Find an equation of the tangent plane to the given parametric surface at the specifie point. \( \begin{array}{l}\text { (a) } r(u, v)=\left(u^{2}+1\right) \mathbf{i}+\left(v^{3}+1\right) \mathbf{j}+(u+v) \mathbf{k} .(5,2,3) \\ \text { (b) } r(u, v)=\sin u \mathbf{i}+\cos u \sin v \mathbf{j}+\sin v \mathbf{k},(0.5, \sqrt{3} / 4,0.5)\end{array} \)

ón \( f_{x}=\frac{1}{2} x^{2}+x \) en el punto de abscisa \( x=a \), PASA \( a=4 \)

(a) Find the volume of the parallelopiped whose coterminus edges are given by the vectors \( \vec{A}=2 \vec{i}- \) \( 3 \vec{j}+4 \vec{k}, \vec{B}=\vec{i}+2 \vec{j}-\vec{k} \) and \( \vec{C}=3 \vec{i}-\vec{j}+2 \vec{k} \). (b) i. If \( \vec{F} \) is a conservative field, prove that curl \( \vec{F}=\nabla \times \vec{F}=\overrightarrow{0} \) (i.e., \( \vec{F} \) is irrotational). (2 marks) ii. Conversely, if \( \nabla \times \vec{F}=\overrightarrow{0} \) (i.e., \( \vec{F} \) is irrotational), prove that \( \vec{F} \) is conservative. (c) If \( \phi=2 x y z^{2}, \quad \vec{F}=x y \vec{i}-z \vec{j}+x^{2} \vec{k} \) and C is the curve \( x=t^{2} . \quad y=2 t . \quad z=t^{3} \) from \( t=0 \)

Calcular la derivada de las a) \( F_{(x)}=\left(x^{5} \cdot \operatorname{sen} x\right) \)

Question 3 finding a potential for it. (a) Find the volume of the parallelopiped whose coterminus edges are given by the vectors \( \left.\vec{A}=2 \vec{i}-e^{x} \cos y+y z\right) \vec{i}+\left(x z-e^{x} \sin y\right) \vec{j}+(x y+z) \vec{k} \) is conservative by \( 3 \vec{j}+4 \vec{k}, \vec{B}=\vec{i}+2 \vec{j}-\vec{k} \) and \( \vec{C}=3 \vec{i}-\vec{j}+2 \vec{k} \).

Encuentre \( y^{\prime} \) si \( y^{5}+3 y^{2}-2 x^{4}=-4 \)