Recitation Worksheet: 9 and 10
Topic: Parametrizations,level surfaces,limits,continuity.
(1) Convert these points from rectangle (Cartesian) coordinate to cylindrical and spherical coordinates.
(a) (1,−1,√6).
(b) (√3,−1,2√3).
Convert these points from cylindrical coordinate to rectangle coordinates.
(a) (√2,3π
4,2)
(b) (3,−π
3,1)
Convert these points from spherical coordinate to rectangle coordinates.
(a) (2,π
2,π
2)
(b) (4,−π
4,π
3)
(2) Determine which type of quadric surface corresponds to the level surfaces of the following functions.
(a) f(x, y, z) = √x2+z2−y
2
(b) f(x, y, z) = √3z−7x+y
(c) f(x, y, z) = x2−y+ 5z2
(d) f(x, y, z) = x2
4+y2
4+z2
4
(e) f(x, y, z)=4x2+ 49z2+y2
(3) Find the limit, if it exists. If it does not exist, show it does not exist.
(a) f(x, y) = x2−2xy
x2−4y2: lim
(x,y)→(2,1)
x2−2xy
x2−4y2
(b) f(x, y) = x2−y6
xy3: lim
(x,y)→(0,0)
x2−y6
xy3
(c) f(x, y) = 2xy
x2+y2: lim
(x,y)→(0,0)
2xy
x2+y2
(d) f(x, y) = 2x3+3y3
x2+y2: lim
(x,y)→(0,0)
2x3+3y3
x2+y2.Hint: Try converting to polar coordinates?
(4) Let f(x, y) = 2x3
x2+y2. Prove the limit exists as (x,y )→(0,0). Then, show that
g(x, y) = (2x3
x2+y2,if (x, y)= (0,0)
0,if (x, y) = (0,0)
is continuous at (x, y) = (0,0).
(5) (a) Let
f(x, y) = (x3−y3
x3+y3,if (x, y)= (0,0)
0,if (x, y) = (0,0)
Is f(x, y) continuous at (x, y) = (0,0)? Explain.
(b) Let
f(x, y) = (sin(xy)
xy ,if (x, y)= (0,0)
0,if (x, y) = (0,0)
Is f(x, y) continuous at (x, y) = (0,0)? Explain.
