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|00000030| 67 68 74 7b 43 68 61 70 | 74 65 72 20 31 3a 20 57 |ght{Chap|ter 1: W|
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|00000060| 42 66 7b 43 68 61 70 74 | 65 72 20 31 3a 20 41 6e |Bf{Chapt|er 1: An|
|00000070| 73 77 65 72 73 20 31 20 | 5c 68 66 69 6c 6c 20 4a |swers 1 |\hfill J|
|00000080| 61 63 6b 20 4b 2e 20 43 | 6f 68 65 6e 20 5c 68 66 |ack K. C|ohen \hf|
|00000090| 69 6c 6c 20 43 6f 6c 6f | 72 61 64 6f 20 53 63 68 |ill Colo|rado Sch|
|000000a0| 6f 6f 6c 20 6f 66 20 4d | 69 6e 65 73 7d 0a 0a 5c |ool of M|ines}..\|
|000000b0| 62 65 67 69 6e 7b 65 6e | 75 6d 65 72 61 74 65 7d |begin{en|umerate}|
|000000c0| 0a 5c 69 74 65 6d 20 28 | 31 2e 35 36 29 20 20 24 |.\item (|1.56)  $|
|000000d0| 41 20 3d 20 50 5e 32 20 | 2f 20 31 36 24 2e 0a 5c |A = P^2 |/ 16$..\|
|000000e0| 62 65 67 69 6e 7b 66 69 | 67 75 72 65 7d 5b 68 74 |begin{fi|gure}[ht|
|000000f0| 62 5d 0a 5c 65 70 73 66 | 79 73 69 7a 65 20 37 30 |b].\epsf|ysize 70|
|00000100| 70 74 0a 5c 63 65 6e 74 | 65 72 6c 69 6e 65 7b 5c |pt.\cent|erline{\|
|00000110| 65 70 73 66 66 69 6c 65 | 7b 61 6e 73 31 70 31 2e |epsffile|{ans1p1.|
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|00000140| 73 20 61 20 66 75 6e 63 | 74 69 6f 6e 20 6f 66 20 |s a func|tion of |
|00000150| 70 65 72 69 6d 65 74 65 | 72 20 2e 7d 0a 5c 65 6e |perimete|r .}.\en|
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|00000170| 6d 20 28 31 2e 35 38 29 | 20 24 56 20 3d 24 20 7b |m (1.58)| $V =$ {|
|00000180| 5c 4c 61 72 67 65 20 24 | 5c 66 72 61 63 7b 53 5e |\Large $|\frac{S^|
|00000190| 7b 32 2f 33 7d 7d 7b 36 | 20 5c 73 71 72 74 7b 5c |{2/3}}{6| \sqrt{\|
|000001a0| 70 69 7d 7d 24 7d 2e 0a | 0a 5c 62 65 67 69 6e 7b |pi}}$}..|.\begin{|
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|000001c0| 73 66 79 73 69 7a 65 20 | 37 35 70 74 0a 5c 63 65 |sfysize |75pt.\ce|
|000001d0| 6e 74 65 72 6c 69 6e 65 | 7b 5c 65 70 73 66 66 69 |nterline|{\epsffi|
|000001e0| 6c 65 7b 61 6e 73 31 70 | 32 2e 65 70 73 7d 7d 0a |le{ans1p|2.eps}}.|
|000001f0| 5c 63 61 70 74 69 6f 6e | 7b 50 72 6f 62 6c 65 6d |\caption|{Problem|
|00000200| 20 32 3a 20 56 6f 6c 75 | 6d 65 20 61 73 20 61 20 | 2: Volu|me as a |
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|00000220| 61 63 65 20 61 72 65 61 | 20 2e 7d 0a 5c 65 6e 64 |ace area| .}.\end|
|00000230| 7b 66 69 67 75 72 65 7d | 0a 0a 0a 0a 5c 69 74 65 |{figure}|....\ite|
|00000240| 6d 20 20 46 6f 72 20 24 | 61 20 20 3e 20 30 24 2c |m  For $|a  > 0$,|
|00000250| 20 20 24 7c 20 61 7c 20 | 3d 20 61 24 2e 20 20 4d |  $| a| |= a$.  M|
|00000260| 6f 72 65 6f 76 65 72 20 | 20 24 7c 2d 61 7c 20 3d |oreover | $|-a| =|
