/ NeXT Education Software Sampler 1992 Fall / NeXT Education Software Sampler 1992 Fall.iso / Mathematics / Notebooks / CSOMinesCalculus / Chapter2 / ans2a.tex

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|00000000| 5c 64 6f 63 75 6d 65 6e | 74 73 74 79 6c 65 5b 31 |\documen|tstyle[1|
|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 32 3a 20 41 6e 73 77 | 65 72 73 20 34 7d 0a 5c | 2: 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 32 3a 20 |.\Bf{Cha|pter 2: |
|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 20 | Jack K.| Cohen  |
|00000090| 5c 68 66 69 6c 6c 20 43 | 6f 6c 6f 72 61 64 6f 20 |\hfill C|olorado |
|000000a0| 53 63 68 6f 6f 6c 20 6f | 66 20 4d 69 6e 65 73 7d |School o|f 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 28 32 2e 33 2e 31 36 |e}.\item| (2.3.16|
|000000d0| 29 20 46 6f 72 20 24 78 | 20 3c 20 37 24 2c 20 74 |) For $x| < 7$, t|
|000000e0| 68 65 20 71 75 61 6e 74 | 69 74 79 20 24 78 20 2d |he quant|ity $x -|
|000000f0| 20 37 24 20 69 73 20 5c | 45 6d 7b 6e 65 67 61 74 | 7$ is \|Em{negat|
|00000100| 69 76 65 7d 2c 20 73 6f | 20 24 7c 78 20 2d 20 37 |ive}, so| $|x - 7|
|00000110| 7c 20 3d 20 2d 28 78 20 | 2d 20 37 29 20 3d 20 37 || = -(x |- 7) = 7|
|00000120| 20 2d 20 78 24 2e 20 20 | 48 65 6e 63 65 20 24 5c | - x$.  |Hence $\|
|00000130| 6c 69 6d 5f 7b 78 20 5c | 72 69 67 68 74 61 72 72 |lim_{x \|rightarr|
|00000140| 6f 77 20 37 5e 2d 7d 24 | 20 7b 5c 4c 61 72 67 65 |ow 7^-}$| {\Large|
|00000150| 20 24 5c 66 72 61 63 7b | 37 20 2d 20 78 7d 7b 7c | $\frac{|7 - x}{||
|00000160| 78 20 2d 20 37 7c 7d 24 | 7d 20 24 20 3d 20 5c 6c |x - 7|}$|} $ = \l|
|00000170| 69 6d 5f 7b 78 20 5c 72 | 69 67 68 74 61 72 72 6f |im_{x \r|ightarro|
|00000180| 77 20 37 5e 2d 7d 24 20 | 7b 5c 4c 61 72 67 65 20 |w 7^-}$ |{\Large |
|00000190| 24 5c 66 72 61 63 7b 37 | 20 2d 20 78 7d 7b 37 20 |$\frac{7| - x}{7 |
|000001a0| 2d 20 78 7d 24 7d 20 24 | 20 3d 20 31 24 2e 20 20 |- x}$} $| = 1$.  |
|000001b0| 41 73 20 61 20 63 68 65 | 63 6b 3a 0a 0a 20 5c 54 |As a che|ck:.. \T|
|000001c0| 74 7b 4c 69 6d 69 74 54 | 61 62 6c 65 5b 28 37 20 |t{LimitT|able[(7 |
|000001d0| 2d 20 78 29 2f 41 62 73 | 5b 78 20 2d 20 37 5d 2c |- x)/Abs|[x - 7],|
|000001e0| 20 5c 7b 78 2c 20 37 2c | 20 36 2c 20 2d 31 5c 7d | \{x, 7,| 6, -1\}|
|000001f0| 5d 20 2f 2f 54 61 62 6c | 65 46 6f 72 6d 7d 20 67 |] //Tabl|eForm} g|
