Multivariate Calculus: multivariate calculus – #27254


(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]

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