Categoria: Funções definidas por ramos

As funções de Heaviside e rampa 0

As funções de Heaviside e rampa

Mais funções: Aleph 11 - Volume 2 Pág. 139 Ex. 12

Enunciado

 As funções de Heaviside e rampa são definidas, respetivamente, por: \[\begin{array}{*{20}{c}}
  {H\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
  0& \Leftarrow &{x < 0} \\
  {\frac{1}{2}}& \Leftarrow &{x = 0} \\
  1& \Leftarrow &{x > 0}
\end{array}} \right.}&{\text{e}}&{R\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
  0& \Leftarrow &{x \leqslant 0} \\
  x& \Leftarrow &{x > 0}
\end{array}} \right.}
\end{array}\]

Mostre que:

  1. $R\left( x \right) = x\,H\left( x \right)$
     
  2. $R\left( x \right) = \frac{{x + \left| x \right|}}{2}$
     
  3. $\left( {R \circ R} \right)\left( x \right) = R\left( x \right)$

Resolução >> Resolução

\[\begin{array}{*{20}{c}}
  {H\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
  0& \Leftarrow &{x < 0} \\
  {\frac{1}{2}}& \Leftarrow &{x = 0} \\
  1& \Leftarrow &{x > 0}
\end{array}} \right.}&{\text{e}}&{R\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
  0& \Leftarrow &{x \leqslant 0} \\
  x& \Leftarrow &{x > 0}
\end{array}} \right.}
\end{array}\]

 

  1. \[\begin{array}{*{20}{c}}
      {x\,H\left( x \right)}& = &{x \times \left\{ {\begin{array}{*{20}{c}}
      0& \Leftarrow &{x < 0} \\
      {\frac{1}{2}}& \Leftarrow &{x = 0} \\
      1& \Leftarrow &{x > 0}
    \end{array}} \right.}& = &{\left\{ {\begin{array}{*{20}{c}}
      {x \times 0}& \Leftarrow &{x < 0} \\
      {x \times \frac{1}{2}}& \Leftarrow &{x = 0} \\
      {x \times 1}& \Leftarrow &{x > 0}
    \end{array}} \right.}& = &{\left\{ {\begin{array}{*{20}{c}}
      0& \Leftarrow &{x < 0} \\
      0& \Leftarrow &{x = 0} \\
      x& \Leftarrow &{x > 0}
    \end{array}} \right.}& = &{R\left( x \right)}
    \end{array}\]
     
  2. \[\begin{array}{*{20}{c}}
      {\frac{{x + \left| x \right|}}{2}}& = &{\left\{ {\begin{array}{*{20}{c}}
      {\frac{{x + \left( { – x} \right)}}{2}}& \Leftarrow &{x \leqslant 0} \\
      {\frac{{x + x}}{2}}& \Leftarrow &{x > 0}
    \end{array}} \right.}& = &{\left\{ {\begin{array}{*{20}{c}}
      0& \Leftarrow &{x \leqslant 0} \\
      x& \Leftarrow &{x > 0}
    \end{array}} \right.}& = &{R\left( x \right)}
    \end{array}\]
     
  3. \[\begin{array}{*{20}{c}}
      {\left( {R \circ R} \right)\left( x \right)}& = &{R\left( {R\left( x \right)} \right)}& = &{R\left( {\left\{ {\begin{array}{*{20}{c}}
      0& \Leftarrow &{x \leqslant 0} \\
      x& \Leftarrow &{x > 0}
    \end{array}} \right.} \right)}& = &{R\left( x \right)}
    \end{array}\]

 

<< Enunciado
0

A função de Heaviside

Mais funções: Aleph 11 - Volume 2 Pág. 138 Ex. 9

Enunciado

Oliver Heaviside (Londres, 18 de maio de 1850 — Torquay, 3 de fevereiro de 1925) foi um matemático inglês.

A função de Heaviside, muito usada na Física e na Engenharia, é definida por: \[H\left( x \right) = \left\{ {\begin{array}{*{20}{c}}
  0& \Leftarrow &{x < 0} \\
  {\frac{1}{2}}& \Leftarrow &{x = 0} \\
  1& \Leftarrow &{x > 0}
\end{array}} \right.\]

  1. Esboce o gráfico da função.
0

Gráfico de $f$

Mais funções: Aleph 11 - Volume 2 Pág. 137 Ex. 5

Enunciado

Considere a função $f$, cuja representação gráfica se apresenta na figura ao lado.

