home *** CD-ROM | disk | FTP | other *** search
open in: 
MacOS 8.1
|
Win98
|
DOS
browse contents |
view JSON data
|
view as text
This file was processed as: LaTeX Document
(document/latex).
| Confidence | Program | Detection | Match Type | Support
|
|---|
| 100%
| dexvert
| LaTeX Document (document/latex)
| magic
| Supported |
| 90%
| dexvert
| Hypertext Markup Language File (text/html)
| magic
| Supported |
| 1%
| dexvert
| Text File (text/txt)
| fallback
| Supported |
| 100%
| file
| HTML document text
| default (weak)
| |
| 99%
| file
| LaTeX document text
| default
| |
| 98%
| file
| exported SGML document text
| default
| |
| 97%
| file
| exported SGML document, ASCII text, with very long lines (731)
| default
| |
| 80%
| TrID
| HyperText Markup Language with DOCTYPE
| default
| |
| 19%
| TrID
| HyperText Markup Language
| default
| |
| 100%
| checkBytes
| Printable ASCII
| default
| |
| 100%
| perlTextCheck
| Likely Text (Perl)
| default
| |
| 100%
| siegfried
| fmt/281 LaTeX (Subdocument)
| default
| |
| 100%
| gt2
| HTML (Hyper Text Markup Language) Datei
| default
| |
| 100%
| detectItEasy
| Format: plain text[LF]
| default (weak)
| |
| 100%
| xdgMime
| text/html
| default
|
|
hex view+--------+-------------------------+-------------------------+--------+--------+
|00000000| 3c 21 44 4f 43 54 59 50 | 45 20 48 54 4d 4c 20 50 |<!DOCTYP|E HTML P|
|00000010| 55 42 4c 49 43 20 22 2d | 2f 2f 57 33 43 2f 2f 44 |UBLIC "-|//W3C//D|
|00000020| 54 44 20 48 54 4d 4c 20 | 33 2e 32 20 46 69 6e 61 |TD HTML |3.2 Fina|
|00000030| 6c 2f 2f 65 6e 22 3e 0a | 0a 3c 21 2d 2d 43 6f 6e |l//en">.|.<!--Con|
|00000040| 76 65 72 74 65 64 20 77 | 69 74 68 20 4c 61 54 65 |verted w|ith LaTe|
|00000050| 58 32 48 54 4d 4c 20 32 | 30 32 32 20 28 52 65 6c |X2HTML 2|022 (Rel|
|00000060| 65 61 73 65 64 20 4a 61 | 6e 75 61 72 79 20 31 2c |eased Ja|nuary 1,|
|00000070| 20 32 30 32 32 29 20 2d | 2d 3e 0a 3c 48 54 4d 4c | 2022) -|->.<HTML|
|00000080| 20 6c 61 6e 67 3d 22 65 | 6e 22 3e 0a 3c 48 45 41 | lang="e|n">.<HEA|
|00000090| 44 3e 0a 3c 54 49 54 4c | 45 3e 43 6f 6e 74 65 6e |D>.<TITL|E>Conten|
|000000a0| 74 73 20 6f 66 20 61 6e | 73 33 5f 63 74 3c 2f 54 |ts of an|s3_ct</T|
