Problems in Mathematical Analysis.pdf - pwp.net.ipl.pt

_Answers__435
parabola y = ~- x*(x* + y > 0); ,k) the entire j/-plane; 1) the entire *t/-plane,
with the exception of the coordinate origin; m) that part of the plane located
above the parabola y* = x and to the right of the (/-axis, including the points
of the t/-axis and excluding the points of the parabola (*:^0, y > V x)\
n) the entire place except points of the straight lines *=1 and t/ = 0; o) the
family of concentric circles 2nk < x2 + y 2 < n (2k + 1 ) ( = 0, 1, 2, ...).
1793. a) First octant (including boundary); b) First, Third, Sixth and Eighth
octants (excluding the boundary); c) a cube bounded by the planes x= 1,
1 y~ and z 1, including its faces; d) a sphere of radius 1 with centre
at the origin, including its surface 1794. a) a plane; the level lines are
straight lines parallel to the straight line *-f = */ 0; b) a paraboloid of revolution;
the level lines are concentric circles with centre at the origin;
c) a hyperbolic paraboloid; the level lines are equilateral hyperbolas;
d) second-order cone; the level lines are equilateral hyperbolas; e) a parabolic
cylinder, the generatrices of which are parallel to the straight line x + t/rf- 1 0;
the level lines are parallel lines; f) the lateral surface of a quadrangular
pyramid; the level lines are the outlines of squares; g)_level
lines are parabolas
y^-Cx z
\ h) the level lines are parabolas y C ]fx ; i) the level Ifnes
2 2
are the circles C (* + y ) = 2*. 1795. a) Parabolas y^Cx* (C > 0); b) hyperbolas
xy^C(\ C |< 1); c) circles jt 2
-f*/ 2 = C 2 ; d) straight lines y = ax-{-C;
c) straight lines y-=Cx(x^Q). 1796. a) Planes parallel to the plane
x-\-y-\-z^=Q\ b) concentric spheres with centre at origin; c) for u > 0,
one-sheet hyperboloids of revolution about the z-axis; for u < 0, two-sheet
hyperboloids of revolution about the same axis; both families of surfaces
are divided by the cone *2 4-r/ 2
z a = (u = 0). 1797. a) 0; b) 0;c) 2;
d) ek \ e) limit does not exist; f) limit does not exist. Hint. In Item(b)
pass to polar coordinates In Items (e) and (f), consider the variation of x
and y along the straight lines y kx and show that the given expression
may tend to different limits, depending an the choice of k. 1798. Continuous.
1799. a) Discontinuity at je = 0, y 0; b) all points of the straight line
x = y (line of discontinuity); c) line of discontinuity is the circle
2 = l; d) the tines of discontinuity are the coordinate axes.
1800 Hint. Putting y = =^
y l const, we get the function = (?,(*) , which
is continuous everywhere, since for y l ^ the denominator * 2 2
-|-f/ ^0, and
when f/ 1 ^0, q^M^O. Similarly, when jt = = *, const, the function