14.4 Differentiability, Tangent planes, and linear approximation In single-variable calculus, a function F is differentiable if the derivative f’ exists. By extension, one might expect that a function f(x,y) would be differentiable if the partial derivatives fx(x,y) and fy (x,y) If f(x,y) has partial derivatives fx(a,b) and fy(a,b) at (a,b), then these derivatives determine lines that are tangent to the graph of f(x,y) at p These two tangent lines determine a plane that is certainly a good candidate for a tangent plane to the graph. We refer to this plane as the plane determined by fx and fy to identify a condition that guarantees that the plane is determined by fx and fy is tangent to the graph, we first need an equation of this plane We begin by finding a normal vector to the plane determined by fx and fy. to do that, we find direction vectors for the tangent lines for fx and¯fy and then take their cross product ¯ ¯i ¯ ¯ W £ V = ¯0 ¯ ¯ j k ¯¯ 1 fy (a; b)¯¯ = hfx (a; b) ; fy (a; b) ; ¡1i ¯ 0 fx (a; b)¯ fx (a; b) (x ¡ a) + fy (a; b) (y ¡ b) ¡ (z ¡ f (a; b)) = 0 z = f (a; b) + fx (a; b) (x ¡ a) + fy (a; b) (y ¡ b) As (x,y) approaches (a,b), this error approaches zero because the two functions are continuous and agree at (a,b). If the error goes to zero fast enough that the graph of f(x,y) flattens and becomes approximately a plane, then the plane determined by fx and fy is an actual tangent plane If f(x,y) is differentiable at all points in a domain D, we say that f(x,y) is differentiable on D in single-variable calculus, a function f that is differentiable at a has a graph Theorem 1: that, as you continuously zoom in at (a, f(a)), appears more and more like the tangent line to the graph at (a, f(a)) If fx(x,y) and fy(x,y) exist and Differentiability at (a,b) implies that in a small region around P, the graph of are continuous on an open disk f(x,y) is nearly indistinguishable from the tangent plane at P. That is to say, we D, then f(x,y) is differentiable can approximate f(x,y) near (a,b) by the linearization l(x,y) on D Like functions of a single variable, the approximation by linearization has associated linear approximation formulas for the change in f ¢f ¼ fx (a; b) ¢x + fy (a; b) ¢y