ĂƐƐŽĐŝĂĚŽ�Ž�
ŽƉĞƌĂĚŽƌ�ĂĚũƵŶƚŽ�ĚĞ�T͕�ĚĞŶŽƚĂĚŽ�ƉŽƌ�T* : V o�V͕�
ƋƵĞ�ƐĞ�ƌĞůĂĐŝŽŶĂ�ĐŽŵ�T�ƉŽƌ� ŝŶƚĞƌŵĠĚŝŽ�ĚŽ�ƉƌŽĚƵƚŽ�
ŝŶƚĞƌŶŽ�ĚĞĮŶŝĚŽ�Ğŵ�V͘�hƐĂŶĚŽ�ĞƐƐĞƐ�ĚŽŝƐ�ŽƉĞƌĂĚŽƌĞƐ͕�Ġ� ƉŽƐƐşǀĞů� ĚĞĮŶŝƌ� ĚŝǀĞƌƐĂƐ� ĐůĂƐƐĞƐ� ĚĞ� ŽƉĞƌĂĚŽƌĞƐ�
ůŝŶĞĂƌĞƐ�ƋƵĞ�ƉŽƐƐƵĞŵ�ĂƉůŝĐĂĕƁĞƐ�ŶĂ�DĂƚĞŵĄƟĐĂ�Ğ�ŶĂ�
&şƐŝĐĂ͕�ĞŶƚƌĞ�ĂƐ�ƋƵĂŝƐ�ƐĞ�ŝŶĐůƵĞŵ͗�
ͻ� ŽƉĞƌĂĚŽƌĞƐ� ĂƵƚŽĂĚũƵŶƚŽƐ� ƐĆŽ� ĂƋƵĞůĞƐ� ƚĂŝƐ� ƋƵĞ�
T* = T͖�
ͻ� ŽƉĞƌĂĚŽƌĞƐ�ĂŶƟͲĂƵƚŽĂĚũƵŶƚŽƐ�ƐĆŽ�ĂƋƵĞůĞƐ� ƚĂŝƐ�ƋƵĞ�
T* = �T͖�
ͻ� ŽƉĞƌĂĚŽƌĞƐ� ƵŶŝƚĄƌŝŽƐ� ƐĆŽ� ĂƋƵĞůĞƐ� ƚĂŝƐ� ƋƵĞ��
T* = T -1.
�ŽŶƐŝĚĞƌĂŶĚŽ� O� � C � Ƶŵ� ĂƵƚŽǀĂůŽƌ� ĐŽŵƉůĞdžŽ�
ƋƵĂůƋƵĞƌ� ĚŽ� ŽƉĞƌĂĚŽƌ� ůŝŶĞĂƌ� T e V� Ƶŵ� ĞƐƉĂĕŽ�
ǀĞƚŽƌŝĂů� ĐŽŵ� ƉƌŽĚƵƚŽ� ŝŶƚĞƌŶŽ͕� ƐŽďƌĞ� Ž� ĐŽƌƉŽ� C͕�
ĂǀĂůŝĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�Ă�ƐĞŐƵŝƌ͘
/͘� K� ŶƷŵĞƌŽ� ĐŽŵƉůĞdžŽ�O� ƚĂŵďĠŵ� Ġ� ĂƵƚŽǀĂůŽƌ�
ĚĞ�T*.
//͘� ^Ğ�T�Ġ�ĂƵƚŽĂĚũƵŶƚŽ͕�ĞŶƚĆŽ�O��R .
///͘� ^Ğ�T�Ġ�ĂŶƟͲĂƵƚŽĂĚũƵŶƚŽ͕�ĞŶƚĆŽ�O�Ġ�Ƶŵ�ŶƷŵĞƌŽ�
ŝŵĂŐŝŶĄƌŝŽ�ƉƵƌŽ͕�ŝƐƚŽ�Ġ͕�O�=�bi͕�ĐŽŵ�b�ŶƷŵĞƌŽ�
ƌĞĂů�ŶĆŽ�ŶƵůŽ͘
/s͘ � ^Ğ�T Ġ�ƵŶŝƚĄƌŝŽ͕�ĞŶƚĆŽ�O�= 1�ŽƵ�O�=��1 .
��ĐŽƌƌĞƚŽ�ĂƉĞŶĂƐ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /͘
B //͘
C /�Ğ�/s͘
D //�Ğ�///͘
E ///�Ğ�/s͘
ÁREA LIVRE
QUESTÃO 26
WĂƌĂ�ƐĞ�ĐƌŝĂƌ�Ƶŵ�ŵĂƉĂͲŵƷŶĚŝ͕�ŝƐƚŽ�Ġ͕�ƵŵĂ�ŵĂƉĂ�ĚĂ�
dĞƌƌĂ͕�Ġ�ƉƌĞĐŝƐŽ�ĨĂnjĞƌ�ƵŵĂ�ƉƌŽũĞĕĆŽ�ĐĂƌƚŽŐƌĄĮĐĂ͕�ƋƵĞ�
Ġ�ƵŵĂ�ĐŽƌƌĞƐƉŽŶĚġŶĐŝĂ�ĞŶƚƌĞ�ŽƐ�ƉŽŶƚŽƐ�ĚĂ�dĞƌƌĂ�;Ğŵ�
ŐĞƌĂů�ƵŵĂ�ĞƐĨĞƌĂͿ�Ğ�ŽƐ�ƉŽŶƚŽƐ�ĚŽ�ŵĂƉĂ�;Ƶŵ�ƉůĂŶŽͿ�ĚĞ�
ŵŽĚŽ�ƋƵĞ�ĂƐ�ĚŝƐƚąŶĐŝĂƐ�ƐĞũĂŵ�ƉƌĞƐĞƌǀĂĚĂƐ͘��džŝƐƚĞŵ�
ĚŝǀĞƌƐĂƐ�ƉƌŽũĞĕƁĞƐ�ĐĂƌƚŽŐƌĄĮĐĂƐ�ĚĂ�dĞƌƌĂ�ĚĞƐƟŶĂĚĂƐ�
ă�ĐŽŶĨĞĐĕĆŽ�ĚĞ�ŵĂƉĂƐͲŵƷŶĚŝ͕� ĨĞŝƚĂƐ�Ă�ƉĂƌƟƌ�ĚĞ�Ƶŵ�
ŵŽĚĞůŽ�ĞƐĨĠƌŝĐŽ�ĚĂ�dĞƌƌĂ͘
EĞƐƐĞ� ĐŽŶƚĞdžƚŽ͕� ĂǀĂůŝĞ� ĂƐ� ƐĞŐƵŝŶƚĞƐ� ĂƐƐĞƌĕƁĞƐ� Ğ� Ă�
ƌĞůĂĕĆŽ�ƉƌŽƉŽƐƚĂ�ĞŶƚƌĞ�ĞůĂƐ͘
