Consider the real-valued function of two variables with rule f(x,y) = xexp{-x2 -
ID: 2983492 • Letter: C
Question
Consider the real-valued function of two variables with rule f(x,y) = xexp{-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 y0 = 0 and y0 = plusminus 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 , and write it in set notation. (Assume for now that z0 epsilon 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 = plusminus 0.1, plusminus 0.2, plusminus 0.3, plusminus 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.)Explanation / Answer
a)i) domain(-inf,inf)
ii)no, for y=1 or -1 the value of the function remain the same. therefore not one one
iii) f(-x,y)=-f(x,y)
f(x,-y)=f(x,y)