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|00000000| 5c 64 6f 63 75 6d 65 6e | 74 73 74 79 6c 65 5b 66 |\documen|tstyle[f|
|00000010| 6c 65 71 6e 2c 65 70 73 | 66 2c 63 61 6c 63 5d 7b |leqn,eps|f,calc]{|
|00000020| 61 72 74 69 63 6c 65 7d | 0a 5c 6d 61 72 6b 72 69 |article}|.\markri|
|00000030| 67 68 74 7b 43 68 61 70 | 74 65 72 20 33 3a 20 57 |ght{Chap|ter 3: W|
|00000040| 6f 72 6b 73 68 65 65 74 | 20 31 30 7d 0a 5c 62 65 |orksheet| 10}.\be|
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|00000060| 42 66 7b 43 68 61 70 74 | 65 72 20 33 3a 20 57 6f |Bf{Chapt|er 3: Wo|
|00000070| 72 6b 73 68 65 65 74 20 | 31 30 20 20 5c 68 66 69 |rksheet |10  \hfi|
|00000080| 6c 6c 20 4a 61 63 6b 20 | 4b 2e 20 43 6f 68 65 6e |ll Jack |K. Cohen|
|00000090| 20 5c 68 66 69 6c 6c 20 | 43 6f 6c 6f 72 61 64 6f | \hfill |Colorado|
|000000a0| 20 53 63 68 6f 6f 6c 20 | 6f 66 20 4d 69 6e 65 73 | School |of Mines|
|000000b0| 7d 0a 0a 5c 76 73 70 61 | 63 65 7b 31 2e 35 65 78 |}..\vspa|ce{1.5ex|
|000000c0| 7d 0a 5c 43 62 7b 4e 65 | 77 74 6f 6e 27 73 20 4d |}.\Cb{Ne|wton's M|
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|000000e0| 53 6f 6c 76 69 6e 67 20 | 77 69 74 68 20 5c 4d 6d |Solving |with \Mm|
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|00000100| 7d 0a 0a 5c 6e 6f 69 6e | 64 65 6e 74 20 5c 42 66 |}..\noin|dent \Bf|
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|000001f0| 73 70 68 65 72 65 20 77 | 68 6f 73 65 20 72 61 64 |sphere w|hose rad|
|00000200| 69 75 73 20 69 73 20 24 | 61 24 20 61 73 20 24 56 |ius is $|a$ as $V|
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|00000220| 72 61 63 7b 5c 70 69 20 | 78 7d 7b 36 7d 24 7d 20 |rac{\pi |x}{6}$} |
|00000230| 24 28 33 72 5e 32 20 2b | 20 78 5e 32 29 24 20 28 |$(3r^2 +| x^2)$ (|
|00000240| 73 65 65 20 46 69 67 75 | 72 65 20 31 29 2e 20 20 |see Figu|re 1).  |
|00000250| 54 68 65 20 73 61 6d 65 | 20 76 6f 6c 75 6d 65 20 |The same| volume |
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|00000270| 20 6c 61 73 74 20 77 6f | 72 6b 73 68 65 65 74 20 | last wo|rksheet |
|00000280| 61 73 20 24 56 20 3d 20 | 24 20 7b 5c 4c 61 72 67 |as $V = |$ {\Larg|
|00000290| 65 20 24 5c 66 72 61 63 | 7b 5c 70 69 20 78 5e 32 |e $\frac|{\pi x^2|
|000002a0| 7d 7b 33 7d 24 7d 20 24 | 28 33 61 20 2d 20 78 29 |}{3}$} $|(3a - x)|
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|00000330| 5c 62 65 67 69 6e 7b 65 | 6e 75 6d 65 72 61 74 65 |\begin{e|numerate|
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|000003b0| 79 20 24 79 24 20 61 6e | 64 20 24 68 24 20 69 6e |y $y$ an|d $h$ in|
|000003c0| 20 6f 74 68 65 72 20 70 | 72 6f 62 6c 65 6d 73 20 | other p|roblems |
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|00000440| 41 70 70 65 61 6c 20 74 | 6f 20 41 72 63 68 69 6d |Appeal t|o Archim|
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|00000550| 34 24 20 61 6e 64 20 24 | 61 20 3d 20 31 24 2c 20 |4$ and $|a = 1$, |