|000000b0| 49 54 4c 45 3e 0a 0a 3c | 4d 45 54 41 20 48 54 54 |ITLE>..<|META HTT|
|000000c0| 50 2d 45 51 55 49 56 3d | 22 43 6f 6e 74 65 6e 74 |P-EQUIV=|"Content|
|000000d0| 2d 54 79 70 65 22 20 43 | 4f 4e 54 45 4e 54 3d 22 |-Type" C|ONTENT="|
|000000e0| 74 65 78 74 2f 68 74 6d | 6c 3b 20 63 68 61 72 73 |text/htm|l; chars|
|000000f0| 65 74 3d 75 74 66 2d 38 | 22 3e 0a 3c 4d 45 54 41 |et=utf-8|">.<META|
|00000100| 20 4e 41 4d 45 3d 22 76 | 69 65 77 70 6f 72 74 22 | NAME="v|iewport"|
|00000110| 20 43 4f 4e 54 45 4e 54 | 3d 22 77 69 64 74 68 3d | CONTENT|="width=|
|00000120| 64 65 76 69 63 65 2d 77 | 69 64 74 68 2c 20 69 6e |device-w|idth, in|
|00000130| 69 74 69 61 6c 2d 73 63 | 61 6c 65 3d 31 2e 30 22 |itial-sc|ale=1.0"|
|00000140| 3e 0a 3c 4d 45 54 41 20 | 4e 41 4d 45 3d 22 47 65 |>.<META |NAME="Ge|
|00000150| 6e 65 72 61 74 6f 72 22 | 20 43 4f 4e 54 45 4e 54 |nerator"| CONTENT|
|00000160| 3d 22 4c 61 54 65 58 32 | 48 54 4d 4c 20 76 32 30 |="LaTeX2|HTML v20|
|00000170| 32 32 22 3e 0a 0a 3c 4c | 49 4e 4b 20 52 45 4c 3d |22">..<L|INK REL=|
|00000180| 22 53 54 59 4c 45 53 48 | 45 45 54 22 20 48 52 45 |"STYLESH|EET" HRE|
|00000190| 46 3d 22 61 6e 73 33 5f | 63 74 2e 63 73 73 22 3e |F="ans3_|ct.css">|
|000001a0| 0a 0a 3c 2f 48 45 41 44 | 3e 0a 20 0a 3c 42 4f 44 |..</HEAD|>. .<BOD|
|000001b0| 59 20 62 67 63 6f 6c 6f | 72 3d 22 23 66 66 66 66 |Y bgcolo|r="#ffff|
|000001c0| 66 66 22 20 74 65 78 74 | 3d 22 23 30 30 30 30 30 |ff" text|="#00000|
|000001d0| 30 22 20 6c 69 6e 6b 3d | 22 23 39 39 34 34 45 45 |0" link=|"#9944EE|
|000001e0| 22 20 76 6c 69 6e 6b 3d | 22 23 30 30 30 30 66 66 |" vlink=|"#0000ff|
|000001f0| 22 20 61 6c 69 6e 6b 3d | 22 23 30 30 66 66 30 30 |" alink=|"#00ff00|
|00000200| 22 3e 0a 0a 26 6c 74 3b | 21 44 4f 43 54 59 50 45 |">..<|!DOCTYPE|
|00000210| 20 48 54 4d 4c 20 50 55 | 42 4c 49 43 20 22 2d 2f | HTML PU|BLIC "-/|
|00000220| 2f 57 33 43 2f 2f 44 54 | 44 20 48 54 4d 4c 20 33 |/W3C//DT|D HTML 3|
|00000230| 2e 32 20 46 69 6e 61 6c | 2f 2f 65 6e 22 26 67 74 |.2 Final|//en">|
|00000240| 3b 0a 0a 3c 50 3e 0a 26 | 6c 74 3b 21 26 6e 64 61 |;..<P>.&|lt;!&nda|
|00000250| 73 68 3b 43 6f 6e 76 65 | 72 74 65 64 20 77 69 74 |sh;Conve|rted wit|
|00000260| 68 20 4c 61 54 65 58 32 | 48 54 4d 4c 20 32 30 32 |h LaTeX2|HTML 202|