/͘� EĆŽ�ĞdžŝƐƚĞ�ƵŵĂ�ƉƌŽũĞĕĆŽ�ĐĂƌƚŽŐƌĄĮĐĂ�ƉĞƌĨĞŝƚĂ�
ĚĞ� ƚŽĚĂ� Ă� dĞƌƌĂ͕� ŝƐƚŽ� Ġ͕� ƵŵĂ� ŝƐŽŵĞƚƌŝĂ� ƋƵĞ�
ĂƐƐŽĐŝĞ�Ă�ĐĂĚĂ�ƉŽŶƚŽ�ĚĞ�ƵŵĂ�ĞƐĨĞƌĂ�Ƶŵ�ƉŽŶƚŽ�
ŶŽ�ƉůĂŶŽ͘
PORQUE
//͘� ��ĐƵƌǀĂƚƵƌĂ�ŐĂƵƐƐŝĂŶĂ�ĚĂ�ĞƐĨĞƌĂ�Ġ�ƉŽƐŝƚŝǀĂ͕�
ĞŶƋƵĂŶƚŽ� Ă� ĐƵƌǀĂƚƵƌĂ� ŐĂƵƐƐŝĂŶĂ� ĚŽ� ƉůĂŶŽ�
Ġ�njĞƌŽ͘
��ƌĞƐƉĞŝƚŽ�ĚĞƐƐĂƐ�ĂƐƐĞƌĕƁĞƐ͕�ĂƐƐŝŶĂůĞ�Ă�ŽƉĕĆŽ�ĐŽƌƌĞƚĂ͘
A �Ɛ�ĂƐƐĞƌĕƁĞƐ�/�Ğ�//�ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ�ǀĞƌĚĂĚĞŝƌĂƐ͕�Ğ�
Ă�ƐĞŐƵŶĚĂ�Ġ�ƵŵĂ�ũƵƐƟĮĐĂƟǀĂ�ĐŽƌƌĞƚĂ�ĚĂ�ƉƌŝŵĞŝƌĂ͘
B �Ɛ�ĂƐƐĞƌĕƁĞƐ� /� Ğ� //� ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ� ǀĞƌĚĂĚĞŝƌĂƐ͕�
ŵĂƐ�Ă�ƐĞŐƵŶĚĂ�ŶĆŽ�Ġ�ƵŵĂ�ũƵƐƟĮĐĂƟǀĂ�ĐŽƌƌĞƚĂ�ĚĂ�
ƉƌŝŵĞŝƌĂ͘
C �� ĂƐƐĞƌĕĆŽ� /� Ġ� ƵŵĂ� ƉƌŽƉŽƐŝĕĆŽ� ǀĞƌĚĂĚĞŝƌĂ͕� Ğ� Ă�
ĂƐƐĞƌĕĆŽ�//�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ĨĂůƐĂ͘
D ��ĂƐƐĞƌĕĆŽ�/�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ĨĂůƐĂ͕�Ğ�Ă�ĂƐƐĞƌĕĆŽ�
//�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ǀĞƌĚĂĚĞŝƌĂ͘
E �Ɛ�ĂƐƐĞƌĕƁĞƐ�/�Ğ�//�ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ�ĨĂůƐĂƐ͘
ÁREA LIVRE
*R33201420*
21
D�d�D�d/��
QUESTÃO 27
�Ž�ƐĞ�ƚƌĂďĂůŚĂƌ�ĐŽŵ�ĨƵŶĕƁĞƐ�ĚĞ�ǀĂƌŝĄǀĞŝƐ�ĐŽŵƉůĞdžĂƐ�Ă�
ǀĂůŽƌĞƐ�ĐŽŵƉůĞdžŽƐ�Ġ�ƉŽƐƐşǀĞů͕�ƉŽƌ�ŵĞŝŽ�ĚĂƐ�ĞƋƵĂĕƁĞƐ�
ĚĞ� �ĂƵĐŚLJͲZŝĞŵĂŶŶ͕� ƵƟůŝnjĂƌ� ĂůŐƵŵĂƐ� ƉƌŽƉƌŝĞĚĂĚĞƐ�
ĞƐƚƵĚĂĚĂƐ�ŶŽ�ĐĄůĐƵůŽ�ĚŝĨĞƌĞŶĐŝĂů�ƉĂƌĂ�ĨƵŶĕƁĞƐ�ĚĞ�ĚƵĂƐ�
ǀĂƌŝĄǀĞŝƐ�ƌĞĂŝƐ�Ă�ǀĂůŽƌĞƐ�ƌĞĂŝƐ͘�WĂƌĂ�ŝƐƐŽ͕�ĐŽŶƐŝĚĞƌĞ�
U �C ��Ƶŵ�ƐƵďĐŽŶũƵŶƚŽ�ĂďĞƌƚŽ͕�ŶĆŽ�ǀĂnjŝŽ�Ğ�ĐŽŶĞdžŽ�
ĚŽ�ƉůĂŶŽ�ĐŽŵƉůĞdžŽ�C e f ͗�U �C �o�C �ƵŵĂ�ĨƵŶĕĆŽ�
ĚĞĮŶŝĚĂ�ƉŽƌ� f(x + iy) = u(x͕ y) + iv(x͕ y)͕�ƉĂƌĂ�
ƋƵĂŝƐƋƵĞƌ�x + iyU͕�ŽŶĚĞ�u͕�v ͗ D � R2 o��R ƐĆŽ�
ĨƵŶĕƁĞƐ�ƌĞĂŝƐ�ĚĞ�ǀĂƌŝĄǀĞŝƐ�ƌĞĂŝƐ�Ğ�D�Ġ�Ƶŵ�ƐƵďĐŽŶũƵŶƚŽ�
ĂďĞƌƚŽ�ĚĞ�R2.
