This is a graph of the surface z = sin(x+y) + cos(x - 3y)



We can compute the tangent plane (linear approximation) at the point (0,0) to f(x,y)=sin(x+y)+cos(x-3y). We get the plane z=1+x+y. Here's a graph of surface along with its tangent plane at (0,0).



The best quadratic approximation, given by Taylor's formula, is 1+x+y-1/2*x+3x*y-9/2*y^2. Here are two plots of the original surface along with the surface described by the quadratic approximation. (The second is zoomed out a little more; you can see that as you get further away from (0,0), the quadratic surface becomes a worse approximation to the original surface!)