|00000270| 32 20 28 52 65 6c 65 61 | 73 65 64 20 4a 61 6e 75 |2 (Relea|sed Janu|
|00000280| 61 72 79 20 31 2c 20 32 | 30 32 32 29 20 26 6e 64 |ary 1, 2|022) &nd|
|00000290| 61 73 68 3b 26 67 74 3b | 0a 26 6c 74 3b 48 54 4d |ash;>|.<HTM|
|000002a0| 4c 20 6c 61 6e 67 3d 22 | 65 6e 22 26 67 74 3b 0a |L lang="|en">.|
|000002b0| 26 6c 74 3b 48 45 41 44 | 26 67 74 3b 0a 26 6c 74 |<HEAD|>.<|
|000002c0| 3b 54 49 54 4c 45 26 67 | 74 3b 43 6f 6e 74 65 6e |;TITLE&g|t;Conten|
|000002d0| 74 73 20 6f 66 20 61 6e | 73 33 26 6c 74 3b 2f 54 |ts of an|s3</T|
|000002e0| 49 54 4c 45 26 67 74 3b | 0a 0a 3c 50 3e 0a 26 6c |ITLE>|..<P>.&l|
|000002f0| 74 3b 4d 45 54 41 20 48 | 54 54 50 2d 45 51 55 49 |t;META H|TTP-EQUI|
|00000300| 56 3d 22 43 6f 6e 74 65 | 6e 74 2d 54 79 70 65 22 |V="Conte|nt-Type"|
|00000310| 20 43 4f 4e 54 45 4e 54 | 3d 22 74 65 78 74 2f 68 | CONTENT|="text/h|
|00000320| 74 6d 6c 3b 20 63 68 61 | 72 73 65 74 3d 75 74 66 |tml; cha|rset=utf|
|00000330| 2d 38 22 26 67 74 3b 0a | 26 6c 74 3b 4d 45 54 41 |-8">.|<META|
|00000340| 20 4e 41 4d 45 3d 22 76 | 69 65 77 70 6f 72 74 22 | NAME="v|iewport"|
|00000350| 20 43 4f 4e 54 45 4e 54 | 3d 22 77 69 64 74 68 3d | CONTENT|="width=|
|00000360| 64 65 76 69 63 65 2d 77 | 69 64 74 68 2c 20 69 6e |device-w|idth, in|
|00000370| 69 74 69 61 6c 2d 73 63 | 61 6c 65 3d 31 2e 30 22 |itial-sc|ale=1.0"|
|00000380| 26 67 74 3b 0a 26 6c 74 | 3b 4d 45 54 41 20 4e 41 |>.<|;META NA|
|00000390| 4d 45 3d 22 47 65 6e 65 | 72 61 74 6f 72 22 20 43 |ME="Gene|rator" C|
|000003a0| 4f 4e 54 45 4e 54 3d 22 | 4c 61 54 65 58 32 48 54 |ONTENT="|LaTeX2HT|
|000003b0| 4d 4c 20 76 32 30 32 32 | 22 26 67 74 3b 0a 0a 3c |ML v2022|">..<|
|000003c0| 50 3e 0a 26 6c 74 3b 4c | 49 4e 4b 20 52 45 4c 3d |P>.<L|INK REL=|
|000003d0| 22 53 54 59 4c 45 53 48 | 45 45 54 22 20 48 52 45 |"STYLESH|EET" HRE|
|000003e0| 46 3d 22 61 6e 73 33 2e | 63 73 73 22 26 67 74 3b |F="ans3.|css">|
|000003f0| 0a 0a 3c 50 3e 0a 26 6c | 74 3b 2f 48 45 41 44 26 |..<P>.&l|t;/HEAD&|
|00000400| 67 74 3b 0a 0a 3c 50 3e | 0a 26 6c 74 3b 42 4f 44 |gt;..<P>|.<BOD|
|00000410| 59 20 62 67 63 6f 6c 6f | 72 3d 22 23 66 66 66 66 |Y bgcolo|r="#ffff|
|00000420| 66 66 22 20 74 65 78 74 | 3d 22 23 30 30 30 30 30 |ff" text|="#00000|