��ƉĂƌƟƌ�ĚĞƐƐĂƐ� ŝŶĨŽƌŵĂĕƁĞƐ͕�ĂǀĂůŝĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�Ă�
ƐĞŐƵŝƌ͘
/͘� ^Ğ�f Ġ�ƵŵĂ�ĨƵŶĕĆŽ�ĂŶĂůşƟĐĂ�ƚĂů�ƋƵĞ�f(U) �R͕�
ĞŶƚĆŽ�f Ġ�ƵŵĂ�ĨƵŶĕĆŽ�ĐŽŶƐƚĂŶƚĞ͘
//͘� �^Ğ�f Ġ�ƵŵĂ�ĨƵŶĕĆŽ�ĂŶĂůşƟĐĂ�Ğ�Ă�ƐƵĂ�ĐŽŶũƵŐĂĚĂ�
f �͗�U�C �o�C ƚĂŵďĠŵ�Ġ�ĂŶĂůşƟĐĂ͕�ĞŶƚĆŽ�f Ġ�
ƵŵĂ�ĨƵŶĕĆŽ�ĐŽŶƐƚĂŶƚĞ͘
///͘� �^ Ğ�f Ġ�ƵŵĂ�ĨƵŶĕĆŽ�ĂŶĂůşƟĐĂ�Ğ�Ă�ĨƵŶĕĆŽ�ŵſĚƵůŽ�
ĚĞ�f͕�|f | : U �C�o�C͕�Ġ�ĐŽŶƐƚĂŶƚĞ͕�ĞŶƚĆŽ�f
ƚĂŵďĠŵ�Ġ�ƵŵĂ�ĨƵŶĕĆŽ�ĐŽŶƐƚĂŶƚĞ͘
��ĐŽƌƌĞƚŽ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /͕�ĂƉĞŶĂƐ͘
B //͕�ĂƉĞŶĂƐ͘
C /�Ğ�///͕�ĂƉĞŶĂƐ͘
D //�Ğ�///͕�ĂƉĞŶĂƐ͘
E /͕�//�Ğ�///͘
ÁREA LIVRE
QUESTÃO 28
�Žŵ� ĂƉůŝĐĂĕƁĞƐ� Ğŵ� ĞůĞƚƌŽƐƚĄƟĐĂ� Ğ� ŵĞĐąŶŝĐĂ͕� Ž�
dĞŽƌĞŵĂ�ĚĞ�'ƌĞĞŶ�ƌĞůĂĐŝŽŶĂ� ŝŶƚĞŐƌĂŝƐ�ĚĞ� ůŝŶŚĂ�ĐŽŵ�
ŝŶƚĞŐƌĂŝƐ� ĚƵƉůĂƐ͘� ^ĞƵ� ƌĞƐƵůƚĂĚŽ� ƉŽĚĞ� ƐĞƌ� ĞdžƉƌĞƐƐŽ�
ƉĞůĂ�ŝŐƵĂůĚĂĚĞ∫
r
(F · dr) =
∫ ∫
A
(
∂F2
∂x
− ∂F1
∂y
)
dA
.
EĞƐƐĞ�ĐŽŶƚĞdžƚŽ͕�ĂǀĂůŝĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�Ă�ƐĞŐƵŝƌ͘
/͘�� r� Ġ� ƵŵĂ� ĐƵƌǀĂ� ƉůĂŶĂ� ƐŝŵƉůĞƐ͕� ĨĞĐŚĂĚĂ͕�
ĚŝĨĞƌĞŶĐŝĄǀĞů�ŽƵ�ĚŝĨĞƌĞŶĐŝĄǀĞů�ƉŽƌ�ƉĂƌƚĞƐ�Ğ�A
Ġ�Ă�ƌĞŐŝĆŽ�ĚĞůŝŵŝƚĂĚĂ�ƉŽƌ�r.
//͘� F = (F1, F2)�Ġ�Ƶŵ�ĐĂŵƉŽ�ǀĞƚŽƌŝĂů�ĚĞ�ĐůĂƐƐĞC
1
ĚĞĮŶŝĚŽ� Ğŵ� Ƶŵ� ĐŽŶũƵŶƚŽ� ĂďĞƌƚŽ� ĚŽ� R2� ƋƵĞ�
ĐŽŶƚĠŵ�Ă�ƌĞŐŝĆŽ�A.
