Question

4. A rede elétrica brasileira funciona através da corrente alternada, cuja tensão obedece uma função trigonométrica.
Supondo que a derivada da tensão pelo tempo seja: dV/dt = 4cos(2t) e que a tensão tinha valor nulo quando a rede elétrica foi ligada, encontre a função da tensão.



A.

V(t) = -8sen(2t)

B.

V(t) = 4sen(2t)

C.

V(t) = 2sen(2t)

D.

V(t) = 2sen(t)

E.

V(t) = -4sen(t)

Answer 1

$\large\boxed{\begin{array}{l}\underline{\rm Equac_{\!\!,}\tilde ao\,diferencial.}\\\sf\acute E\,uma\,equac_{\!\!,}\tilde ao\,que\,envolve\,derivadas\\\sf de\,uma\,ou\,mais\,vari\acute aveis\,dependentes\\\sf com\,relac_{\!\!,}\tilde ao\,a\,uma\,ou\,mais\,vari\acute aveis\\\sf\,independentes.\\\underline{\tt exemplos:}\\\boldsymbol{\dfrac{d^2y}{dx^2}+xy\bigg(\dfrac{dy}{dx}\bigg)^2=0}\\\\\boldsymbol{\dfrac{\partial v}{\partial s}+\dfrac{\partial v}{\partial t}=v}\end{array}}$

$\large\boxed{\begin{array}{l}\sf Dada\,a\,equac_{\!\!,}\tilde ao\,\dfrac{dV}{dt}=4cos(2t)\\\sf Deseja-se\,saber\,qual\,func_{\!\!,}\tilde ao\,cuja\,derivada\\\sf \acute e\,4cos(2t).\\\sf dV=4cos(2t)\,dt\longrightarrow isolando\,os\,diferenciais.\\\displaystyle\sf\int dV=\int 4cos(2t)dt\longrightarrow integrando\,nos\,dois\,membros.\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm Integrais\,b\acute asicas}\\\displaystyle\sf\int dv=v+c\\\displaystyle\sf\int [f(x)\pm g(x)]dx=\int f(x)\,dx\pm\int g(x)dx\\\displaystyle\sf\int k\,f(x)dx=k\int f(x)\,dx\\\displaystyle\sf\int u^n\,du=\begin{cases}\sf\dfrac{u^{n+1}}{n+1}+c,se\,n\ne-1\\\\\sf\ln|u|+c,se\,n=-1\end{cases}\end{array}}$

$\large\boxed{\begin{array}{l}\underline{\rm integral\,da\,func_{\!\!,}\tilde ao\,cosseno}\\\displaystyle\sf\int cos(k\,t)dt=\dfrac{1}{k}sen(kt)+c\end{array}}$

$\large\boxed{\begin{array}{l}\sf usando\,as\,regras\,listadas\,anteriormente\\\sf temos:\\\displaystyle\sf\int dV=\int 4cos(2t)\\\sf V+c_1=4\displaystyle\int cos(2t)dt\\\sf V+c_1=\diagup\!\!\!4\cdot\dfrac{1}{\diagup\!\!\!2}sen(2t)+c_2\\\\\sf V+c_1=2sen(2t)+c_2\\\sf V=2sen(2t)+c_2-c_1\\\sf V(t)=2sen(2t)+c,onde\,c=c_1-c_2\\\sf quando\,t=0(a\,rede\,foi\,ligada)\,tens\tilde ao\,se\,anula.\\\sf V(0)=2sen(2\cdot 0)+c\\\sf 0=0+c\implies c=0\\\sf A\,func_{\!\!,}\tilde ao\,desejada\,\acute e\end{array}}$

$\Huge\boxed{\boxed{\boxed{\boxed{\sf V(t)=2sen(2t)}}}}\\\huge\boxed{\boxed{\boxed{\boxed{\sf\blue{\maltese}~\red{alternativa~C}}}}}$