Consider the real-valued function of two variables with rule f(x, y) = x exp(-x2
ID: 2983170 • Letter: C
Question
Consider the real-valued function of two variables with rule f(x, y) = x exp(-x2 - y2). What is the maximal domain, D? Is the function one-to-one? Why or why not? How is f( -x, y) related to f(x, y)? How is f(x, -y) related to f(x, y)? Sketch the cross-sections of f with x0 = 1, -1, 1/10, -1/10 on a single diagram. Sketch the cross-sections of f with yo = 0 and yo = 1 on a single diagram. (You should use a graphical calculator for assistance, or a similar, free resource such as http://www.wolframalpha.com.) Find the level set of height zero, and write it in set notation. Find the general level set of height zo 0, and write it in set notation. (Assume for now that zo ran(f), which you are not asked to find (yet).) Again using your calculator, and the symmetries noted in (a)(iii), sketch the contours for Z0 = 0.1, 0.2, 0.3, 0.4 on a single contour diagram. (An appropriate window is -2 to 2 for both x and y.) Sketch the graph of f. (You should not spend too long on your sketch.) Find f. Evaluate at f ( , 1/2 ), ( ,-1/2), (1/2, 0) and at (1, 0). Mark arrows representing the direction of these vectors at the corresponding points on a diagram of the (x, y)-plane. Calculate the directional derivative of f at the point (1, 1) in the direction alpha = pi/6 What is the direction of greatest increase of / at (1, 1)? Find fxx, fxy, fyy. Find the second order Taylor polynomial of f about the point (1, 0). Write down the tangent plane to f at (1, 0). Also find the tangent plane to f at (0, 0).Explanation / Answer
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