///͘� ∂F2
∂x
− ∂F1
∂y
� Ġ� ƵŵĂ�ĞdžƉƌĞƐƐĆŽ�ƋƵĞ� ƐĞ� ĂŶƵůĂ�
ƉĂƌĂ�ĐĂŵƉŽƐ�ǀĞƚŽƌŝĂŝƐ�ĐŽŶƐĞƌǀĂƟǀŽƐ͘
��ĐŽƌƌĞƚŽ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /͕�ĂƉĞŶĂƐ͘
B ///͕�ĂƉĞŶĂƐ͘
C /�Ğ�//͕�ĂƉĞŶĂƐ͘
D //�Ğ�///͕�ĂƉĞŶĂƐ͘
E /͕�//�Ğ�///͘
ÁREA LIVRE
*R33201421*
22
D�d�D�d/��
QUESTÃO 29
�ŵ� ƚŽƉŽůŽŐŝĂ� ĚŽƐ� ĞƐƉĂĕŽƐ� ŵĠƚƌŝĐŽƐ͕� Ġ� ĐŽŵƵŵ�
Ž� ĞƐƚƵĚŽ� ĚĞ� ƉƌŽƉƌŝĞĚĂĚĞƐ� ƌĞůĂƟǀĂƐ� Ă� ĐŽŶũƵŶƚŽƐ�
ĂďĞƌƚŽƐ͕� ĨĞĐŚĂĚŽƐ͕� ĐŽŵƉĂĐƚŽƐ͕� ĐŽŶĞdžŽƐ͕� ĐŽŶĞdžŽƐ�
ƉŽƌ� ĐĂŵŝŶŚŽƐ͕� ĞŶƚƌĞ� ŽƵƚƌĂƐ͘� �ƐƐĂƐ� ĐĂƌĂĐƚĞƌşƐŝƚĐĂƐ�
ƉĞƌŵŝƚĞŵ�ĐŽŵƉĂƌĂƌ�ĞƐƉĂĕŽƐ�ŵĠƚƌŝĐŽƐ�Ğ�ǀĞƌŝĮĐĂƌ�ƐĞ�
ƐĆŽ�ƚŽƉŽůŽŐŝĐĂŵĞŶƚĞ�ĞƋƵŝǀĂůĞŶƚĞƐ͘���ƌĞƐƉĞŝƚŽ�ĚŝƐƐŽ͕�
ĂǀĂůŝĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�Ă�ƐĞŐƵŝƌ͘
/͘� ^Ğ� Ƶŵ� ĞƐƉĂĕŽ� ŵĠƚƌŝĐŽ�M� ƉŽĚĞ� ƐĞƌ� ĞƐĐƌŝƚŽ�
ĐŽŵŽ�ƵŶŝĆŽ�ĚĞ�ĚŽŝƐ�ĐŽŶũƵŶƚŽƐ�A eB͕�ĚŝƐũƵŶƚŽƐ͕�
ĂďĞƌƚŽƐ�Ğ�ĨĞĐŚĂĚŽƐ͕�ĞŶƚĆŽ�M ŶĆŽ�Ġ�ĐŽŶĞdžŽ͘
//͘� ^Ğ�M Ġ�Ƶŵ�ĞƐƉĂĕŽ�ŵĠƚƌŝĐŽ͕�ĞŶƚĆŽ�Ƶŵ�ĐŽŶũƵŶƚŽ�
K  M Ġ� ĐŽŵƉĂĐƚŽ� ƐĞ͕� Ğ� ƐŽŵĞŶƚĞ� ƐĞ͕�K Ġ�
ĨĞĐŚĂĚŽ�Ğ�ůŝŵŝƚĂĚŽ͘
///͘� ^Ğ� J = [0, 5[� Ġ� Ƶŵ� ŝŶƚĞƌǀĂůŽ� ĐŽŵ� Ă� ŵĠƚƌŝĐĂ�
ŝŶĚƵnjŝĚĂ�ĚĞ�R͕�ĞŶƚĆŽ�Ž�ŝŶƚĞƌǀĂůŽ�[0, 1[�Ġ�ĂďĞƌƚŽ�
Ğŵ�J.
��ĐŽƌƌĞƚŽ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /͕�ĂƉĞŶĂƐ͘
B ///͕�ĂƉĞŶĂƐ͘
C /�Ğ�//͕�ĂƉĞŶĂƐ͘
D //�Ğ�///͕�ĂƉĞŶĂƐ͘
E /͕�//�Ğ�///͘
ÁREA LIVRE
QUESTÃO 30
��ƌĞƐƉĞŝƚŽ�ĚŽ�ǀĂůŽƌ�ĚĂ��ŝŶƚĞŐƌĂů�
∫
γ
(z − a)n dz ͕�Ğŵ�
ƋƵĞ n�Ġ�Ƶŵ� ŝŶƚĞŝƌŽ͕�z�Ġ�ƵŵĂ�ǀĂƌŝĄǀĞů�ĐŽŵƉůĞdžĂ�Ğ�J͕�
ƵŵĂ� ĐƵƌǀĂ� ĨĞĐŚĂĚĂ� ŶŽ� ƉůĂŶŽ� ĐŽŵƉůĞdžŽ͕� ĂǀĂůŝĞ� ĂƐ�
ĂĮƌŵĂĕƁĞƐ�Ă�ƐĞŐƵŝƌ͘
/͘� �� ŶƵůŽ� ƉĂƌĂ� ƚŽĚŽ� n t 0� Ğ� ƋƵĂůƋƵĞƌ� ĐƵƌǀĂ�
ĨĞĐŚĂĚĂ�J͕�ƵŵĂ�ǀĞnj�ƋƵĞ�;z � aͿn�Ġ�Ă�ĚĞƌŝǀĂĚĂ�
ĚĞ�(z − a)n+1
n + 1
.
//͘� ��ŶƵůŽ�ƉĂƌĂ�ƚŽĚŽ�n < 0�͕�n z �1͕��Ğ�ƋƵĂůƋƵĞƌ�
ĐƵƌǀĂ�ĨĞĐŚĂĚĂ�J�ƋƵĞ�ŶĆŽ�ƉĂƐƐĞ�ƉŽƌ�a.
///͘� ��ŶƵůŽ͕�ƐĞ�n = �1 e J��ĨŽƌ�Ă�ĐŝƌĐƵŶĨĞƌġŶĐŝĂ�ĚĂĚĂ�
ƉŽƌ�J;tͿ� a + reit �͕0 d t d 2S.
