 
Summary: Math 2200 Homework 4 (Due: Friday October 15rd)
Problem 1 For f(x, y) = x2
· Sin(y) verify that fxy = fyx by taking the partial
derivatives in both orders.
Solution:
fx = 2x · Sin(y)
fxy = 2x · Cos(y)
fy = x2
· Cos(y)
fyx = 2x · Cos(y)
Problem 2 Find the domain and range of each of the following functions.
(i) f(x, y) = Ln(2x + y + 4),
(ii) g(x, y) = Cos(xy)
(iii) h(x, y) = x2+y2
x2+y2+1
(Hint: the function is symmetric about the origin.
Solution:
(i) Since there is a line where the argument is zero, the domain is D = {(x, y) :
2x + y + 4 > 0} or D = {(x, y) : y > 2x  4} which makes the line easier to
visualize. As we approach the line, Ln diverges to  while, as we let (x, x)