|00000560| 77 65 20 6f 62 74 61 69 | 6e 20 74 68 65 20 65 71 |we obtai|n the eq|
|00000570| 75 61 74 69 6f 6e 20 24 | 33 78 5e 32 20 2d 20 78 |uation $|3x^2 - x|
|00000580| 5e 33 20 3d 20 31 24 20 | 69 6e 20 61 67 72 65 65 |^3 = 1$ |in agree|
|00000590| 6d 65 6e 74 20 77 69 74 | 68 20 74 68 65 20 74 65 |ment wit|h the te|
|000005a0| 78 74 2e 0a 09 5c 69 74 | 65 6d 20 55 73 65 20 74 |xt...\it|em Use t|
|000005b0| 68 65 20 49 6e 74 65 72 | 6d 65 64 69 61 74 65 20 |he Inter|mediate |
|000005c0| 56 61 6c 75 65 20 54 68 | 65 6f 72 65 6d 20 74 6f |Value Th|eorem to|
|000005d0| 20 70 72 6f 76 65 20 74 | 68 61 74 20 74 68 65 20 | prove t|hat the |
|000005e0| 65 78 70 6c 69 63 69 74 | 20 63 75 62 69 63 20 65 |explicit| cubic e|
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|00000600| 33 78 5e 32 20 2b 20 31 | 20 3d 20 30 24 2c 20 6a |3x^2 + 1| = 0$, j|
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|00000650| 64 7b 65 6e 75 6d 65 72 | 61 74 65 7d 0a 0a 5c 69 |d{enumer|ate}..\i|
|00000660| 74 65 6d 20 4f 6e 65 20 | 77 61 79 20 74 6f 20 73 |tem One |way to s|
|00000670| 6f 6c 76 65 20 65 71 75 | 61 74 69 6f 6e 73 20 77 |olve equ|ations w|
|00000680| 69 74 68 20 5c 4d 6d 61 | 5c 20 69 73 20 74 6f 20 |ith \Mma|\ is to |
|00000690| 63 6f 64 65 20 69 6e 20 | 79 6f 75 72 20 6f 77 6e |code in |your own|
|000006a0| 20 4e 65 77 74 6f 6e 27 | 73 20 6d 65 74 68 6f 64 | Newton'|s method|
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|000006c0| 6d 7d 0a 66 5b 78 5f 5d | 20 3a 3d 20 78 5e 33 20 |m}.f[x_]| := x^3 |
|000006d0| 2d 20 33 78 5e 32 20 2b | 20 31 0a 78 20 3d 20 30 |- 3x^2 +| 1.x = 0|
|000006e0| 2e 35 0a 0a 78 20 3d 20 | 78 20 2d 20 66 5b 78 5d |.5..x = |x - f[x]|
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|00000740| 61 6e 63 65 20 77 69 74 | 68 20 5c 4d 6d 61 5c 20 |ance wit|h \Mma\ |
|00000750| 73 79 6e 74 61 78 20 61 | 6e 64 20 64 65 66 69 6e |syntax a|nd defin|
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|00000790| 76 61 6c 20 74 68 61 74 | 20 77 65 20 68 61 76 65 |val that| we have|
|000007a0| 20 70 72 6f 76 65 64 20 | 63 6f 6e 74 61 69 6e 73 | proved |contains|
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|000007e0| 61 6c 75 65 20 6f 66 20 | 24 78 24 20 28 24 78 5f |alue of |$x$ ($x_|
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|00000800| 65 66 74 20 66 72 6f 6d | 20 74 68 65 20 60 60 6f |eft from| the ``o|
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|00000820| 24 29 20 75 73 65 64 20 | 6f 6e 20 74 68 65 20 72 |$) used |on the r|
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|00000840| 27 20 76 61 6c 75 65 20 | 69 73 20 30 2e 35 20 74 |' value |is 0.5 t|
|00000850| 68 65 20 66 69 72 73 74 | 20 74 69 6d 65 20 77 65 |he first| time we|
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|00000880| 27 20 74 68 65 20 74 68 | 69 72 64 20 6c 69 6e 65 |' the th|ird line|