/s͘� �� ŝŐƵĂů�Ă�2Si͕� ƐĞ�n = �1 e J�ĨŽƌ�ƵŵĂ�ĐƵƌǀĂ�
ĨĞĐŚĂĚĂ� ĐŽŶƚŝĚĂ� ŶŽ� ƐĞŵŝͲĞƐƉĂĕŽ� ĚŽ� ƉůĂŶŽ͕�
Imz > 0͕� Ğŵ� ƋƵĞ� a� ŶĆŽ� ƉĞƌƚĞŶĕĂ� Ă� ĞƐƐĞ�
ƐĞŵŝͲĞƐƉĂĕŽ͘
��ĐŽƌƌĞƚŽ�ĂƉĞŶĂƐ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A ///͘
B /s͘
C /�Ğ�//͘
D /�Ğ�///͘
E //�Ğ�/s͘
ÁREA LIVRE
*R33201422*
Ϯϯ
D�d�D�d/��
QUESTÃO 31
��ŝŶƚĞŐƌĂĕĆŽ�ĚĞ�ĨƵŶĕƁĞƐ�Ġ�Ƶŵ�ĚŽƐ�ƉƌŝŶĐŝƉĂŝƐ�ƚſƉŝĐŽƐ�ĚĂ��ŶĄůŝƐĞ�ĞƐƐĞŶĐŝĂůŵĞŶƚĞ�ƚĞſƌŝĐĂ͕�ĞŶƋƵĂŶƚŽ�ĂƐ�ĞƋƵĂĕƁĞƐ�
ĚŝĨĞƌĞŶĐŝĂŝƐ�ƉŽƐƐƵĞŵ�ĚŝǀĞƌƐĂƐ�ĂƉůŝĐĂĕƁĞƐ�ĚĞŶƚƌŽ�Ğ� ĨŽƌĂ�ĚĂ�DĂƚĞŵĄƟĐĂ͘��ŵ�ĂŵďŽƐ�ŽƐ�ĐĂƐŽƐ͕�Ž�ĞƐƚƵĚŽ�ĚĂƐ�
ƐĞƋƵġŶĐŝĂƐ�Ğ�ƐĠƌŝĞƐ�ĚĞ�ĨƵŶĕƁĞƐ�ĐŽŵ�ƌĞůĂĕĆŽ�Ă�ĐŽŶǀĞƌŐġŶĐŝĂ͕�ĚĞƌŝǀĂĕĆŽ�Ğ�ŝŶƚĞŐƌĂĕĆŽ�ƚĞƌŵŽ�Ă�ƚĞƌŵŽ͕�ŽĐƵƉĂ�Ƶŵ�
ƉĂƉĞů�ĞƐƐĞŶĐŝĂů͘��ŝĂŶƚĞ�ĚŝƐƐŽ͕�ĂŶĂůŝƐĞ�ĂƐ�ĂĮƌŵĂĕƁĞƐ�Ă�ƐĞŐƵŝƌ͘
/͘� �ŽŶƐŝĚĞƌĞ�Ă�ƐĞƋƵġŶĐŝĂ��ĚĞ�ĨƵŶĕƁĞƐ�fn : [0, 1]→ R͕�ĚĂĚĂ�ƉŽƌ�fn(x) = x
n
n!
ƉĂƌĂ�ƚŽĚŽ�n = 1, 2, . . ..
^�f(x) =
∞∑
n=1
fn(x)͕�ĞŶƚĆŽ� f(x) �= 0 ƉĂƌĂ�ƚŽĚŽ�x ∈ [0, 1].
//͘� �ŽŶƐŝĚĞƌĞ� J = [0, 1] e g : J → R ĚĂĚĂ� ƉŽƌ� g(x) =
{
1 se x ∈ Q ∩ J
0 se x �∈ Q ∩ J ͘� KďƐĞƌǀĞ� ƋƵĞ� Q ∩ J
ƚĞŵ�ŵĞĚŝĚĂ�ŶƵůĂ͕�ŽƵ�ƐĞũĂ͕�g(x) = 0 Ğŵ�ƋƵĂƐĞ�ƚŽĚŽ�ƉŽŶƚŽ�ĚĞ�J͘�>ŽŐŽ͕�
∫ 1
0
g(x)dx = 0 .
///͘� �� ƐĞƋƵġŶĐŝĂ� ĚĞ� ĨƵŶĕƁĞƐ� fn : [0, 1]→ R ͕� ĚĂĚĂ� ƉŽƌ� fn(x) = x
n
n!
ƉĂƌĂ� ƚŽĚŽ� n = 1, 2, . . .� ƐĂƟƐĨĂnj�
∞∑
n=1
∫ 1
0
fn(x)dx =
∫ 1
0
∞∑
n=1
fn(x)dx .
��ĐŽƌƌĞƚŽ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /͕�ĂƉĞŶĂƐ͘
B ///͕�ĂƉĞŶĂƐ͘
C /�Ğ�//͕�ĂƉĞŶĂƐ͘
D //�Ğ�///͕�ĂƉĞŶĂƐ
E /͕�//�Ğ�///͘
ÁREA LIVRE
*R33201423*
24
D�d�D�d/��
QUESTÃO 32
K�dĞŽƌĞŵĂ�ĚĞ�'ƌĞĞŶ�Ġ�ƵŵĂ�ŐƌĂŶĚĞ�ĨĞƌƌĂŵĞŶƚĂ�ƉĂƌĂ�
Ž�ĐĄůĐƵůŽ�ĚĞ�ŝŶƚĞŐƌĂŝƐ�ĚĞ�ůŝŶŚĂ͘�^ĞƵ�ƌĞƐƵůƚĂĚŽ�ƉĞƌŵŝƚĞ�
ƌĞůĂĐŝŽŶĂƌ� ƵŵĂ� ŝŶƚĞŐƌĂů� ĚĞ� ůŝŶŚĂ� ĂŽ� ůŽŶŐŽ� ĚĞ� Ƶŵ�
ĐĂŵŝŶŚŽ� ĨĞĐŚĂĚŽ� ĐŽŵ� ƵŵĂ� ŝŶƚĞŐƌĂů� ĚƵƉůĂ� ƐŽďƌĞ� Ă�
ƌĞŐŝĆŽ�ĚĞůŝŵŝƚĂĚĂ�ƉŽƌ�ĞƐƐĞ�ĐĂŵŝŶŚŽ͘
hƟůŝnjĂŶĚŽ�Ž�dĞŽƌĞŵĂ�ĚĞ�'ƌĞĞŶ͕�ĐŽŶĐůƵŝͲƐĞ�ƋƵĞ�Ă�ŝŶƚĞŐƌĂů�
ĚĞ�ůŝŶŚĂ�ĚŽ�ĐĂŵƉŽ�ǀĞƚŽƌŝĂů͕�F�͗�R2 o R2͕�ĚĞĮŶŝĚŽ�ƉŽƌ�
F(x, y) = (y, �x)͕�ĂŽ�ůŽŶŐŽ�ĚŽ�ĐĂŵŝŶŚŽ�ĨĞĐŚĂĚŽ�ĚĞĮŶŝĚŽ�
ƉĞůĂƐ�ĐƵƌǀĂƐ�h(x) = ln(x)�͖�y = 0 ; x = 2͕�Ġ�ŝŐƵĂů�Ă
A �2 ln(2) + 1.
B 2 ln(2) � 1.
C �4 ln(2) + 2.
D 4 ln(2) � 2.
