local linearity ... the use of it in calculus?

it is either when you zoom into a function so far that the function becomes linear or it is when you select two points and draw a secant line between them. Both are used to estimate the slope of a curve, also known as a derivative.

Local Linearity

The definition of L is (dx0,dy0,dz0).(x - x0,y - y0, z - z0) where . is the dot product and (dx0,dy0,dy0) is gradf evaluated at P. Now (dx,dy,dz) = (y0, x0, 2z0) from the definition of f, so we have that: y0(x - x0) + x0(y - y0) + 2z0(z - z0) == y + 8z - 16 By examining the coefficients, y0 = 0 and hence x0 = 1, and z0 = 4. So the point is (1,0,4).