Labware - MA35 Multivariable Calculus - Three Variable Calculus



Tangent Hyperplanes and Normal Vectors


Consider a point P=(x0,y0,z0) in the domain of f(x,y,z). If at this point the xy-, yz-, and xz-slice curves are differentiable, then their tangent lines determine a hyperplane consisting of the points (x0,y0,z0,f(x0,y0,z0)) that is tangent to the hypersurface at P.

This hyperplane is defined as the tangent hyperplane to P and has the equation

w = fx(x0,y0,z0)(x - x0) + fy(x0,y0,z0)(y - y0) + fz(x0,y0,z0)(z - z0) + f(x0,y0,z0).



  • 1. Find equations for the tangent hyperplanes at the point (x0, y0, z0) for each of the following functions:
    • f(x, y, z) = x2 + y2 + z2
    • f(x, y, z) = x + y
    • f(x, y, z) = xy + 2xz + 3yz
    • f(x, y, z) = ex + 2y + sin(z)
  • 2. Describe the xy, xz, and yz slices of the tangent hyperplane at a critical point of a function.