|00000200| 69 76 65 73 3a 0a 5c 62 | 65 67 69 6e 7b 76 65 72 |ives:.\b|egin{ver|
|00000210| 62 61 74 69 6d 7d 0a 36 | 2e 20 20 20 20 20 20 20 |batim}.6|.       |
|00000220| 20 31 2e 0a 36 2e 39 20 | 20 20 20 20 20 20 31 2e | 1..6.9 |      1.|
|00000230| 0a 36 2e 39 39 20 20 20 | 20 20 20 31 2e 0a 36 2e |.6.99   |   1..6.|
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|00000250| 39 20 20 20 20 31 2e 0a | 36 2e 39 39 39 39 39 20 |9    1..|6.99999 |
|00000260| 20 20 31 2e 0a 5c 65 6e | 64 7b 76 65 72 62 61 74 |  1..\en|d{verbat|
|00000270| 69 6d 7d 0a 28 54 68 69 | 73 20 69 73 20 6f 75 72 |im}.(Thi|s is our|
|00000280| 20 69 6d 70 6c 65 6d 65 | 6e 74 61 74 69 6f 6e 20 | impleme|ntation |
|00000290| 6f 66 20 74 68 65 20 72 | 65 63 65 6e 74 20 60 60 |of the r|ecent ``|
|000002a0| 50 72 6f 67 72 61 6d 20 | 6f 66 20 74 68 65 20 57 |Program |of the W|
|000002b0| 65 65 6b 22 2e 20 20 54 | 68 65 20 6c 61 73 74 20 |eek".  T|he last |
|000002c0| 74 77 6f 20 6e 75 6d 65 | 72 69 63 61 6c 20 61 72 |two nume|rical ar|
|000002d0| 67 75 6d 65 6e 74 73 20 | 72 65 73 70 65 63 74 69 |guments |respecti|
|000002e0| 76 65 6c 79 20 67 69 76 | 65 20 74 68 65 20 6e 75 |vely giv|e the nu|
|000002f0| 6d 62 65 72 20 6f 66 20 | 74 61 62 6c 65 20 65 6e |mber of |table en|
|00000300| 74 72 69 65 73 20 61 6e | 64 20 74 68 65 20 73 74 |tries an|d the st|
|00000310| 61 72 74 69 6e 67 20 70 | 6f 69 6e 74 20 6f 66 66 |arting p|oint off|
|00000320| 73 65 74 20 66 72 6f 6d | 20 74 68 65 20 6c 69 6d |set from| the lim|
|00000330| 69 74 20 70 6f 69 6e 74 | 20 28 73 6f 20 74 68 65 |it point| (so the|
|00000340| 20 74 61 62 6c 65 20 73 | 74 61 72 74 73 20 77 69 | table s|tarts wi|
|00000350| 74 68 20 37 20 2d 20 31 | 20 3d 20 36 20 69 6e 20 |th 7 - 1| = 6 in |
|00000360| 74 68 65 20 70 72 65 73 | 65 6e 74 20 63 61 73 65 |the pres|ent case|
|00000370| 29 2e 0a 0a 5c 69 74 65 | 6d 20 28 32 2e 33 2e 32 |)...\ite|m (2.3.2|
|00000380| 30 29 20 24 5c 73 71 72 | 74 7b 28 34 20 2b 20 78 |0) $\sqr|t{(4 + x|
|00000390| 29 5e 32 7d 20 3d 20 7c | 34 20 2b 20 78 7c 24 2e |)^2} = ||4 + x|$.|
|000003a0| 20 46 6f 72 20 24 78 20 | 3c 20 2d 34 24 20 28 65 | For $x |< -4$ (e|
|000003b0| 2e 67 2e 20 24 78 20 3d | 20 2d 35 24 29 2c 20 74 |.g. $x =| -5$), t|
|000003c0| 68 65 20 71 75 61 6e 74 | 69 74 79 20 24 34 20 2b |he quant|ity $4 +|
|000003d0| 20 78 24 20 69 73 20 5c | 45 6d 7b 6e 65 67 61 74 | x$ is \|Em{negat|