  1. Encontre uma expressão que permita definir a função $f$.
     
  2. Determine, algebricamente, a função definida por $g\left( x \right) = f\left( {x + 2} \right) + 1$.
    Esboce o gráfico de $g$.
     
  3. Transforme o gráfico de $f$, de forma a obter o gráfico da função definida por $h\left( x \right) =  – f\left( x \right) + 1$.
0

Considere a função cujo gráfico está representado na figura

Mais funções: Aleph 11 - Volume 2 Pág. 136 Ex. 3

Enunciado

Considere a função $f$, de domínio $\left] { – \infty , – 1} \right[ \cup \left[ {1, + \infty } \right[$, cujo gráfico está representado na figura.

Determine um expressão que defina a função.

Resolução >> Resolução

O troço da esquerda é um arco de parábola, a qual pode ser definida por $y = a\left( {x + \frac{7}{4}} \right)\left( {x – {x_2}} \right)$, pois $ – \frac{7}{4}$ é um zero da função.…

0

A continuidade da função

Mais funções: Aleph 11 - Volume 2 Pág. 116 Ex. 7

Enunciado

Com a ajuda da calculadora gráfica, estude a continuidade das seguintes funções de acordo com os valores que o parâmetro real $m$ toma.

\[\begin{array}{*{20}{c}}
  {h\left( x \right) = \left\{ {\begin{array}{*{20}{l}}
  {\frac{m}{x}}& \Leftarrow &{0 < x \leqslant 2} \\
  { – {x^2} + 10x + 3}& \Leftarrow &{x > 2}
\end{array}} \right.}&{}&{\text{e}}&{}&{p\left( x \right) = \left\{ {\begin{array}{*{20}{l}}
  {\frac{3}{x}}& \Leftarrow &{0 < x \leqslant 1} \\
  {1 + mx}& \Leftarrow &{x > 1}
\end{array}} \right.}
\end{array}\]

Resolução >> Resolução

 

\[{h\left( x \right) = \left\{ {\begin{array}{*{20}{l}}
  {\frac{m}{x}}& \Leftarrow &{0 < x \leqslant 2} \\
  { – {x^2} + 10x + 3}& \Leftarrow &{x > 2}
\end{array}} \right.}\]

 

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\[{p\left( x \right) = \left\{ {\begin{array}{*{20}{l}}
  {\frac{3}{x}}& \Leftarrow &{0 < x \leqslant 1} \\
  {1 + mx}& \Leftarrow &{x > 1}
\end{array}} \right.}\]

 

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<< Enunciado
0

Defina a função por ramos

Mais funções: Aleph 11 - Volume 2 Pág. 116 Ex. 6

Enunciado

Representação gráfica da função $f$

Considere uma função $f$, real de variável real, de domínio $\mathbb{R}$, cuja representação gráfica se apresenta ao lado.

  1. Complete a tabela:
     
    $x$        
    $f\left( x \right)$ $0$ $1$ $3$ $5$

     

  2. Determine a equação reduzida de cada uma das retas: AB, BC e CD.
0

Defina sem usar o símbolo de módulo

Mais funções: Aleph 11 - Volume 2 Pág. 116 Ex. 5

Enunciado

Defina, sem usar o símbolo de módulo, e represente graficamente, cada uma das seguintes funções:

  1. $f(x) = \left| {x – 1} \right| + 2$
     
  2. $g(x) =  – \left| {3{x^2} – 2x – 1} \right|$
     
  3. $h(x) =  – \left| {x\left( {x – 2} \right)\left( {x + 1} \right)} \right|$

Resolução >> Resolução

  1.  

    Gráfico de $f$

    \[\begin{array}{*{20}{l}}
      {f(x)}& = &{\left| {x – 1} \right| + 2} \\
      {}& = &{\left\{ {\begin{array}{*{20}{l}}
      {\left( {x – 1} \right) + 2}& \Leftarrow &{x – 1 \geqslant 0} \\
      { – \left( {x – 1} \right) + 2}& \Leftarrow &{x – 1 < 0}
    \end{array}} \right.} \\
      {}& = &{\left\{ {\begin{array}{*{20}{l}}
      { – x + 3}& \Leftarrow &{x < 1} \\
      {x + 1}& \Leftarrow &{x \geqslant 1}
    \end{array}} \right.}
    \end{array}\]
     
     
    Para a representação gráfica de $f$ recorreu-se aos gráficos das funções auxiliares ${y_1} =  – x + 3$ e ${y_2} = x + 1$.