Module MA1132 (Frolov), Advanced Calculus
Tutorial Sheet 3
To be solved during the tutorial session Thursday/Friday, 4/5 February 2016
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) = ln(x2 − y 2 )
p
(b) f (x, y) = (x2 + 2x + y 2 )(−1 − x2 − 2x − 4y 2 + 4y)
2. Sketch the level curve z = k for the specified values of k
z = x2 + 2x + 4y 2 − 4y ,
k = −2, −1, 2 .
3. Determine whether the limit exists. If so, find its value
(a)
lim
(x,y)→(0,0)
sin(x2 − y 2 )
p
1 + x2 + y 2 − 1
(b)
lim
arctan(xy) ln(ex
2 +y 2
− 1)
(x,y)→(0,0)
(c)
lim
(x,y)→(1,1)
1 − cosh(x2 − y 2 )
x−y
sin(x − y)(1 − exp( x+y
))
4. Show that the hyperboloid of two sheets x2 + y 2 − z 2 = −1 is not a connected surface that
is there are two points of the hyperboloid which cannot be connected by a curve which
lies on the hyperboloid.
5. Sketch the graph of the function and identify it
p
(a) z = x2 + 2x + y 2
(b) z = 2x − x2 − y 2
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