|00000270| 20 61 24 20 74 6f 6f 2e | 20 20 4e 6f 74 65 20 74 | a$ too.|  Note t|
|00000280| 68 61 74 20 24 7c 2d 61 | 7c 20 3d 20 7c 61 7c 24 |hat $|-a|| = |a|$|
|00000290| 20 5c 45 6d 7b 61 6c 6c | 7d 20 74 68 65 20 74 69 | \Em{all|} the ti|
|000002a0| 6d 65 21 0a 0a 5c 69 74 | 65 6d 20 20 46 6f 72 20 |me!..\it|em  For |
|000002b0| 24 61 20 3c 20 30 24 2c | 20 20 24 7c 20 61 7c 20 |$a < 0$,|  $| a| |
|000002c0| 3d 20 2d 61 24 3b 20 20 | 24 7c 2d 61 7c 20 3d 20 |= -a$;  |$|-a| = |
|000002d0| 2d 61 24 2e 0a 0a 5c 69 | 74 65 6d 20 20 46 6f 72 |-a$...\i|tem  For|
|000002e0| 20 24 78 20 5c 67 65 71 | 20 61 24 2c 20 74 68 65 | $x \geq| a$, the|
|000002f0| 20 69 6e 65 71 75 61 6c | 69 74 79 20 72 65 61 64 | inequal|ity read|
|00000300| 73 20 24 62 20 5c 6c 65 | 71 20 78 20 2d 20 61 20 |s $b \le|q x - a |
|00000310| 5c 6c 65 71 20 63 24 2e | 20 20 41 64 64 69 6e 67 |\leq c$.|  Adding|
|00000320| 20 24 61 24 20 65 76 65 | 72 79 77 68 65 72 65 20 | $a$ eve|rywhere |
|00000330| 67 69 76 65 73 20 24 61 | 20 2b 20 62 20 5c 6c 65 |gives $a| + b \le|
|00000340| 71 20 78 20 5c 6c 65 71 | 20 61 20 2b 20 63 24 2e |q x \leq| a + c$.|
|00000350| 20 20 53 69 6e 63 65 20 | 24 63 20 5c 67 65 71 20 |  Since |$c \geq |
|00000360| 62 24 2c 20 74 68 69 73 | 20 69 6e 65 71 75 61 6c |b$, this| inequal|
|00000370| 69 74 79 20 63 61 6e 20 | 62 65 20 73 61 74 69 73 |ity can |be satis|
|00000380| 66 69 65 64 20 62 79 20 | 61 20 6e 6f 6e 2d 65 6d |fied by |a non-em|
|00000390| 70 74 79 20 73 65 74 20 | 61 6e 64 20 73 69 6e 63 |pty set |and sinc|
|000003a0| 65 20 24 62 20 5c 67 65 | 71 20 30 24 20 74 68 65 |e $b \ge|q 0$ the|
|000003b0| 20 6c 6f 77 65 72 20 6c | 69 6d 69 74 20 69 73 20 | lower l|imit is |
|000003c0| 63 6f 6d 70 61 74 69 62 | 6c 65 20 77 69 74 68 20 |compatib|le with |
|000003d0| 74 68 65 20 61 73 73 75 | 6d 70 74 69 6f 6e 20 24 |the assu|mption $|
|000003e0| 78 20 5c 67 65 71 20 61 | 24 2e 20 20 54 68 75 73 |x \geq a|$.  Thus|
|000003f0| 20 70 61 72 74 20 6f 66 | 20 74 68 65 20 73 6f 6c | part of| the sol|
|00000400| 75 74 69 6f 6e 20 73 65 | 74 20 69 73 20 74 68 65 |ution se|t is the|
|00000410| 20 63 6c 6f 73 65 64 20 | 69 6e 74 65 72 76 61 6c | closed |interval|
|00000420| 2c 20 20 24 5b 61 20 2b | 20 62 2c 20 61 20 2b 20 |,  $[a +| b, a + |
|00000430| 63 5d 24 2e 20 20 53 69 | 6d 69 6c 61 72 20 61 6e |c]$.  Si|milar an|
|00000440| 61 6c 79 73 69 73 20 6f | 66 20 74 68 65 20 24 78 |alysis o|f the $x|
|00000450| 20 3c 20 61 24 20 72 61 | 6e 67 65 20 67 69 76 65 | < a$ ra|nge give|
|00000460| 73 20 74 68 65 20 6f 74 | 68 65 72 20 70 61 72 74 |s the ot|her part|
