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EXAMPLE 13 (a) Eliminate the variable z; (b) eliminate the variables y and z from the following equations:

2w + 2 x + 5 y + z = 6 3w + 6 x + 2 y + 2 z = 1 (a) Eliminate[{w + x + y + z 3, 2 w + 2 x + 5 y + z 6, 3 w + 6 x + 2 y + 3 z 1}, z] w 3 x 4 y && 3 x 8 + y (b) Eliminate[{w + x + y + z 3, 2 w + 2 x + 5 y + z 6, 3 w + 6 x + 2 y + 3 z 1}, {y, z}] 29 13 x w

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Not all algebraic equations are solvable by Mathematica, even if theoretical solutions exist. If Mathematica is unable to solve an equation, it will represent the solution in a symbolic form. For the most part, such solutions are useless and a numerical approximation is more appropriate. Numerical approximations are obtained with the command NSolve.

f 3x2 6yz i 2y 3xz j 1 4xyz2 kg dxi dyj dzk C 3x2 6yz dx 2y 3xz dy 1 4xyz2 dz

Samba supports UNIX client access to remote CIFS/SMB servers via the smbclient, smbwrapper, smbtar, and smbprint routines.

NSolve[equations, variables] solves equations numerically for variables. NSolve[equations, variables, n] solves equations numerically for variables to n digits of precision.

Solve[x5 + x4 + x3 + x2 + x + 2 0]

(a) If x t; y t2 ; z t3 , points 0; 0; 0 and 1; 1; 1 correspond to t 0 and t 1, respectively.

Root[2 + #1 + #12 + #13 + #14 + #15&, 1]}, Root[2 + #1 + #12 + #13 + #14 + #15&, 2]}, Root[2 + #1 + #12 + #13 + #14 + #15&, 3]}, Root[2 + #1 + #12 + #13 + #14 + #15&, 4]}, Root[2 + #1 + #12 + #13 + #14 + #15&, 5]}}

Mathematica cannot solve this equation exactly, so it returns a symbolic solution. However, we can obtain a numerical approximation. NSolve[x5 + x4 + x3 + x2 + x + 2 0]

- 185 -

1.21486}, {x 0.522092 1.06118 },{x 0.522092 + 1.06118 }, {x 0.629523 0.883585 }, {x 0.629523 + 0.883585 }}

by the rst mean value theorem for integrals (Page 93). Then if x is any point interior to a; b , F 0 x lim

An extraneous solution is a number that is technically not a solution of the equation, but evolves from the solution process. When solving radical equations, one typically encounters extraneous solutions. For example, when solving x = 3, which has no real solution, the squaring process yields x = 9.

VerifySolutions is an option that determines whether Mathematica should verify if solutions obtained are extraneous. The default, VerifySolutions True, eliminates extraneous solutions from the solution list. If such solutions are desired, the option VerifySolutions False should be used.

smbclient is an "ftp-like" CIFS/SMB command-line client application. Capabilities include moving files to and from file shares, creating and removing files and directories, printing files on remote printers and exercising NetBIOS name-to-address resolution. $ smbclient L <server> smbwrapper is an experimental tool that allows UNIX clients limited file operations on CIFS/SMB file services. Uses a loadable, shared library invoked by smbsh subshell. It is not installed by default. See the distribution document ./source/smbwrapper/Readme. $ smbsh smbtar is a wrapper script for smbclient that simplifies using smbclient's tar options. $ smbtar s <server> -c|x <chare> <device> smbprint is a printcap filter that pipes print input into _smbclient for distribution to remote printer shares. Other CIFS/SMB clients include: Windows 9x, Windows NT: Integrated NetBEUI over TCP/IP. DOS, Windows 3x, WWG: MS Network Client, DOS LAN Manager, TCP/IP Stack: ftp://ftp.microsoft.com/BusSys/Clients OS/2 Warp: peer-to-peer networking Macintosh: DAVE Thursby Software Systems, Inc.; http://www.thursby.com

Solve[x + x 5]

since f is continuous. If x a or x b, we use right- or left-hand limits, respectively, and the result holds in these cases as well.

{{x 1 (11 2

21 )

}} }}

5.11. Prove the fundamental theorem of the calculus, Part 2 (Pages 94 and 95).

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