Taylor Series
Text Taylor series are polynomials that approximate functions.
For functions of three variables, Taylor series depend on first, second, etc. partial derivatives at some point (x_{0}, y_{0}, z_{0}).
The tangent hyperparaboloid at a point P = (x_{0},y_{0},z_{0}) is the second order approximation to the hypersurface.
We expand the hypersurface in a Taylor series around the point P
f(x,y,z) = 
f(x_{0},y_{0},z_{0}) + (x  x_{0})f_{x}(x_{0},y_{0},z_{0}) + (y  y_{0})f_{y}(x_{0},y_{0},z_{0}) + (z  z_{0})f_{z}(x_{0},y_{0},z_{0})
+ 1/2( (x  x_{0})^{2}f_{xx}(x_{0},y_{0},z_{0}) + (x  x_{0})(y  y_{0})f_{xy}(x_{0},y_{0},z_{0}) + (x  x_{0})(z  z_{0})f_{xz}(x_{0},y_{0},z_{0})
+ (x  x_{0})(y  y_{0})f_{yx}(x_{0},y_{0},z_{0}) + (y  y_{0})^{2} f_{yy}(x_{0},y_{0},z_{0}) + (y  y_{0})(z  z_{0})f_{yz}(x_{0},y_{0},z_{0})
+ (z  z_{0})(x  x_{0})f_{zx}(x_{0},y_{0},z_{0}) + (z  z_{0}(y  y_{0})f_{zy}(x_{0},y_{0},z_{0})
+ (z  z_{0})^{2}f_{zz}(x_{0},y_{0},z_{0}) ) + R_{3}(x,y,z) 
and then we drop the terms of order 3 or higher to get the tangent hyperparaboloid PB(x,y,z)
PB(x,y,z) = 
f(x_{0},y_{0},z_{0}) + (x  x_{0})f_{x}(x_{0},y_{0},z_{0}) + (y  y_{0})f_{y}(x_{0},y_{0},z_{0}) + (z  z_{0})f_{z}(x_{0},y_{0},z_{0})
+ 1/2 ( (x  x_{0})^{2}f_{xx}(x_{0},y_{0},z_{0}) + (x  x_{0})(y  y_{0})f_{xy}(x_{0},y_{0},z_{0}) + (x  x_{0})(z  z_{0})f_{xz}(x_{0},y_{0},z_{0})
+ (x  x_{0})(y  y_{0})f_{yx}(x_{0},y_{0},z_{0}) + (y  y_{0})^{2}f_{yy}(x_{0},y_{0},z_{0}) + (y  y_{0})(z  z_{0})f_{yz}(x_{0},y_{0},z_{0})
+ (z  z_{0})(x  x_{0})f_{zx}(x_{0},y_{0},z_{0}) + (z  z_{0}(y  y_{0})f_{zy}(x_{0},y_{0},z_{0}) + (z  z_{0})^{2}f_{zz}(x_{0},y_{0},z_{0}) ) 
In general, the nth order taylor approximation for a function f(x, y, z) is the polynomial that has the same nth and lower partial derivatives as the function f(x, y, z) at the point (x_{0}, y_{0}, z_{0}).
Demos
Slicing the Tangent Hyperparaboloid
 
Observe that the tangent hyperparaboloid PB depends on the same number of variables as the function f, i.e. on 3 variables. Therefore, the tangent hyperparaboloid to a function of three variables cannot be visualized directly; it lies in 4space.
However, we can slice the tangent hyperparaboloid along the x, y and zaxes. If each point (x,y,z,f(x,y,z)) has a tangent hyperparaboloid then slicing the surface together with the tangent hyperparaboloid at a given point yields three paraboids that are tangent to the respective slice surface of the hypersurface.
In this demo, you can view these x, y and zslices of the tangent hyperparaboloid.

Exercises 1. Try entering for f(x, y, z) various polynomial functions of degree ≤ 2. What do you notice? Why should this result be expected?
2. Enter for f(x, y, z) the expresssion x^{2} + y^{2} + z^{3}. Describe the accuracy of each of the three slices of the Taylor approximation.
3. How does the Hessian matrix of the tangent hyperparaboloid at some point (x_{0}, y_{0}, z_{0}) of the function f(x, y, z) relate to the Hessian matrix of the function itself at that point?
