Sketch the contour graph of the function of two variables f
(x,y)
=
(x − 1)
2+
(y− 2)
2.
Which of the following functions could correspond to the following contour
graphs? z = x2 + 3y2,z = 3x2 + y2,z = x2− y2,z = x + y.
PICT
Which of the following functions could correspond to the following contour graphs?
z = x2− 3y2,z = y2 + 3x2,z = x − y,z = x + y.
PICT
Which of the following functions could correspond to the following contour graphs?
z = sin(x + y),z = x + y,z = (x + y)2,z = x2− y.
PICT
Find the following limits if they exist. If they do not exist, explain why.
(a) lim
(x,y)
→
(0,0)
2 2
xx2−+yy2
(b) lim
(x,y)
→
(0,0)
2x3+xy22−-x22−2y2
x +2y
(c) lim
(x,y)
→
(0,0)
sin(x2+y2)
x2+y2
(d) lim
(x,y)
→
(0,0)
sin(x2+2y2)
x2+2y2
(e) lim
(x,y)
→
(0,0)
sin(x2+2y2)
--2x2+y2-
(f) lim
(x,y)
→
(0,0)
(x2−y4)2
(x2+y4)2
Find the following limits if they exist. If they do not exist, tell why.
(a) lim
(x,y)
→
(0,0)
x
(x2−y4)2
(x2+y4)2-
(b) lim
(x,y)
→
(0,0)
x-sin(x22+22y2)
2x +y
(c) lim
(x,y)
→
(0,0)
--xy-
x2+y2
(d) lim
(x,y)
→
(1,0)
x3−3x2+3x− 1− y2x+y2
----x2−2x+1+y2----
∗Suppose f is a function defined on a set D and that a ∈ D is not a limit point of
D. Show that if I define the notion of limit in the same way as above, then
limx→af
(x)
= 5. Show that it is also the case that limx→af
(x)
= 7. In other
words, the concept of limit is totally meaningless. This is why the insistence that
the point a be a limit point of D.
∗Show that the definition of continuity at a ∈ D
(f)
is not dependent
on a being a limit point of D
(f)
. The concept of limit and the concept
of continuity are related at those points a which are limit points of the
domain.