/ NeXT Education Software Sampler 1992 Fall / NeXT Education Software Sampler 1992 Fall.iso / Mathematics / Notebooks / CSOMinesCalculus / Chapter1 / ans4.tex

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|00000010| 31 70 74 2c 66 6c 65 71 | 6e 2c 65 70 73 66 2c 63 |1pt,fleq|n,epsf,c|
|00000020| 61 6c 63 5d 7b 61 72 74 | 69 63 6c 65 7d 0a 5c 6d |alc]{art|icle}.\m|
|00000030| 61 72 6b 72 69 67 68 74 | 7b 43 68 61 70 74 65 72 |arkright|{Chapter|
|00000040| 20 31 3a 20 41 6e 73 77 | 65 72 73 20 34 7d 0a 5c | 1: Answ|ers 4}.\|
|00000050| 62 65 67 69 6e 7b 64 6f | 63 75 6d 65 6e 74 7d 0a |begin{do|cument}.|
|00000060| 0a 5c 42 66 7b 43 68 61 | 70 74 65 72 20 31 3a 20 |.\Bf{Cha|pter 1: |
|00000070| 41 6e 73 77 65 72 73 20 | 34 20 5c 68 66 69 6c 6c |Answers |4 \hfill|
|00000080| 20 4a 61 63 6b 20 4b 2e | 20 43 6f 68 65 6e 20 5c | Jack K.| Cohen \|
|00000090| 68 66 69 6c 6c 20 43 6f | 6c 6f 72 61 64 6f 20 53 |hfill Co|lorado S|
|000000a0| 63 68 6f 6f 6c 20 6f 66 | 20 4d 69 6e 65 73 7d 0a |chool of| Mines}.|
|000000b0| 0a 5c 62 65 67 69 6e 7b | 65 6e 75 6d 65 72 61 74 |.\begin{|enumerat|
|000000c0| 65 7d 0a 5c 69 74 65 6d | 20 20 24 72 20 5c 61 70 |e}.\item|  $r \ap|
|000000d0| 70 72 6f 78 20 35 30 2c | 30 30 38 2e 33 33 24 2e |prox 50,|008.33$.|
|000000e0| 20 20 41 20 72 65 61 6c | 20 70 69 74 66 61 6c 6c |  A real| pitfall|
|000000f0| 20 68 65 72 65 20 69 73 | 20 67 65 74 74 69 6e 67 | here is| getting|
|00000100| 20 74 72 61 70 70 65 64 | 20 69 6e 74 6f 20 69 6e | trapped| into in|
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|00000120| 72 65 67 69 6d 65 20 77 | 68 65 72 65 20 24 72 20 |regime w|here $r |
|00000130| 3c 20 73 20 3d 20 31 30 | 30 30 24 2d 2d 2d 60 60 |< s = 10|00$---``|
|00000140| 6f 62 76 69 6f 75 73 6c | 79 27 27 20 74 68 65 20 |obviousl|y'' the |
|00000150| 72 61 64 69 75 73 20 6f | 66 20 74 68 65 20 61 73 |radius o|f the as|
|00000160| 74 65 72 6f 69 64 20 77 | 6f 75 6c 64 20 73 65 6e |teroid w|ould sen|
|00000170| 73 69 62 6c 79 20 62 65 | 20 6d 75 63 68 20 6c 61 |sibly be| much la|
|00000180| 72 67 65 72 20 74 68 61 | 6e 20 24 73 24 2e 20 20 |rger tha|n $s$.  |
|00000190| 20 42 79 20 74 68 65 20 | 77 61 79 2c 20 74 68 65 | By the |way, the|
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|000001b0| 79 27 27 20 69 6e 20 6d | 61 74 68 65 6d 61 74 69 |y'' in m|athemati|