|00000890| 20 6f 76 65 72 20 61 6e | 64 20 6f 76 65 72 20 61 | over an|d over a|
|000008a0| 67 61 69 6e 20 74 6f 20 | 67 65 6e 65 72 61 74 65 |gain to |generate|
|000008b0| 20 61 20 73 65 71 75 65 | 6e 63 65 20 6f 66 20 61 | a seque|nce of a|
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|00000980| 74 68 69 73 20 61 70 70 | 72 6f 61 63 68 20 74 6f |this app|roach to|
|00000990| 20 74 68 65 20 66 6f 6c | 6c 6f 77 69 6e 67 20 70 | the fol|lowing p|
|000009a0| 72 6f 62 6c 65 6d 73 3a | 0a 09 5c 62 65 67 69 6e |roblems:|..\begin|
|000009b0| 7b 65 6e 75 6d 65 72 61 | 74 65 7d 0a 09 5c 69 74 |{enumera|te}..\it|
|000009c0| 65 6d 20 28 33 2e 39 2e | 31 29 20 24 78 5e 32 20 |em (3.9.|1) $x^2 |
|000009d0| 2d 20 35 20 3d 20 30 3b | 20 5c 71 75 61 64 20 20 |- 5 = 0;| \quad  |
|000009e0| 78 5f 30 20 3d 20 32 2e | 35 24 20 28 69 2e 65 2e |x_0 = 2.|5$ (i.e.|
|000009f0| 20 66 69 6e 64 20 74 68 | 65 20 70 6f 73 69 74 69 | find th|e positi|
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|00000a20| 2e 32 29 20 24 78 5e 33 | 20 2d 20 32 20 3d 20 30 |.2) $x^3| - 2 = 0|
|00000a30| 3b 20 5c 71 75 61 64 20 | 20 78 5f 30 20 3d 20 31 |; \quad | x_0 = 1|
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|00000a50| 68 65 20 63 75 62 65 20 | 72 6f 6f 74 20 6f 66 20 |he cube |root of |
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|00000ac0| 20 69 6e 20 74 68 65 20 | 70 72 6f 63 65 73 73 20 | in the |process |
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|00000b40| 33 78 5e 32 20 2b 20 31 | 0a 78 20 3d 20 30 2e 35 |3x^2 + 1|.x = 0.5|
|00000b50| 3b 0a 44 6f 5b 0a 20 20 | 20 20 78 20 3d 20 78 20 |;.Do[.  |  x = x |
|00000b60| 2d 20 66 5b 78 5d 2f 66 | 27 5b 78 5d 3b 0a 20 20 |- f[x]/f|'[x];.  |
|00000b70| 20 20 50 72 69 6e 74 5b | 78 5d 2c 0a 7b 38 7d 5d |  Print[|x],.{8}]|
|00000b80| 0a 5c 65 6e 64 7b 76 65 | 72 62 61 74 69 6d 7d 0a |.\end{ve|rbatim}.|
|00000b90| 77 68 69 63 68 20 67 69 | 76 65 73 20 74 68 65 20 |which gi|ves the |
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|00000c80| 2e 0a 0a 20 5c 42 66 7b | 49 6d 70 6f 72 74 61 6e |... \Bf{|Importan|
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|00000cd0| 61 63 74 20 61 72 69 74 | 68 6d 65 74 69 63 2e 20 |act arit|hmetic. |
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|00000d60| 74 6f 20 73 69 67 6e 61 | 6c 20 74 6f 20 5c 4d 6d |to signa|l to \Mm|
|00000d70| 61 5c 20 74 68 61 74 20 | 69 6e 66 69 6e 69 74 65 |a\ that |infinite|
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|00001490| 20 2d 20 32 61 20 78 5e | 32 20 2d 20 34 20 78 20 | - 2a x^|2 - 4 x |
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|00001a30| 20 36 38 78 5e 32 20 2b | 20 31 31 30 30 78 20 2d | 68x^2 +| 1100x -|
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|000021c0| 0a                      |                         |.       |        |
+--------+-------------------------+-------------------------+--------+--------+