CALCULO 1 - UN6 - AVALIAÇÃO OBJETIVA 2

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Minhas Disciplinas / Meus cursos / 422391 / Unidade 6: Aplicações da Derivada / UN 6 - Avaliação Objetiva
Cálculo
Iniciado em terça, 31 ago 2021, 08:32
Estado Finalizada
Concluída em terça, 31 ago 2021, 08:38
Tempo
empregado
5 minutos 43 segundos
Avaliar 1,70 de um máximo de 1,70(100%)
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Questão 1
Correto
Atingiu 0,34 de 0,34
A �gura abaixo exibe o grá�co da função f´(x) no intervalo [-4,3].
Fonte: Elaborado pela autora
O grá�co da função f(x) pode ser representado por:
 
Escolha uma opção:
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Sua resposta está correta.
 Feedback da resposta correta
A resposta correta é: 
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Questão 2
Correto
Atingiu 0,34 de 0,34
A prefeitura de uma cidade cedeu uma área para que lojistas pudessem expor suas mercadorias. Foram
disponibilizados uma área aproximada de 20 m², os expositores devem arcar com o custo para cercar cada stand,
todos devem seguir o mesmo padrão de forma que:
• duas laterais serão cercadas utilizando madeira a um custo de R$ 15,00 por metro de comprimento.
• A terceira lateral, a que será o fundo do stand será construída utilizando bloco de cimento custo de R$ 26,00 por
metro de comprimento.
•A altura de cada divisão deve ser de 2m.
 
 
 
 
Determine as dimensões aproximadas que minimizarão o custo da construção dos stands.
 
Escolha uma opção:
5,1 e 3,9
4,3 e 4,7
4,6 e 4,3
4,2 e 4,8 
5,3 e 3,8
Sua resposta está correta.
 
Temos que a área do stand é dada por
x.y=20
onde
Custo da madeira = área dos lados de madeira x custo por metro
Custo da madeira =4x.15=60x
 
Custo bloco de cimento= área do lado do bloco de cimento x custo por metro
Custo bloco de cimento=2y.26=52y
 
O custo total dos materiais será dado por
Custo= custo madeira + custo bloco cimento
C=60x+52 y
Como a área é dada por x.y=20 , isolando y temos y=20/x
C= 60x+52.20/x
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C=60x+1040/x
Vamos analisar o pontos criticos, onde c´=0
C´=60-1040/x²
Fazendo c´=0, temos
60 -1040/x²=0
-1040/x²=-60
X²=17,3
X=4,16 ou x=-4,16 (com x>0 esse valor não nos interessa)
 
Se x=4,16 o valor de y encontramos em y=20/x, ou seja y@ 4,81
Portanto, aproximadamente temos 4,2 e 4,8 as dimensões do stand.
 
Use este campo para justi�car a razão da alternativa apontada acima, ser a correta, serve também como
feedback para o aluno.
A resposta correta é: 4,2 e 4,8
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Questão 3
Correto
Atingiu 0,34 de 0,34
Calcule a área aproximada nos intervalos entre [0,2] utilizando uma estimativa inferior entre o grá�co da função
f(x)=x³+1 e o eixo x. Utilize 5 retângulos, conforme a �gura abaixo:
 
Fonte: Elaborado pela autora.
 
O valor aproximado da área será:
 
Escolha uma opção:
4,40 u.a
4,56 u.a 
4,32 u.a
4,70 u.a
4,76 u.a
Sua resposta está correta.
Justi�cativa do Gabarito:
A resposta correta é: 4,56 u.a
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Questão 4
Correto
Atingiu 0,34 de 0,34
A �gura abaixo exibe o grá�co da função f´(x) no intervalo [-2,3]. 
   
 Fonte: Elaborado pela autora 
 São feitas as seguintes a�rmações: 
I-Os pontos criticos ocorrem quando x=0 e x=2. 
II- em x=2 temos um ponto de mínimo relativo. 
III- em x=1 temos um ponto de in�exão.
É correto o que se a�rma em:
Escolha uma opção:
I, apenas.
II e III, apenas.
II, apenas
I, II e III.
I e III, apenas. 
Sua resposta está correta.
  Pelo grá�co da função derivada f´(x), pode se observar que :
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A a�rmação I – está correta pois nos pontos onde x=1 e x=2 temos os pontos criticos.
A a�rmação II- é falsa pois, em até x=2 a função f(x) será crescente e depois decrescente, poranto nesse ponto
temos um máximo relativo.
A a�rmação III- é correta pois em x=1 temos um ponto de in�exão pois os valores de f´(x) passam de crescentes
para decrescentes.
 
A resposta correta é: I e III, apenas.
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Questão 5
Correto
Atingiu 0,34 de 0,34
As derivadas permitem o estudo do comportamento de uma função. Os pontos onde f`(x) =0, temos a tangente
paralela ao eixo x. Neste pontos, temos o que denominamos de pontos criticos, onde possivelmente temos um
ponto de máximo relativo ou minimo relativo. Assim, dadas as funções f(x)=x³+2x²+4 e g(x)=3x²+4, são feitas as
seguintes a�rmações:
 
I-f(x) e g(x) tem ponto minimo iguais.
II- para x>0 f(x) e g(x) são crescentes.
III-f(x) tem um ponto de in�exão em x=2/3
 
É correto o que se a�rma em:
 
Escolha uma opção:
II e III, apenas.
I, II e III.
I e II, apenas. 
II, apenas
I, apenas.
Sua resposta está correta.
Seja f(x)=x³+2x²+4 e g(x)=3x²+4
f´(x)=3x²+4x, para f´(x)=0®3x²+4x=0® os pontos criticos serão x=0 e x=-4/3
f´´(x)=6x+4
f´´(0)=4> 0 ponto minimo
f´´(-4/3)=-4<0 ponto máximo
 
f´´(x)=0®x=-2/3 temos um ponto de in�exão , a a�rmação III está incorreta
 
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Para g(x)=3x²+4, temos:
g`(x)=6x , para g´(x)=0®x=0
g´´(x)=6>0 ponto mínimo
 
Portanto as a�rmações I e II são corretas
 
Use este campo para justi�car a razão da alternativa apontada acima, ser a correta, serve também como
feedback para o aluno.
A resposta correta é: I e II, apenas.
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