|000001c0| 63 73 20 6d 65 61 6e 73 | 20 60 60 63 6c 65 61 72 |cs means| ``clear|
|000001d0| 20 77 68 65 6e 20 79 6f | 75 20 74 68 69 6e 6b 20 | when yo|u think |
|000001e0| 61 62 6f 75 74 20 69 74 | 20 66 6f 72 20 61 77 68 |about it| for awh|
|000001f0| 69 6c 65 27 27 2e 20 20 | 46 6f 72 20 65 78 61 6d |ile''.  |For exam|
|00000200| 70 6c 65 2c 20 73 6f 6d | 65 20 72 65 61 6c 20 73 |ple, som|e real s|
|00000210| 6d 61 72 74 20 70 65 6f | 70 6c 65 20 68 61 76 65 |mart peo|ple have|
|00000220| 20 62 65 65 6e 20 6b 6e | 6f 77 6e 20 74 6f 20 68 | been kn|own to h|
|00000230| 61 76 65 20 70 75 74 20 | 75 70 20 61 20 66 69 72 |ave put |up a fir|
|00000240| 73 74 20 5c 54 74 7b 50 | 6c 6f 74 7d 20 66 6f 72 |st \Tt{P|lot} for|
|00000250| 20 76 61 6c 75 65 73 20 | 6f 66 20 24 72 24 20 6e | values |of $r$ n|
|00000260| 65 61 72 20 30 20 69 6e | 20 73 6f 6c 76 69 6e 67 |ear 0 in| solving|
|00000270| 20 74 68 65 20 70 72 65 | 73 65 6e 74 20 70 72 6f | the pre|sent pro|
|00000280| 62 6c 65 6d 2e 0a 0a 5c | 6e 6f 69 6e 64 65 6e 74 |blem...\|noindent|
|00000290| 20 41 6e 6f 74 68 65 72 | 20 70 72 6f 62 6c 65 6d | Another| problem|
|000002a0| 20 69 73 20 74 68 61 74 | 20 77 65 20 68 61 76 65 | is that| we have|
|000002b0| 20 6e 6f 20 69 64 65 61 | 20 77 68 65 72 65 20 74 | no idea| where t|
|000002c0| 6f 20 68 75 6e 74 20 66 | 6f 72 20 74 68 65 20 60 |o hunt f|or the `|
|000002d0| 60 70 68 79 73 69 63 61 | 6c 6c 79 20 73 65 6e 73 |`physica|lly sens|
|000002e0| 69 62 6c 65 27 27 20 72 | 6f 6f 74 2e 20 20 41 73 |ible'' r|oot.  As|
|000002f0| 20 70 72 6f 6d 69 73 65 | 64 20 69 6e 20 74 68 65 | promise|d in the|
|00000300| 20 70 72 6f 62 6c 65 6d | 20 73 74 61 74 65 6d 65 | problem| stateme|
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|00000330| 74 68 69 73 20 69 6e 20 | 61 20 66 65 77 20 77 65 |this in |a few we|
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|00000350| 32 34 29 20 41 20 73 6b | 65 74 63 68 20 72 65 76 |24) A sk|etch rev|
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|00000370| 69 67 68 74 20 74 72 69 | 61 6e 67 6c 65 73 3a 20 |ight tri|angles: |
|00000380| 24 64 20 3d 20 5c 73 71 | 72 74 7b 32 7d 24 2e 0a |$d = \sq|rt{2}$..|
|00000390| 0a 5c 69 74 65 6d 20 20 | 49 74 20 63 68 65 63 6b |.\item  |It check|
|000003a0| 73 2e 0a 0a 5c 69 74 65 | 6d 20 4e 6f 20 73 6f 6c |s...\ite|m No sol|
|000003b0| 75 74 69 6f 6e 20 67 69 | 76 65 6e 20 74 6f 20 68 |ution gi|ven to h|