|000003e0| 69 76 65 7d 2c 20 74 68 | 75 73 20 24 7c 34 20 2b |ive}, th|us $|4 +|
|000003f0| 20 78 7c 20 3d 20 2d 28 | 34 20 2b 20 78 29 24 20 | x| = -(|4 + x)$ |
|00000400| 61 6e 64 20 73 6f 20 74 | 68 65 20 6c 69 6d 69 74 |and so t|he limit|
|00000410| 20 69 73 20 2d 31 2e 20 | 20 41 6c 74 65 72 6e 61 | is -1. | Alterna|
|00000420| 74 65 6c 79 2c 0a 5c 62 | 65 67 69 6e 7b 76 65 72 |tely,.\b|egin{ver|
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|00000450| 28 34 20 2b 20 78 29 5e | 32 5d 2c 20 7b 78 2c 20 |(4 + x)^|2], {x, |
|00000460| 2d 34 2c 20 36 2c 20 2d | 31 7d 5d 20 2f 2f 54 61 |-4, 6, -|1}] //Ta|
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|000004a0| 20 2d 31 2e 0a 2d 34 2e | 30 30 31 20 20 20 20 20 | -1..-4.|001     |
|000004b0| 2d 31 2e 0a 2d 34 2e 30 | 30 30 31 20 20 20 20 2d |-1..-4.0|001    -|
|000004c0| 31 2e 0a 2d 34 2e 30 30 | 30 30 31 20 20 20 2d 31 |1..-4.00|001   -1|
|000004d0| 2e 0a 5c 65 6e 64 7b 76 | 65 72 62 61 74 69 6d 7d |..\end{v|erbatim}|
|000004e0| 0a 0a 5c 69 74 65 6d 20 | 28 32 2e 33 2e 32 36 29 |..\item |(2.3.26)|
|000004f0| 20 24 78 20 3d 20 35 24 | 20 69 73 20 74 68 65 20 | $x = 5$| is the |
|00000500| 70 6f 69 6e 74 20 77 68 | 65 72 65 20 74 68 65 20 |point wh|ere the |
|00000510| 6c 69 6d 69 74 73 20 64 | 6f 6e 27 74 20 65 78 69 |limits d|on't exi|
|00000520| 73 74 2e 20 20 54 68 65 | 20 66 75 6e 63 74 69 6f |st.  The| functio|
|00000530| 6e 20 62 65 63 6f 6d 65 | 73 20 70 6f 73 69 74 69 |n become|s positi|
|00000540| 76 65 20 6c 61 72 67 65 | 20 6e 6f 20 6d 61 74 74 |ve large| no matt|
|00000550| 65 72 20 77 68 69 63 68 | 20 73 69 64 65 20 77 65 |er which| side we|
|00000560| 20 61 70 70 72 6f 61 63 | 68 20 35 20 66 72 6f 6d | approac|h 5 from|
|00000570| 3a 20 24 5c 6c 69 6d 5f | 7b 78 20 5c 72 69 67 68 |: $\lim_|{x \righ|
|00000580| 74 61 72 72 6f 77 20 35 | 5e 2d 7d 20 3d 20 5c 6c |tarrow 5|^-} = \l|
|00000590| 69 6d 5f 7b 78 20 5c 72 | 69 67 68 74 61 72 72 6f |im_{x \r|ightarro|
|000005a0| 77 20 35 5e 2b 7d 20 3d | 20 2b 5c 69 6e 66 74 79 |w 5^+} =| +\infty|
|000005b0| 24 2e 0a 0a 0a 5c 65 6e | 64 7b 65 6e 75 6d 65 72 |$....\en|d{enumer|
|000005c0| 61 74 65 7d 0a 5c 65 6e | 64 7b 64 6f 63 75 6d 65 |ate}.\en|d{docume|
|000005d0| 6e 74 7d 0a             |                         |nt}.    |        |
+--------+-------------------------+-------------------------+--------+--------+