Osculating Paraboloid equation

Question

I know how to find tangent plane but I don't know about how to find osculating paraboloid and then approximate N. Can somebody help me out?

Answer 1

The equation of the osculating paraboloid can be written as $$ z=F(x,y)=a(x-1)^2+b(x-1)(y-1)+c(y-1)^2+d(x-1)+e(y-1)+1. $$ You can find the coefficients by imposing that all partial derivatives of $F$, up to second order, are the same as the derivatives of $x^9y^7$ at $(1,1)$: $$ {\partial F\over\partial x}(1,1)= {\partial\over\partial x}(x^9y^7)\bigg\vert_{(1,1)} \quad {\partial F\over\partial y}(1,1)= {\partial\over\partial y}(x^9y^7)\bigg\vert_{(1,1)} $$ $$ {\partial^2 F\over\partial x^2}(1,1)= {\partial^2\over\partial x^2}(x^9y^7)\bigg\vert_{(1,1)} \quad {\partial^2 F\over\partial x\partial y}(1,1)= {\partial^2\over\partial x\partial y}(x^9y^7)\bigg\vert_{(1,1)} \quad {\partial^2 F\over\partial y^2}(1,1)= {\partial^2\over\partial y^2}(x^9y^7)\bigg\vert_{(1,1)} $$

Answer 2

A paraboloid is a quadratic surface, i.e. of the form $z=ax^2+bxy+cy^2+dx+ey+z$, a quadratic polynomial in $x,y$.

For a paraboloid to be osculating, all its coefficients must match those of the Taylor development to the second order. (Similarly, the tangent plane is an osculating plane.)