|000003c0| 6f 6e 6f 72 20 70 72 6f | 62 6c 65 6d 73 2e 0a 0a |onor pro|blems...|
|000003d0| 5c 69 74 65 6d 20 28 31 | 2e 32 2e 32 35 29 20 20 |\item (1|.2.25)  |
|000003e0| 24 64 20 3d 20 24 20 7b | 5c 4c 61 72 67 65 24 5c |$d = $ {|\Large$\|
|000003f0| 66 72 61 63 7b 34 7d 7b | 31 33 7d 24 7d 20 24 5c |frac{4}{|13}$} $\|
|00000400| 73 71 72 74 7b 32 36 7d | 24 2e 0a 0a 5c 69 74 65 |sqrt{26}|$...\ite|
|00000410| 6d 20 20 0a 20 20 20 20 | 20 20 20 20 5c 62 65 67 |m  .    |    \beg|
|00000420| 69 6e 7b 65 6e 75 6d 65 | 72 61 74 65 7d 0a 20 20 |in{enume|rate}.  |
|00000430| 20 20 20 20 20 20 5c 69 | 74 65 6d 20 49 74 20 63 |      \i|tem It c|
|00000440| 68 65 63 6b 73 20 28 64 | 65 73 70 69 74 65 20 74 |hecks (d|espite t|
|00000450| 68 65 20 61 6e 73 77 65 | 72 20 69 6e 20 74 68 65 |he answe|r in the|
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|00000480| 6d 20 54 68 65 20 66 6f | 72 6d 20 24 79 20 3d 20 |m The fo|rm $y = |
|00000490| 6d 78 20 2b 62 5f 31 2c | 20 79 20 3d 20 6d 78 20 |mx +b_1,| y = mx |
|000004a0| 2b 62 5f 32 24 20 64 6f | 65 73 6e 27 74 20 69 6e |+b_2$ do|esn't in|
|000004b0| 63 6c 75 64 65 20 74 68 | 65 20 63 61 73 65 20 6f |clude th|e case o|
|000004c0| 66 20 74 77 6f 20 5c 45 | 6d 7b 76 65 72 74 69 63 |f two \E|m{vertic|
|000004d0| 61 6c 7d 20 6c 69 6e 65 | 73 2e 0a 20 20 20 20 20 |al} line|s..     |
|000004e0| 20 20 20 5c 65 6e 64 7b | 65 6e 75 6d 65 72 61 74 |   \end{|enumerat|
|000004f0| 65 7d 0a 0a 5c 69 74 65 | 6d 0a 20 20 20 20 20 20 |e}..\ite|m.      |
|00000500| 20 20 5c 62 65 67 69 6e | 7b 65 6e 75 6d 65 72 61 |  \begin|{enumera|
|00000510| 74 65 7d 0a 20 20 20 20 | 20 20 20 20 5c 69 74 65 |te}.    |    \ite|
|00000520| 6d 20 24 79 20 3d 20 33 | 78 20 2d 20 32 24 2e 0a |m $y = 3|x - 2$..|
|00000530| 20 20 20 20 20 20 20 20 | 0a 20 20 20 20 20 20 20 |        |.       |
|00000540| 20 5c 69 74 65 6d 20 24 | 79 20 3d 20 32 2e 31 78 | \item $|y = 2.1x|
|00000550| 20 2d 20 31 2e 31 24 2e | 0a 20 20 20 20 20 20 20 | - 1.1$.|.       |
|00000560| 20 0a 20 20 20 20 20 20 | 20 20 5c 69 74 65 6d 20 | .      |  \item |
|00000570| 24 79 20 3d 20 32 2e 30 | 31 78 20 2d 20 31 2e 30 |$y = 2.0|1x - 1.0|
|00000580| 31 24 2e 0a 20 20 20 20 | 20 20 20 20 0a 20 20 20 |1$..    |    .   |
|00000590| 20 20 20 20 20 5c 69 74 | 65 6d 20 20 53 6c 6f 70 |     \it|em  Slop|
