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Show[g1, g2, g3, AspectRatio Automatic]

4.10 The curve traced by a point on a circle as the circle rolls along a straight line is called a cycloid and has parametric equations x = r ( sin ), y = r (1 cos ) where r represents the radius of the circle. Plot the cycloid formed as a circle of radius 1 makes four complete revolutions.

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Choosing the p sign as de ning the principal value and replacing y by x, we have cosh 1 x ln x x2 1 . The choice x A 1 is made so that the inverse function is real.

ParametricPlot[{p Sin[p], 1 Cos[p]}, {p, 0, 8 o}, Ticks {Automatic, {0, 1, 2}}]

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4.11 Let P be a point at a distance a from the center of a circle of radius r. (Imagine the point being placed on a spoke of a bicycle wheel.) The curve traced by P as the circle rolls along a straight line is called a trochoid. Its parametric equations are x = r a sin , y = r a cos . Sketch the trochoid with 1 r = 1, a = as the circle makes four revolutions. What would the graph look like if r = 1, a = 2 so 2 that the point is outside the circle

(a) We must show that given any > 0 we can nd > 0 (depending on in general) such that jx2 4j < when 0 < jx 2j < . Choose @ 1 so that 0 < jx 2j < 1 or 1 < x < 3, x 6 2. Then jx2 4j j x 2 x 2 j jx 2jjx 2j < jx 2j < 5 . Take as 1 or =5, whichever is smaller. Then we have jx2 4j < whenever 0 < jx 2j < and the required result is proved. It is of interest to consider some numerical values. If for example we wish to make jx2 4j < :05, we can choose =5 :05=5 :01. To see that this is actually the case, note that if 0 < jx 2j < :01 then 1:99 < x < 2:01 x 6 2 and so 3:9601 < x2 < 4:0401, :0399 < x2 4 < :0401 and certainly jx2 4j < :05 x2 6 4 . The fact that these inequalities also happen to hold at x 2 is merely coincidental. If we wish to make jx2 4j < 6, we can choose 1 and this will be satis ed. (b) There is no di erence between the proof for this case and the proof in (a), since in both cases we exclude x 2.

implemented using either shared memory (fast) or file locks (slow) because there is no general UNIX counterpart for share mode operations. Samba implements DENY_DOS, DENY_ALL, DENY_READ, DENY_WRITE, DENY_NONE, and DENY_FCB lock requests (Table 11.5). share modes = <yes, no> Table 11.5 Samba Share Modes

r = 1; a = 1/2; ParametricPlot[{r p a Sin[p], r a Cos[p]}, {p, 0, 8 o}, PlotRange {Automatic, {0, 2}}, Ticks {Automatic, {0, 1, 2}}]

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2x4 6x3 x2 3 We must show that for any > 0 we can nd > 0 such that 8 < when x 1

r = 1; a = 2; ParametricPlot[{r p a Sin[p], r a Cos[p]}, {p, 0, 8 o}]

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DOS compatibility mode. Additional opens by the same client are allowed. File compatibility mode. DENY_DOS open then set open to read/write. If first open client is writing then no other client reads are allowed. If first client is read-only, then additional client reads are allowed. All other open requests are denied. Write-only opens are allowed. Read- only opens are allowed. Additional open requests are allowed.

4.12 A circle of radius b rolls on the inside of a larger circle of radius a. The curve traced out by a fixed point initially at (a, 0) is called a hypocycloid and has equations

x!3 x!3

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