Question

Dada a função f(x,y)=yeˆx determine o seu gradiente

Answer 1

✅ Após resolver os cálculos, concluímos que o vetor gradiente é:

   $\Large\displaystyle\text{ \begin{gathered}\boxed{\boxed{\:\:\: \vec{\nabla} f(x, y) = (ye^{x},\,e^{x}) \:\:\:}}\end{gathered} }$

Seja a função em duas variáveis:

               $\Large\displaystyle\begin{gathered} f(x, y) = ye^{x}\end{gathered}$

Calculando o vetor gradiente da referida função que representa um superfície no espaço:

   $\Large\displaystyle\begin{gathered} \vec{\nabla} f(x, y) = \langle f_{x}(x, y),\,f_{y}(x, y)\rangle\end{gathered}$

                        $\Large\displaystyle\begin{gathered} = \frac{\partial f}{\partial x}\,\vec{i} + \frac{\partial f}{\partial y}\,\vec{j}\end{gathered}$

                        $\Large\displaystyle\begin{gathered} = y\cdot e^{x}\,\vec{i} + 1\cdot y^{1 - 1}\cdot e^{x}\,\vec{i}\end{gathered}$

                        $\Large\displaystyle\begin{gathered} = ye^{x}\,\vec{i} + e^{x}\,\vec{j}\end{gathered}$

                        $\Large\displaystyle\begin{gathered} = (ye^{x},\,e^{x})\end{gathered}$

✅ Portanto, o vetor gradiente é:

    $\Large\displaystyle\begin{gathered} \vec{\nabla} f(x, y) = (ye^{x},\,e^{x})\end{gathered}$

   

$\LARGE\displaystyle\text{ \begin{gathered} \underline{\boxed{\boldsymbol{\:\:\:Bons \:estudos!!\:\:\:Boa\: sorte!!\:\:\:}}}\end{gathered} }$

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