E ln(2) � 1.
ÁREA LIVRE
QUESTÃO 33
K� ĐŝůŝŶĚƌŽ� ƌĞƚŽ� ƐŽďƌĞ� Ž� ĐşƌĐƵůŽ� x2 + y2 = 1
ĂĚŵŝƚĞ� ƉĂƌĂŵĞƚƌŝnjĂĕĆŽ� χ : U → R3 ͕� Ğŵ� ƋƵĞ
χ(u, v) = (cos(u), sen(u), v)
e
U = {(u, v) ∈ R2 : 0 < u < 2π,−∞ < v < +∞}.
WĂƌĂ�ƐĞ�ĞŶĐŽŶƚƌĂƌ�Ă�ƉƌŝŵĞŝƌĂ�ĨŽƌŵĂ�ĨƵŶĚĂŵĞŶƚĂů�ĚĞƐƐĂ�
ƐƵƉĞƌİĐŝĞ�ƌĞŐƵůĂƌ͕ �I = Edu2 + 2Fdudv +Gdv2
ĨŽƌĂŵ�ƵƟůŝnjĂĚĂƐ�ĂƐ�ĞdžƉƌĞƐƐƁĞƐ�Ă�ƐĞŐƵŝƌ͘/͘� χu = (sen(u),−cos(u), 0) .
//͘� χv = (0, 0, 1)
///͘� E(u, v) = 1
��ĐŽƌƌĞƚŽ�Ž�ƋƵĞ�ƐĞ�ĂĮƌŵĂ�Ğŵ
A /͕�ĂƉĞŶĂƐ͘
B //͕�ĂƉĞŶĂƐ͘
C /�Ğ�///͕�ĂƉĞŶĂƐ͘
D //�Ğ�///͕�ĂƉĞŶĂƐ͘
E /͕�//�Ğ�///͘
ÁREA LIVRE
*R33201424*
25
D�d�D�d/��
QUESTÃO 34
K�ƚĞŽƌĞŵĂ� ĨƵŶĚĂŵĞŶƚĂů�ĚŽ�ĐĄůĐƵůŽ�Ġ�ƵŵĂ�ƉŽĚĞƌŽƐĂ�
ĨĞƌƌĂŵĞŶƚĂ� ƉĂƌĂ� Ă�DĂƚĞŵĄƟĐĂ͕� ƉŽŝƐ� Ġ� ĂŵƉůĂŵĞŶƚĞ�
ƵƟůŝnjĂĚŽ�Ğŵ�ƐƵĂƐ�ĚŝǀĞƌƐĂƐ�ƐƵďĄƌĞĂƐ �͘�Ž�ůŽŶŐŽ�ĚĂ�ŚŝƐƚſƌŝĂ͕�
ĚŝǀĞƌƐĂƐ�ǀĞƌƐƁĞƐ�ĚĞƐƐĞ�ƚĞŽƌĞŵĂ�ĨŽƌĂŵ�ĚĞŵŽŶƐƚƌĂĚĂƐ͘�
&ĂnjĞŶĚŽ�ƵƐŽ�ůŝǀƌĞŵĞŶƚĞ�ĚĞƐƐĂƐ�ǀĞƌƐƁĞƐ �͕�ĐŽŶƐŝĚĞƌĂŶĚŽ��
ĚƵĂƐ�ĨƵŶĕƁĞƐ�f, g : [a, b] o R�ĐŽŶơŶƵĂƐ�ŶŽ�ŝŶƚĞƌǀĂůŽ�
ĨĞĐŚĂĚŽ�Ğ�ůŝŵŝƚĂĚŽ�[a, b]�Ğ�ĚĞĮŶŝŶĚŽ�F, G : [a, b]o R�ƉŽƌ��
F(x) = a +
a
x
f(t)dt e G(x) = b +
a
x
g(t)dt͕
ƉĂƌĂ�ƚŽĚŽ�x [a, b]͕�ĂǀĂůŝĞ�ĂƐ�ƐĞŐƵŝŶƚĞƐ�ĂƐƐĞƌĕƁĞƐ�Ğ�Ă�
ƌĞůĂĕĆŽ�ƉƌŽƉŽƐƚĂ�ĞŶƚƌĞ�ĞůĂƐ͘
WĂƌĂ�ƚŽĚŽ�x  [a, b]͕�ƚĞŵͲƐĞ�ƋƵĞ
F(x) G(x) - F(a)G(a) =
a
x
[f(t)G(t) + F(t)g(t)]dt]
PORQUE
^Ğ�<�͗�[a, b] o�R�ƉŽƐƐƵŝ�ĚĞƌŝǀĂĚĂ�ŝŶƚĞŐƌĄǀĞů͕�ĞŶƚĆŽ�
<(b) � <(a) =
a
b
d
�<(t)dt.
