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smbstatus [-b] [-d] [-L] [-p] [-S] [-s configuration file] [-u username] smbstatus is a very simple program to list the current Samba connections.

8.15 Verify the Mean Value Theorem for the function f ( x ) = x + sin 2 x on the interval [0, ].

f (x) is continuous and differentiable everywhere. Define a = 0, b = and solve the equation f (b) f (a) = f ' (c)(b a) for c. To approximate their values, we look at the graph with the endpoints connected by a line segment. f[x_] = x + Sin[2 x] ; a = 0; b = o; m = f[b] f[a]; b a l[x_] = f[a]+ m (x a);

Plot[{f[x], l[x]}, {x, a, b}]

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DIFFERENTIATION OF COMPOSITE FUNCTIONS 6.76. (a) If U x; y; z 2x2 yz xz2 , x 2 sin t, y t2 t 1, z 3e t , nd dU=dt at t 0. H x; y sin 3x y , x3 2y 2t3 , x y2 t2 3t, nd dH=dt. ! 36t2 y 12t 9x2 6t2 6x2 t 18 Ans: a 24; b cos 3x y 6x2 y 2 If F x; y 2x y = y 2x , x 2u 3v, y u 2v, nd (d) @2 F=@v2 ; e @2 F=@u @v, where u 2, v 1. Ans. (a) 7, (b) 14; c 21; d 112; e 49 (a) @F=@u; b @F=@v; (b) if

3.0 2.5 2.0 1.5 1.0 0.5 0.5 1.0 1.5 2.0 2.5 3.0

smbtar -s server [-p password] [-x service] [-X] [-d directory] [-u user] \ [-t tape] [-b blocksize] [-N filename] [-i] [-r] [-l log level] [-v] filenames smbtar is a very small shell script on top of smbclient which dumps SMB shares directly to tape. mount-point is a UNIX umount for normal users. With this program, normal users can unmount smbfilesystems, provided that it is suid root.

It looks like the tangent line will be parallel to the secant when x 1 or x 2.5. Clearly both values lie between 0 and . FindRoot[f[b] f[a] f'[c](b a), {c, 1}] FindRoot[f[b] f[a] f'[c](b a), {c, 2.5}]

c @2 F=@u2 ,

0.785398} 2.35619}

A function f has an absolute (global) maximum over an interval, I, at a point c if f(x) f(c) for all x in I. In other words, f(c) is the largest value of f(x) in I. A similar definition (with the inequality reversed) holds for an absolute minimum. One of the most important applications of differential calculus is optimization, i.e., finding the maximum and minimum values of a function, subject to certain constraints. Not all functions have absolute maxima and minima. However the Extreme Value Theorem gives conditions sufficient to guarantee their existence: If f is continuous on a closed bounded interval, then f has both an absolute maximum and an absolute minimum in that interval. A critical number of a function f is a number c for which f '(c) = 0 or f '(c) fails to exist. It can be shown that if a function is continuous on the closed interval [a, b], then the absolute maximum and minimum will be found either at a critical number or at an endpoint of the interval. We can use Mathematica to help us find the maximum and/or minimum values.

EXAMPLE 15 We wish to find the absolute maximum and minimum values of the function f (x) = x4 4 x3 + 2 x2 + 4 x + 2 on the interval [0, 4]. First we find the critical numbers.

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6.78. 6.79.

f[x_]= x4 4 x3 + 2 x2 + 4 x + 2; Solve[f'[x] 0]

{{x 1},{x 1 2},{x 1 + 2}}

Of these three numbers, only two lie in the interval [0, 4]. We compute the value of the function at these numbers as well as the endpoints of the interval. c1 = 0; c2 = 1; c3 = 1 + 2; c4 = 4; points = {{c1, f[c1]}, {c2, f[c2]}, {c3, f[c3]}, {c4, f[c4]}} //Expand;

If U x2 F y=x , show that under suitable restrictions on F, x @U=@x y @U=@y 2U. If x u cos v sin and y u sin v cos , where is a constant, show that @V=@x 2 @V=@y 2 @V=@u 2 @V=@v 2

TableForm[points, TableHeadings {None, {"x", "f[x]"}}]

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