Remember that we can set the value ofa univarr'afe function g : 1R } IR {in terms of, say, :11) equal to a dependent variable (say, 3;} to obtain the formula for the graph of a curve in 2D (I, y)space, y = 9(3)- Moreover, recall that the tangent line to this curve at a point 30 E R may be written in point-gradient form as y 9(30) = f(30)($ En)- Analogously, we may set the value of a bivariate function f : R2 :~ IR {in terms of, say, :3 and 3;} equal to a dependent variable (say, ,2} to obtain the formula for the graph of a surface in 3D (2, y, z)space, z = its, 2:)- We can nd the tangent plane to a surface at a point [:0 , m.) E R2 using the gradient of 3'. Specically, we have that zftmu.w) = Fm'm] ' (3:33) Notice that the lefthand side is the directional derivative of f at the point [:0 , go) in the direction of the vector [3: 2:0 , y yo }T_ Moreover, if we expand the righthand side and rearrange, we wind up with the Cartesian form 3 = ran,qu + gamma mu) + gimme qu. For example, consider the surface specified by function f given by f(x, y) = (3x2+ 5y2) 19x2 + 24xy + 16y2 +5, for x, y E R. (i) If we let (10, yo) = (1, 1), then the z-coordinate of the corresponding point on the surface z = f(x, y) is 20 = f(1, 1) = (ii) By carefully calculating the partial derivatives of f, we find that the value of the gradient of f at (x0, yo) = (1, 1) is the column vector, Vf ( 1 , 1 ) = (iii) Hence, we find that the tangent plane to the surface z = f(x, y) at the point (TO, yo, 20) = (1, 1, f(1, 1)) has the Cartesian equation 2 = (where the right-hand side is a scalar expression in terms of a and y only)