Question

Poderia me ajudar nessa pergunta por favor?

Answer 1

i)
$f(x,y,z)=z-e^x\sin y\\ \\ \nabla f(x,y,z) = (f_x,f_y,f_y)\\ \\ f_x(x,y,z)=-e^x\sin y\\ f_y(x,y,z)=-e^x\cos y\\ f_z(x,y,z)=1\\ \\ \nabla f(x,y,z)=(-e^x\sin y,-e^x\cos y,1)\\ \\ \boxed{\nabla f\left(\ln 3,\dfrac{3\pi}{2},-3\right)=(3,0,1)}$

ii)
$\nabla f(x,y,z)=(-e^x\sin y,-e^x\cos y,1)\\ \\ D_{\vec u}f(P_0)=\nabla f(P_0)\cdot \dfrac{\vec v}{\|\vec v\|}\\ \\ D_{\vec u}f(P_0)=(3,0,1)\cdot \dfrac{(2,3,6)}{\sqrt{2^2+3^2+6^2}}\\ \\ \boxed{D_{\vec u}f(P_0)=\dfrac{12}{7}}$