|00000470| 20 6f 66 20 74 68 65 20 | 73 6f 6c 75 74 69 6f 6e | of the |solution|
|00000480| 20 73 65 74 2e 0a 0a 5c | 69 74 65 6d 0a 20 20 20 | set...\|item.   |
|00000490| 20 20 20 20 20 5c 62 65 | 67 69 6e 7b 65 6e 75 6d |     \be|gin{enum|
|000004a0| 65 72 61 74 65 7d 0a 20 | 20 20 20 20 20 20 20 5c |erate}. |       \|
|000004b0| 69 74 65 6d 20 54 68 65 | 20 73 6f 6c 75 74 69 6f |item The| solutio|
|000004c0| 6e 20 73 65 74 20 69 73 | 20 65 6d 70 74 79 2e 0a |n set is| empty..|
|000004d0| 20 20 20 20 20 20 20 20 | 5c 69 74 65 6d 20 54 68 |        |\item Th|
|000004e0| 65 20 73 6f 6c 75 74 69 | 6f 6e 20 73 65 74 20 69 |e soluti|on set i|
|000004f0| 73 20 74 68 65 20 73 61 | 6d 65 20 61 73 20 66 6f |s the sa|me as fo|
|00000500| 72 20 24 62 20 3d 20 30 | 24 2e 0a 20 20 20 20 20 |r $b = 0|$..     |
|00000510| 20 20 20 5c 65 6e 64 7b | 65 6e 75 6d 65 72 61 74 |   \end{|enumerat|
|00000520| 65 7d 0a 0a 5c 69 74 65 | 6d 0a 20 20 20 20 20 20 |e}..\ite|m.      |
|00000530| 20 20 5c 62 65 67 69 6e | 7b 65 6e 75 6d 65 72 61 |  \begin|{enumera|
|00000540| 74 65 7d 0a 20 20 20 20 | 20 20 20 20 5c 69 74 65 |te}.    |    \ite|
|00000550| 6d 20 54 68 65 20 73 6f | 6c 75 74 69 6f 6e 20 73 |m The so|lution s|
|00000560| 65 74 20 6f 66 20 24 62 | 20 5c 6c 65 71 20 7c 78 |et of $b| \leq |x|
|00000570| 20 2d 20 61 7c 20 3c 20 | 63 24 20 69 73 20 24 28 | - a| < |c$ is $(|
|00000580| 61 20 2d 20 63 2c 20 61 | 20 2d 20 62 5d 20 5c 63 |a - c, a| - b] \c|
|00000590| 75 70 20 5b 61 20 2b 20 | 62 2c 20 61 20 2b 20 63 |up [a + |b, a + c|
|000005a0| 29 24 2e 0a 20 20 20 20 | 20 20 20 20 5c 69 74 65 |)$..    |    \ite|
|000005b0| 6d 20 54 68 65 20 73 6f | 6c 75 74 69 6f 6e 20 73 |m The so|lution s|
|000005c0| 65 74 20 6f 66 20 24 62 | 20 3c 20 7c 78 20 2d 20 |et of $b| < |x - |
|000005d0| 61 7c 20 5c 6c 65 71 20 | 63 24 20 69 73 20 24 5b |a| \leq |c$ is $[|
|000005e0| 61 20 2d 20 63 2c 20 61 | 20 2d 20 62 29 20 5c 63 |a - c, a| - b) \c|
|000005f0| 75 70 20 28 61 20 2b 20 | 62 2c 20 61 20 2b 20 63 |up (a + |b, a + c|
|00000600| 5d 24 2e 0a 20 20 20 20 | 20 20 20 20 5c 69 74 65 |]$..    |    \ite|
|00000610| 6d 20 54 68 65 20 73 6f | 6c 75 74 69 6f 6e 20 73 |m The so|lution s|
|00000620| 65 74 20 6f 66 20 24 62 | 20 3c 7c 78 20 2d 20 61 |et of $b| <|x - a|
|00000630| 7c 20 5c 20 3c 63 24 20 | 69 73 20 24 28 61 20 2d || \ <c$ |is $(a -|
|00000640| 20 63 2c 20 61 20 2d 20 | 62 29 20 5c 63 75 70 20 | c, a - |b) \cup |
|00000650| 28 61 20 2b 20 62 2c 20 | 61 20 2b 20 63 29 24 2e |(a + b, |a + c)$.|
|00000660| 0a 20 20 20 20 20 20 20 | 20 5c 65 6e 64 7b 65 6e |.       | \end{en|
|00000670| 75 6d 65 72 61 74 65 7d | 0a 0a 5c 69 74 65 6d 20 |umerate}|..\item |