dt
�ĐĞƌĐĂ�ĚĞƐƐĂƐ�ĂƐƐĞƌĕƁĞƐ͕�ĂƐƐŝŶĂůĞ�Ă�ĂůƚĞƌŶĂƟǀĂ�ĐŽƌƌĞƚĂ͘
A �Ɛ�ĂƐƐĞƌĕƁĞƐ�/�Ğ�//�ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ�ǀĞƌĚĂĚĞŝƌĂƐ͕�Ğ�
Ă�//�Ġ�ƵŵĂ�ũƵƐƟĮĐĂƟǀĂ�ĐŽƌƌĞƚĂ�ĚĂ�/͘
B �Ɛ�ĂƐƐĞƌĕƁĞƐ� /� Ğ� //� ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ� ǀĞƌĚĂĚĞŝƌĂƐ͕�
ŵĂƐ�Ă�//�ŶĆŽ�Ġ�ƵŵĂ�ũƵƐƟĮĐĂƟǀĂ�ĐŽƌƌĞƚĂ�ĚĂ�/͘
C ��ĂƐƐĞƌĕĆŽ�/�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ǀĞƌĚĂĚĞŝƌĂ͕�Ğ�Ă�//�Ġ�
ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ĨĂůƐĂ͘
D ��ĂƐƐĞƌĕĆŽ�/�Ġ�ƵŵĂ�ƉƌŽƉŽƐŝĕĆŽ�ĨĂůƐĂ͕�Ğ�Ă�//�Ġ�ƵŵĂ�
ƉƌŽƉŽƐŝĕĆŽ�ǀĞƌĚĂĚĞŝƌĂ͘
E �Ɛ�ĂƐƐĞƌĕƁĞƐ�/�Ğ�//�ƐĆŽ�ƉƌŽƉŽƐŝĕƁĞƐ�ĨĂůƐĂƐ͘
QUESTÃO 35
KƐ�ĂŶĠŝƐ�ƋƵŽĐŝĞŶƚĞ�ƐĆŽ�ĨƌĞƋƵĞŶƚĞŵĞŶƚĞ�ƵƚŝůŝnjĂĚŽƐ�
ƉĂƌĂ� ƐĞ� ŽďƚĞƌ� ĞdžƚĞŶƐƁĞƐ� ĚĞ� ĐŽƌƉŽƐ͘� ^Ğ� F� Ġ� Ƶŵ�
ĐŽƌƉŽ� Ğ�P� Ġ� Ƶŵ� ƉŽůŝŶƀŵŝŽ� ŝƌƌĞĚƵƚşǀĞů� Ğŵ�F[X]͕�
ĞŶƚĆŽ�L = F[X]/<P>�Ġ�Ƶŵ�ĐŽƌƉŽ�ĐƵũŽ�ƉŽůŝŶƀŵŝŽ�
ŵŝŶŝŵĂů�ƐŽďƌĞ�F�Ġ�P.
�ŽŶƐŝĚĞƌĂŶĚŽ� ĞƐƐĂƐ� ŝŶĨŽƌŵĂĕƁĞƐ͕� ĂƐƐŝŶĂůĞ� Ă� ŽƉĕĆŽ�
ƋƵĞ�ƌĞƉƌĞƐĞŶƚĂ�Ƶŵ�ĐŽƌƉŽ�ĐƵũŽ�ƉŽůŝŶƀŵŝŽ�ŵŝŶŝŵĂů�Ġ�
ŝŶĚŝĐĂĚŽ�ƉĞůŽ�ŐĞƌĂĚŽƌ�ĚŽ�ŝĚĞĂů͘
A [x]/ < x2 � 1 >
B [x]/ < x2 + 2 >
C [x]/ < x2 + 3 >
D [x]/ < x2 � 4 >
E [x]/ < x2 � 5 >
ÁREA LIVRE
*R33201425*
26
D�d�D�d/��
QUESTÃO 1
YƵĂů� Ž� ŐƌĂƵ� ĚĞ� ĚŝĮĐƵůĚĂĚĞ� ĚĞƐƚĂ� ƉƌŽǀĂ� ŶĂ� ƉĂƌƚĞ� ĚĞ�
&ŽƌŵĂĕĆŽ�'ĞƌĂů͍
A�DƵŝƚŽ�ĨĄĐŝů͘
B�&ĄĐŝů͘
C�DĠĚŝŽ͘
D��ŝİĐŝů͘
E�DƵŝƚŽ�ĚŝİĐŝů͘
QUESTÃO 2 
YƵĂů� Ž� ŐƌĂƵ� ĚĞ� ĚŝĮĐƵůĚĂĚĞ� ĚĞƐƚĂ� ƉƌŽǀĂ� ŶĂ� ƉĂƌƚĞ� ĚĞ�
�ŽŵƉŽŶĞŶƚĞ��ƐƉĞĐşĮĐŽ͍
A�DƵŝƚŽ�ĨĄĐŝů͘
B�&ĄĐŝů͘
C�DĠĚŝŽ͘
D��ŝİĐŝů͘
E�DƵŝƚŽ�ĚŝİĐŝů͘
QUESTÃO 3 
�ŽŶƐŝĚĞƌĂŶĚŽ�Ă�ĞdžƚĞŶƐĆŽ�ĚĂ�ƉƌŽǀĂ͕�Ğŵ�ƌĞůĂĕĆŽ�ĂŽ�ƚĞŵƉŽ�
ƚŽƚĂů͕�ǀŽĐġ�ĐŽŶƐŝĚĞƌĂ�ƋƵĞ�Ă�ƉƌŽǀĂ�ĨŽŝ
A�ŵƵŝƚŽ�ůŽŶŐĂ͘
B�ůŽŶŐĂ͘
C�ĂĚĞƋƵĂĚĂ͘
D�ĐƵƌƚĂ͘
E�ŵƵŝƚŽ�ĐƵƌƚĂ͘
QUESTÃO 4 
KƐ� ĞŶƵŶĐŝĂĚŽƐ� ĚĂƐ� ƋƵĞƐƚƁĞƐ� ĚĂ� ƉƌŽǀĂ� ŶĂ� ƉĂƌƚĞ� ĚĞ�
