For questions $5-8,$ use the function $f(x,y)=7x+xy$

Find the dot product of the following vectors: $($:$1,5,7$:$)$ and $($:$3,7,9$:$)$

Find the cross product of the following vectors: $($:$1,5,7$:$)$ and $($:$3,7,9$:$)$

Find the equation of a line through the following points: $(1,5,7)$ and $(3,7,9)$

Find the equation of a plane through the following points: $(1,5,7)$ and $(3,7,9)$ and $(0,1,2)$

Find $f_x,f_y,f_{},f_{yy},f_{xy}$

Find the tangent plane to $f(x,y)$ at the point $(0,0,0).$

Let $x=sin(2t)$ and $y=cos(3t).$ Use the chain rule to find $\frac{dz}{dt}$ at $t=0.$

Find max$(D_{u}f(x,y))$ at $(1,2).$

Calculate the double integral over the rectangle:

${}_R[3x^2+x{y}^3]dA,R=\{(x,y)|-2x7,2y3\}$

Calculate the double integral over the general region:

${}_R[x^3]dA,R=\{(x,y)|1x5,0yln(x)\}$

Given that $V(E)={}_{E}dV,$ find the volume of the tetrahedron bounded by the coordinate planes and $x+y+z=3.$

Given $F(x,y)=x{y}^{2}i+x^{2}yj$ and $C$:$r(t)=$ where $0t1,$ find $($a$)$ the potential function of $F$ and $($b$)$ use the potential function to calculate the line integral ${\displaystyle \underset{C}{\overset{\phantom{}}{}}}F*dr.$