|00000680| 28 31 2e 31 39 29 20 41 | 66 74 65 72 20 64 69 76 |(1.19) A|fter div|
|00000690| 69 64 69 6e 67 20 62 79 | 20 35 2c 20 77 65 20 73 |iding by| 5, we s|
|000006a0| 65 65 20 74 68 61 74 20 | 24 61 20 3d 20 34 2f 35 |ee that |$a = 4/5|
|000006b0| 2c 20 62 20 3d 20 32 2f | 35 2c 20 63 20 3d 20 34 |, b = 2/|5, c = 4|
|000006c0| 2f 35 24 2e 20 20 54 68 | 75 73 20 74 68 65 20 73 |/5$.  Th|us the s|
|000006d0| 6f 6c 75 74 69 6f 6e 20 | 73 65 74 20 24 5b 61 20 |olution |set $[a |
|000006e0| 2d 20 63 2c 20 61 20 2d | 62 5d 20 5c 63 75 70 20 |- c, a -|b] \cup |
|000006f0| 5b 61 20 2b 20 62 2c 20 | 61 20 2b 20 63 5d 24 20 |[a + b, |a + c]$ |
|00000700| 62 65 63 6f 6d 65 73 20 | 24 5b 30 2c 20 32 2f 35 |becomes |$[0, 2/5|
|00000710| 5d 20 5c 63 75 70 20 5b | 36 2f 35 2c 20 38 2f 35 |] \cup [|6/5, 8/5|
|00000720| 5d 24 2e 0a 0a 5c 69 74 | 65 6d 20 28 31 2e 31 37 |]$...\it|em (1.17|
|00000730| 29 20 49 66 20 24 63 20 | 5c 72 69 67 68 74 61 72 |) If $c |\rightar|
|00000740| 72 6f 77 20 2b 5c 69 6e | 66 74 79 24 2c 20 74 68 |row +\in|fty$, th|
|00000750| 65 6e 20 74 68 65 20 73 | 6f 6c 75 74 69 6f 6e 20 |en the s|olution |
|00000760| 73 65 74 20 62 65 63 6f | 6d 65 73 20 24 28 2d 5c |set beco|mes $(-\|
|00000770| 69 6e 66 74 79 2c 20 61 | 20 2d 62 5d 20 5c 63 75 |infty, a| -b] \cu|
|00000780| 70 20 5b 61 20 2b 20 62 | 2c 20 2b 5c 69 6e 66 74 |p [a + b|, +\inft|
|00000790| 79 29 24 2e 20 20 49 6e | 65 71 75 61 6c 69 74 79 |y)$.  In|equality|
|000007a0| 20 24 7c 31 20 2d 20 33 | 78 7c 20 3e 20 32 24 20 | $|1 - 3|x| > 2$ |
|000007b0| 63 61 6e 20 62 65 20 72 | 65 77 72 69 74 74 65 6e |can be r|ewritten|
|000007c0| 20 61 73 20 24 32 2f 33 | 20 3c 20 7c 78 20 2d 20 | as $2/3| < |x - |
|000007d0| 31 2f 33 7c 24 2c 20 73 | 6f 20 24 61 20 3d 20 31 |1/3|$, s|o $a = 1|
|000007e0| 2f 33 2c 20 62 20 3d 20 | 32 2f 33 24 2e 20 20 4e |/3, b = |2/3$.  N|
|000007f0| 6f 74 69 6e 67 20 74 68 | 65 20 73 74 72 69 63 74 |oting th|e strict|
|00000800| 20 69 6e 65 71 75 61 6c | 69 74 69 65 73 20 69 6e | inequal|ities in|
|00000810| 20 74 68 69 73 20 70 72 | 6f 62 6c 65 6d 2c 20 74 | this pr|oblem, t|
|00000820| 68 65 20 73 6f 6c 75 74 | 69 6f 6e 20 73 65 74 20 |he solut|ion set |
|00000830| 62 65 63 6f 6d 65 73 20 | 24 28 2d 5c 69 6e 66 74 |becomes |$(-\inft|
|00000840| 79 2c 20 2d 31 2f 33 29 | 20 5c 63 75 70 28 31 2c |y, -1/3)| \cup(1,|
|00000850| 20 2b 5c 69 6e 66 74 79 | 29 24 2e 0a 0a 5c 69 74 | +\infty|)$...\it|
|00000860| 65 6d 20 28 31 2e 31 35 | 29 20 20 49 66 20 24 62 |em (1.15|)  If $b|
|00000870| 20 3d 20 30 24 2c 20 74 | 68 65 6e 20 74 68 65 20 | = 0$, t|hen the |