&ŽƌŵĂĕĆŽ�'ĞƌĂů�ĞƐƚĂǀĂŵ�ĐůĂƌŽƐ�Ğ�ŽďũĞƟǀŽƐ͍
A�^ŝŵ͕�ƚŽĚŽƐ͘
B�^ŝŵ͕�Ă�ŵĂŝŽƌŝĂ͘
C��ƉĞŶĂƐ�ĐĞƌĐĂ�ĚĂ�ŵĞƚĂĚĞ͘
D�WŽƵĐŽƐ͘
E�EĆŽ͕�ŶĞŶŚƵŵ͘
QUESTÃO 5 
KƐ� ĞŶƵŶĐŝĂĚŽƐ� ĚĂƐ� ƋƵĞƐƚƁĞƐ� ĚĂ� ƉƌŽǀĂ� ŶĂ� ƉĂƌƚĞ� ĚĞ�
�ŽŵƉŽŶĞŶƚĞ��ƐƉĞĐşĮĐŽ�ĞƐƚĂǀĂŵ�ĐůĂƌŽƐ�Ğ�ŽďũĞƟǀŽƐ͍
A�^ŝŵ͕�ƚŽĚŽƐ͘
B�^ŝŵ͕�Ă�ŵĂŝŽƌŝĂ͘
C��ƉĞŶĂƐ�ĐĞƌĐĂ�ĚĂ�ŵĞƚĂĚĞ͘
D�WŽƵĐŽƐ͘
E�EĆŽ͕�ŶĞŶŚƵŵ͘
QUESTÃO 6
�Ɛ� ŝŶĨŽƌŵĂĕƁĞƐͬŝŶƐƚƌƵĕƁĞƐ� ĨŽƌŶĞĐŝĚĂƐ� ƉĂƌĂ� Ă� ƌĞƐŽůƵĕĆŽ�
ĚĂƐ�ƋƵĞƐƚƁĞƐ�ĨŽƌĂŵ�ƐƵĮĐŝĞŶƚĞƐ�ƉĂƌĂ�ƌĞƐŽůǀġͲůĂƐ͍
A�^ŝŵ͕�ĂƚĠ�ĞdžĐĞƐƐŝǀĂƐ͘
B�^ŝŵ͕�Ğŵ�ƚŽĚĂƐ�ĞůĂƐ͘
C�^ŝŵ͕�ŶĂ�ŵĂŝŽƌŝĂ�ĚĞůĂƐ͘
D�^ŝŵ͕�ƐŽŵĞŶƚĞ�Ğŵ�ĂůŐƵŵĂƐ͘
E�EĆŽ͕�Ğŵ�ŶĞŶŚƵŵĂ�ĚĞůĂƐ͘
QUESTÃO 7 
sŽĐġ�ƐĞ�ĚĞƉĂƌŽƵ�ĐŽŵ�ĂůŐƵŵĂ�ĚŝĮĐƵůĚĂĚĞ�ĂŽ�ƌĞƐƉŽŶĚĞƌ�
ă�ƉƌŽǀĂ͘�YƵĂů͍
A��ĞƐĐŽŶŚĞĐŝŵĞŶƚŽ�ĚŽ�ĐŽŶƚĞƷĚŽ͘
B�&ŽƌŵĂ�ĚŝĨĞƌĞŶƚĞ�ĚĞ�ĂďŽƌĚĂŐĞŵ�ĚŽ�ĐŽŶƚĞƷĚŽ͘
C��ƐƉĂĕŽ�ŝŶƐƵĮĐŝĞŶƚĞ�ƉĂƌĂ�ƌĞƐƉŽŶĚĞƌ�ăƐ�ƋƵĞƐƚƁĞƐ͘
D�&ĂůƚĂ�ĚĞ�ŵŽƟǀĂĕĆŽ�ƉĂƌĂ�ĨĂnjĞƌ�Ă�ƉƌŽǀĂ͘
E�EĆŽ�ƟǀĞ�ƋƵĂůƋƵĞƌ�ƟƉŽ�ĚĞ�ĚŝĮĐƵůĚĂĚĞ�ƉĂƌĂ�ƌĞƐƉŽŶĚĞƌ�
ă�ƉƌŽǀĂ͘
QUESTÃO 8 
�ŽŶƐŝĚĞƌĂŶĚŽ� ĂƉĞŶĂƐ� ĂƐ� ƋƵĞƐƚƁĞƐ� ŽďũĞƟǀĂƐ� ĚĂ� ƉƌŽǀĂ͕�
ǀŽĐġ�ƉĞƌĐĞďĞƵ�ƋƵĞ
A�ŶĆŽ�ĞƐƚƵĚŽƵ�ĂŝŶĚĂ�Ă�ŵĂŝŽƌŝĂ�ĚĞƐƐĞƐ�ĐŽŶƚĞƷĚŽƐ͘
B�ĞƐƚƵĚŽƵ�ĂůŐƵŶƐ�ĚĞƐƐĞƐ�ĐŽŶƚĞƷĚŽƐ͕�ŵĂƐ�ŶĆŽ�ŽƐ�ĂƉƌĞŶĚĞƵ͘
C�ĞƐƚƵĚŽƵ�Ă�ŵĂŝŽƌŝĂ�ĚĞƐƐĞƐ�ĐŽŶƚĞƷĚŽƐ �͕ŵĂƐ�ŶĆŽ�ŽƐ�ĂƉƌĞŶĚĞƵ͘
D�ĞƐƚƵĚŽƵ�Ğ�ĂƉƌĞŶĚĞƵ�ŵƵŝƚŽƐ�ĚĞƐƐĞƐ�ĐŽŶƚĞƷĚŽƐ͘
E�ĞƐƚƵĚŽƵ�Ğ�ĂƉƌĞŶĚĞƵ�ƚŽĚŽƐ�ĞƐƐĞƐ�ĐŽŶƚĞƷĚŽƐ͘
QUESTÃO 9 
YƵĂů�ĨŽŝ�Ž�ƚĞŵƉŽ�ŐĂƐƚŽ�ƉŽƌ�ǀŽĐġ�ƉĂƌĂ�ĐŽŶĐůƵŝƌ�Ă�ƉƌŽǀĂ͍
A�DĞŶŽƐ�ĚĞ�ƵŵĂ�ŚŽƌĂ͘
B��ŶƚƌĞ�ƵŵĂ�Ğ�ĚƵĂƐ�ŚŽƌĂƐ͘
C��ŶƚƌĞ�ĚƵĂƐ�Ğ�ƚƌġƐ�ŚŽƌĂƐ͘
D��ŶƚƌĞ�ƚƌġƐ�Ğ�ƋƵĂƚƌŽ�ŚŽƌĂƐ͘
E�YƵĂƚƌŽ�ŚŽƌĂƐ͕�Ğ�ŶĆŽ�ĐŽŶƐĞŐƵŝ�ƚĞƌŵŝŶĂƌ͘
QUESTIONÁRIO DE PERCEPÇÃO DA PROVA
�Ɛ�ƋƵĞƐƚƁĞƐ�ĂďĂŝdžŽ�ǀŝƐĂŵ�ůĞǀĂŶƚĂƌ�ƐƵĂ�ŽƉŝŶŝĆŽ�ƐŽďƌĞ�Ă�ƋƵĂůŝĚĂĚĞ�Ğ�Ă�ĂĚĞƋƵĂĕĆŽ�ĚĂ�ƉƌŽǀĂ�ƋƵĞ�ǀŽĐġ�ĂĐĂďŽƵ�ĚĞ�ƌĞĂůŝnjĂƌ͘ �
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