As discussed in the text, a function f (x, y) may have partial derivatives fx (a, b) and

Chapter 2, Problem 57

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As discussed in the text, a function f (x, y) may have partial derivatives fx (a, b) and f y (a, b) yet fail to be differentiable at (a, b). Geometrically, if a function f (x, y) is differentiable at (a, b), then, as we zoom in on the point(a, b, f (a, b)), the graph of z = f (x, y) will flatten out and look like the plane given by equation (4) in this section. For the functions f (x, y) given in Exercises 5357, (a) calculate fx (a, b) and f y (a, b) at the indicated point(a, b) and write the equation for the plane given by formula (4) of this section, (b) use a computer to graph the equation z = f (x, y)together with the plane calculated in part (a). Zoom in near the point (a, b, f (a, b)) and discuss whether or not f (x, y) is differentiable at (a, b). (c) Give an analytic (i.e., nongraphical) argument for your answer in part (b).f (x, y) = x 2 sin y + y2 cos x, (a, b) = ! 3 , 4 #