|000005a0| 65 20 69 73 20 24 6d 20 | 3d 20 32 24 2c 20 74 61 |e is $m |= 2$, ta|
|000005b0| 6e 67 65 6e 74 20 6c 69 | 6e 65 20 69 73 20 24 79 |ngent li|ne is $y|
|000005c0| 20 3d 20 32 78 20 2d 20 | 31 24 2e 0a 0a 20 20 20 | = 2x - |1$...   |
|000005d0| 20 20 20 20 20 5c 69 74 | 65 6d 20 20 44 69 64 20 |     \it|em  Did |
|000005e0| 79 6f 75 20 6e 6f 74 69 | 63 65 20 74 68 61 74 20 |you noti|ce that |
|000005f0| 74 68 65 20 63 75 72 76 | 65 20 66 6f 72 20 24 79 |the curv|e for $y|
|00000600| 20 3d 20 78 5e 32 24 20 | 77 61 73 20 61 6c 6d 6f | = x^2$ |was almo|
|00000610| 73 74 20 6c 69 6e 65 61 | 72 20 69 6e 20 74 68 65 |st linea|r in the|
|00000620| 20 60 60 7a 6f 6f 6d 65 | 64 27 27 20 70 6c 6f 74 | ``zoome|d'' plot|
|00000630| 3f 20 0a 20 20 20 20 20 | 20 20 20 0a 20 20 20 20 |? .     |   .    |
|00000640| 20 20 20 20 5c 69 74 65 | 6d 20 54 68 65 20 70 72 |    \ite|m The pr|
|00000650| 65 63 69 73 65 20 64 65 | 66 69 6e 69 74 69 6f 6e |ecise de|finition|
|00000660| 20 72 65 71 75 69 72 65 | 73 20 63 61 6c 63 75 6c | require|s calcul|
|00000670| 75 73 20 63 6f 6e 63 65 | 70 74 73 20 74 68 61 74 |us conce|pts that|
|00000680| 20 77 69 6c 6c 20 62 65 | 20 64 69 73 63 75 73 73 | will be| discuss|
|00000690| 65 64 20 73 6f 6f 6e 2e | 20 20 28 41 74 74 65 6d |ed soon.|  (Attem|
|000006a0| 70 74 73 20 74 6f 20 67 | 69 76 65 20 61 20 64 65 |pts to g|ive a de|
|000006b0| 66 69 6e 69 74 69 6f 6e | 20 69 6e 20 74 65 72 6d |finition| in term|
|000006c0| 73 20 6f 66 20 6f 6e 6c | 79 20 61 6c 67 65 62 72 |s of onl|y algebr|
|000006d0| 61 20 61 6e 64 20 67 65 | 6f 6d 65 74 72 79 20 63 |a and ge|ometry c|
|000006e0| 6f 6e 63 65 70 74 73 20 | 61 72 65 20 73 75 72 65 |oncepts |are sure|
|000006f0| 20 74 6f 20 62 65 20 69 | 6e 61 64 65 71 75 61 74 | to be i|nadequat|
|00000700| 65 2e 29 20 20 20 49 74 | 20 5c 45 6d 7b 69 73 7d |e.)   It| \Em{is}|
|00000710| 20 61 20 73 65 63 61 6e | 74 20 6c 69 6e 65 2e 20 | a secan|t line. |
|00000720| 0a 20 20 20 20 20 20 20 | 20 5c 65 6e 64 7b 65 6e |.       | \end{en|
|00000730| 75 6d 65 72 61 74 65 7d | 20 0a 20 20 20 20 20 20 |umerate}| .      |
|00000740| 20 20 0a 5c 65 6e 64 7b | 65 6e 75 6d 65 72 61 74 |  .\end{|enumerat|
|00000750| 65 7d 0a 5c 65 6e 64 7b | 64 6f 63 75 6d 65 6e 74 |e}.\end{|document|
|00000760| 7d 0a                   |                         |}.      |        |
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