# Multivariate Calculus: multivariate calculus – #27254

Question:

 (a) Part of a landscape generated in a computer game is defined by the quadric surface $z=3{{x}^{2}}+2xy+{{y}^{2}}+10x+2y+5$. (i) Obtain the critical point of the surface and determine if it is a maximum, a minimum or a saddle point. (ii) Determine the equation of the tangent plane to the surface at the point where $x=-1$ and $y=0$. (iii) Find or state a vector that is parallel to the normal to the surface where $x=-1$ and $y=0$. [11] (b) (i) Construct the bilinear surface patch based on the line segments joining $A\left( 1,\,1,\,0 \right)$ with $B\left( 2,\,1,\,1 \right)$ and $C\left( 1,\,2,\,0 \right)$ with $D\left( 2,\,2,\,0 \right)$. (ii) Obtain the parametric equations of the curve joining A to D. (iii) Show that the curve joining B to C is a parabola lying in the plane $x+y=3$. [9]