Homework should be turned in at the BEGINNING OF CLASS. All problem numbers refer to McCallum's book, the required text for the course. You should write up solutions neatly to all problems, making sure to SHOW ALL YOUR WORK. A nonempty subset will be graded. You are strongly encouraged to work on these problems with other classmates, although the solutions you turn in should be your own work.
Problems: 2, 6, 9, 13, 16, 18, 19
Problems: 2, 3, 4, 6, 15, 16
Note: In problem #16 you are asked to compute the linear L(x,y) and quadratic Q(x,y) approximations to f(x,y) as well as find an upper bound on the error function in each case. In part (c) of the problem, compute the exact values f(0.1,0.1), L(0.1,0.1) and Q(0.1,0.1). Then, looking at your error estimates in parts (a) and (b), you should find that the error estimate is larger than the actual error in each case.
Hint: For part (a) use Maple and print out a contour plot of f(x,y). It is easier to do part (c) before part (b). You should actually compute f_x and f_y in part (c). For part (d), show that f(x,y) is not continuous at the origin and therefore f(x,y) is not differentiable at the origin either. For part (e), use the limit definition of partial derivatives to calculate f_x(0,0) and f_y(0,0). To check continuity of the first partials at (0,0), compute the limit as (x,y) approaches (0,0) for your expressions from part (c).
Be thankful that I didn't assign any more problems from this section. These problems are tough!