|00000880| 73 6f 6c 75 74 69 6f 6e | 20 73 65 74 20 24 5b 61 |solution| set $[a|
|00000890| 20 2d 20 63 2c 20 61 20 | 2d 20 62 5d 20 5c 63 75 | - c, a |- b] \cu|
|000008a0| 70 20 5b 61 20 2b 20 62 | 2c 20 61 20 2b 20 63 5d |p [a + b|, a + c]|
|000008b0| 24 20 62 65 63 6f 6d 65 | 73 20 24 5b 61 20 2d 20 |$ become|s $[a - |
|000008c0| 63 2c 20 61 20 5d 20 5c | 63 75 70 20 5b 61 2c 20 |c, a ] \|cup [a, |
|000008d0| 61 20 2b 20 63 5d 24 20 | 20 6f 72 20 73 69 6d 70 |a + c]$ | or simp|
|000008e0| 6c 79 20 24 5b 61 20 2d | 20 63 2c 20 20 61 20 2b |ly $[a -| c,  a +|
|000008f0| 20 63 5d 24 2e 20 20 52 | 65 77 72 69 74 65 20 20 | c]$.  R|ewrite  |
|00000900| 24 7c 33 20 2d 20 32 78 | 7c 20 3c 20 35 24 20 61 |$|3 - 2x|| < 5$ a|
|00000910| 73 20 24 7c 78 20 2d 20 | 33 2f 32 7c 20 3c 20 35 |s $|x - |3/2| < 5|
|00000920| 2f 32 24 20 74 6f 20 69 | 64 65 6e 74 69 66 79 20 |/2$ to i|dentify |
|00000930| 24 61 20 3d 20 33 2f 32 | 2c 20 63 20 3d 20 35 2f |$a = 3/2|, c = 5/|
|00000940| 32 24 2e 20 20 20 54 68 | 65 20 73 6f 6c 75 74 69 |2$.   Th|e soluti|
|00000950| 6f 6e 20 73 65 74 20 69 | 73 20 74 68 65 6e 20 24 |on set i|s then $|
|00000960| 28 2d 31 2c 20 34 29 24 | 2e 20 20 20 49 66 20 77 |(-1, 4)$|.   If w|
|00000970| 65 20 68 61 76 65 20 5c | 45 6d 7b 73 74 72 69 63 |e have \|Em{stric|
|00000980| 74 7d 20 69 6e 65 71 75 | 61 6c 69 74 79 20 61 74 |t} inequ|ality at|
|00000990| 20 74 68 65 20 6c 6f 77 | 65 72 20 65 6e 64 20 28 | the low|er end (|
|000009a0| 24 30 20 3c 20 7c 33 20 | 2d 20 32 78 7c 20 3c 20 |$0 < |3 |- 2x| < |
|000009b0| 35 24 29 2c 20 74 68 65 | 6e 20 74 68 65 20 70 6f |5$), the|n the po|
|000009c0| 69 6e 74 20 24 61 20 3d | 20 33 2f 32 24 20 69 73 |int $a =| 3/2$ is|
|000009d0| 20 60 60 6d 69 73 73 69 | 6e 67 27 27 20 66 72 6f | ``missi|ng'' fro|
|000009e0| 6d 20 74 68 65 20 73 6f | 6c 75 74 69 6f 6e 20 73 |m the so|lution s|
|000009f0| 65 74 3a 20 24 28 2d 31 | 2c 20 33 2f 32 29 20 5c |et: $(-1|, 3/2) \|
|00000a00| 63 75 70 20 28 33 2f 32 | 2c 20 34 29 24 2e 20 20 |cup (3/2|, 4)$.  |
|00000a10| 53 75 63 68 20 60 60 70 | 75 6e 63 74 75 72 65 64 |Such ``p|unctured|
|00000a20| 27 27 20 69 6e 74 65 72 | 76 61 6c 73 20 70 6c 61 |'' inter|vals pla|
|00000a30| 79 20 61 20 6b 65 79 20 | 72 6f 6c 65 20 69 6e 20 |y a key |role in |
|00000a40| 74 68 65 20 73 74 75 64 | 79 20 6f 66 20 6c 69 6d |the stud|y of lim|
|00000a50| 69 74 73 2e 0a 0a 5c 65 | 6e 64 7b 65 6e 75 6d 65 |its...\e|nd{enume|
|00000a60| 72 61 74 65 7d 0a 5c 65 | 6e 64 7b 64 6f 63 75 6d |rate}.\e|nd{docum|
|00000a70| 65 6e 74 7d 0a          |                         |ent}.   |        |
+--------+-------------------------+-------------------------+--------+--------+