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Sketch the hypocycloid for a = 4, b = 1 (0 x 2 ) and then again for a = 8, b = 5 (0 x 10 ).

a = 4; b = 1; ParametricPlot[{x[ ], y[ ]}, { , 0, 2 o}]

a = 8; b = 5; ParametricPlot[{x[ ], y[ ]}, { , 0, 10 o}]

2x4 6x3 x2 3 2x3 4x2 3x 3 x 1 2x3 4x2 x 1 x 1 3x 3 on cancelling the common factor x 1 6 0. Then we must show that for any > 0, we can nd > 0 such that j2x3 4x2 3x 5j < when 0 < jx 1j < . Choosing @ 1, we have 0 < x < 2, x 6 1. Now j2x3 4x2 3x 5j jx 1jj2x2 2x 5j < j2x2 2x 5j < j2x2 j j2xj 5 < 8 4 5 17 . Taking as the smaller of 1 and =17, the required result follows. 0 < jx 1j < . Since x 6 1, we can write

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Opportunistic locking is a locking method that allows Windows clients aggressively to cache file data blocks on the local client. Three types of oplocks might be requested by a LAN Manager or Windows client. The first is called an exclusive oplock. This permits a client to obtain exclusive access to a file. The second type is a batch oplock. A file locked with a batch oplock allows the client to keep a file open on the server even though a local application on the client may have closed the file. The third type of oplock is a level II oplock and permits multiple client read access to a file with no writers. Enabling Samba oplock support (default) can realize file IO performance improvements of up to 30 percent. See the smb.conf (5) man page for details on lock parameters. oplocks=<yes, no> Samba release 2.0.5 introduced support for level II oplocks. The level2 oplocks allow a read/write client to downgrade its oplock to readonly when additional clients request access to a file. level2 oplocks = <yes, no> For specific instances when you want to allow multiple clients to request oplocks on the same file, you can direct Samba to permit multiple oplock request by enabling the fake oplocks parameter. Keep in mind that if this option is permitted on writeable shares, you

ContourPlot[y2 x3(2 x), {x, 0, 2}, {y, 2, 2}, Frame False, Axes True]

ContourPlot[y2 x3 + 3x2, {x, 3, 3}, {y, 8, 8}, Axes True, Frame False]

4.15 The polar graph r = q is called the Spiral of Archimedes. Sketch the graph for 0 10 and then again for 10 10 .

x!3

PolarPlot[p, {p, 0, 10 o}]

- 158 -

PolarPlot[p, {p, 10 o, 10 o}]

(a) For x > 3, For x < 3,

10 10

10 10

4.16 The equation r = sin n , where n is a positive integer, represents a family of polar curves called roses. Investigate the behavior of this family and form a conjecture about how the number of loops is related to n.

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