|00000430| 30 22 20 6c 69 6e 6b 3d | 22 23 39 39 34 34 45 45 |0" link=|"#9944EE|
|00000440| 22 20 76 6c 69 6e 6b 3d | 22 23 30 30 30 30 66 66 |" vlink=|"#0000ff|
|00000450| 22 20 61 6c 69 6e 6b 3d | 22 23 30 30 66 66 30 30 |" alink=|"#00ff00|
|00000460| 22 26 67 74 3b 0a 0a 3c | 50 3e 0a 26 6c 74 3b 50 |">..<|P>.<P|
|00000470| 26 67 74 3b 0a 43 68 61 | 70 74 65 72 20 31 3a 20 |>.Cha|pter 1: |
|00000480| 41 6e 73 77 65 72 73 20 | 33 20 4a 61 63 6b 20 4b |Answers |3 Jack K|
|00000490| 2e 20 43 6f 68 65 6e 20 | 43 6f 6c 6f 72 61 64 6f |. Cohen |Colorado|
|000004a0| 20 53 63 68 6f 6f 6c 20 | 6f 66 20 4d 69 6e 65 73 | School |of Mines|
|000004b0| 0a 0a 3c 50 3e 0a 26 6c | 74 3b 50 26 67 74 3b 0a |..<P>.&l|t;P>.|
|000004c0| 0a 3c 50 3e 0a 26 6c 74 | 3b 4f 4c 26 67 74 3b 0a |.<P>.<|;OL>.|
|000004d0| 26 6c 74 3b 4c 49 26 67 | 74 3b 54 68 65 20 41 6c |<LI&g|t;The Al|
|000004e0| 70 68 61 20 6c 65 74 74 | 75 63 65 20 70 61 74 63 |pha lett|uce patc|
|000004f0| 68 3a 0a 0a 3c 50 3e 0a | 26 6c 74 3b 4f 4c 26 67 |h:..<P>.|<OL&g|
|00000500| 74 3b 0a 26 6c 74 3b 4c | 49 26 67 74 3b 60 42 65 |t;.<L|I>`Be|
|00000510| 73 74 27 20 6d 65 61 6e | 73 20 6d 61 78 69 6d 75 |st' mean|s maximu|
|00000520| 6d 20 61 72 65 61 20 28 | 75 73 75 61 6c 6c 79 29 |m area (|usually)|
|00000530| 2e 0a 26 6c 74 3b 2f 4c | 49 26 67 74 3b 0a 26 6c |..</L|I>.&l|
|00000540| 74 3b 4c 49 26 67 74 3b | 49 6e 74 75 69 74 69 76 |t;LI>|Intuitiv|
|00000550| 65 6c 79 20 61 20 73 71 | 75 61 72 65 20 77 6f 75 |ely a sq|uare wou|
|00000560| 6c 64 20 67 69 76 65 20 | 74 68 65 20 6d 61 78 69 |ld give |the maxi|
|00000570| 6d 75 6d 20 61 72 65 61 | 20 61 6d 6f 6e 67 20 61 |mum area| among a|
|00000580| 6c 6c 20 72 65 63 74 61 | 6e 67 75 6c 61 72 20 70 |ll recta|ngular p|
|00000590| 61 74 63 68 65 73 2e 20 | 20 54 68 75 73 20 74 68 |atches. | Thus th|
|000005a0| 65 20 73 69 64 65 20 28 | 26 6c 74 3b 49 26 67 74 |e side (|<I>|
|000005b0| 3b 78 26 6c 74 3b 2f 49 | 26 67 74 3b 29 20 69 73 |;x</I|>) is|
|000005c0| 20 31 30 30 2f 34 20 3d | 20 32 35 20 6d 20 61 6e | 100/4 =| 25 m an|
|000005d0| 64 20 74 68 65 20 6d 61 | 78 69 6d 75 6d 20 61 72 |d the ma|ximum ar|
|000005e0| 65 61 20 69 73 20 36 32 | 35 20 6d 26 6c 74 3b 53 |ea is 62|5 m<S|
