Module MA1132 (Frolov), Advanced Calculus
Homework Sheet 3
Each set of homework questions is worth 100 marks
Due: at the beginning of the tutorial session Thursday/Friday, 11/12 February 2016
Name:
1. Sketch the domain of f (you may use Mathematica). Use solid lines for portions of the
boundary included in the domain and dashed lines for portions not included. Determine
whether the domain is an open set, closed set, or neither.
(a) f (x, y) =
√
x−y
√
x+y
(b) f (x, y) =
ln(x+y)
√
x−y
(c) f (x, y) =
q
x2 +y 2 −1
2−x2 +2x−4y 2 +4y
2. Sketch the level curve z = k for the specified values of k
z = 4x2 + 4x + y 2 − 2y ,
k = −2, −1, 2 .
3. Determine whether the limit exists. If so, find its value
(a)
−√
lim
(x,y)→(0,0)
1
e x2 +y2
ex2 +y2 − 1
(b)
lim
(x,y)→(1,−1)
1 − cosh(x + y)
2x
sin(x2 − y 2 ) ln( x−y
)
(c)
lim
(x,y)→(0,0)
3 + cos(2x) − 4 cosh(y)
p
1 − 4 1 + x2 + y 2
4. A cone is the union of a set of half-lines that start at a common apex point and go
through
a base which can be any parametric curve. Show that the graph of z =
p
2
x − 4x + 4y 2 + 8y + 8 is a cone. Where is its apex? What can be chosen as its base?
Sketch the base.
5. Show that the hyperboloid of one sheet x2 + y 2 − z 2 = 1 is a connected surface, that is any
two points of the hyperboloid can be connected by a curve which lies on the hyperboloid.
Sketch the hyperboloid.
6. Sketch the graph of the function and identify it
p
(a) z = − 2x − x2 − y 2
1
(b) z =
p
2y − y 2
Bonus questions (each bonus question is worth extra 25 marks)
1. Consider the function
Z
f (x, y) =
x
y
q
dt
t2 − x2 y 2 − t2
(a) Sketch the domain of f . Use solid lines for portions of the boundary included in the
domain and dashed lines for portions not included. Determine whether the domain
is an open set, closed set, or neither.
(b) Assume that 0 < x < y, and determine whether the limit exists. If so, find its
value. If the limit does not exist find the leading behaviour
p of the function in a
neighbourhood of the limiting point, e.g. lim(x,y)→(0,0) ln( 1 + (x + y)2 − 1) does
not exist and the leading behaviour of this function in a neighbourhood of (0, 0) is
2 ln(x + y) .
i.
lim
f (x, y)
(x,y)→(0,0)
ii.
lim
f (x, y)
(x,y)→(1,1)
iii.
lim
(x,y)→(0,1)
2
f (x, y)