|000005f0| 55 50 26 67 74 3b 32 26 | 6c 74 3b 2f 53 55 50 26 |UP>2&|lt;/SUP&|
|00000600| 67 74 3b 2e 0a 26 6c 74 | 3b 2f 4c 49 26 67 74 3b |gt;..<|;/LI>|
|00000610| 0a 26 6c 74 3b 4c 49 26 | 67 74 3b 59 6f 75 72 20 |.<LI&|gt;Your |
|00000620| 66 69 67 75 72 65 20 73 | 68 6f 75 6c 64 20 68 61 |figure s|hould ha|
|00000630| 76 65 20 73 68 6f 77 6e | 20 61 20 72 65 63 74 61 |ve shown| a recta|
|00000640| 6e 67 6c 65 20 77 69 74 | 68 20 74 77 6f 20 66 61 |ngle wit|h two fa|
|00000650| 63 69 6e 67 20 73 69 64 | 65 73 20 6c 61 62 65 6c |cing sid|es label|
|00000660| 65 64 20 77 69 74 68 20 | 26 6c 74 3b 49 26 67 74 |ed with |<I>|
|00000670| 3b 78 26 6c 74 3b 2f 49 | 26 67 74 3b 20 61 6e 64 |;x</I|> and|
|00000680| 20 74 68 65 20 6f 74 68 | 65 72 20 74 77 6f 20 77 | the oth|er two w|
|00000690| 69 74 68 20 26 6c 74 3b | 21 26 6e 64 61 73 68 3b |ith <|!–|
|000006a0| 20 4d 41 54 48 0a 20 3c | 21 2d 2d 20 4d 41 54 48 | MATH. <|!-- MATH|
|000006b0| 0a 20 24 31 30 30 20 2d | 20 32 78 20 3d 20 32 28 |. $100 -| 2x = 2(|
|000006c0| 35 30 20 2d 20 78 29 24 | 0a 20 2d 2d 3e 0a 31 30 |50 - x)$|. -->.10|
|000006d0| 30 20 2d 20 32 3c 49 3e | 78 3c 2f 49 3e 20 3d 20 |0 - 2<I>|x</I> = |
|000006e0| 32 28 35 30 20 2d 20 3c | 49 3e 78 3c 2f 49 3e 29 |2(50 - <|I>x</I>)|
|000006f0| 0a 20 26 6e 64 61 73 68 | 3b 26 67 74 3b 0a 31 30 |. &ndash|;>.10|
|00000700| 30 20 2d 20 32 26 6c 74 | 3b 49 26 67 74 3b 78 26 |0 - 2<|;I>x&|
|00000710| 6c 74 3b 2f 49 26 67 74 | 3b 20 3d 20 32 28 35 30 |lt;/I>|; = 2(50|
|00000720| 20 2d 20 26 6c 74 3b 49 | 26 67 74 3b 78 26 6c 74 | - <I|>x<|
|00000730| 3b 2f 49 26 67 74 3b 29 | 20 2e 0a 26 6c 74 3b 2f |;/I>)| ..</|
|00000740| 4c 49 26 67 74 3b 0a 26 | 6c 74 3b 4c 49 26 67 74 |LI>.&|lt;LI>|
|00000750| 3b 26 6c 74 3b 49 26 67 | 74 3b 78 26 6c 74 3b 2f |;<I&g|t;x</|
|00000760| 49 26 67 74 3b 20 69 73 | 20 6f 6e 65 20 73 69 64 |I> is| one sid|
|00000770| 65 20 6f 66 20 74 68 65 | 20 72 65 63 74 61 6e 67 |e of the| rectang|
|00000780| 6c 65 2c 20 26 6c 74 3b | 49 26 67 74 3b 41 26 6c |le, <|I>A&l|
|00000790| 74 3b 2f 49 26 67 74 3b | 20 69 73 20 69 74 73 20 |t;/I>| is its |
|000007a0| 61 72 65 61 2e 20 20 54 | 68 65 20 76 61 6c 69 64 |area. T|he valid|
|000007b0| 20 26 6c 74 3b 49 26 67 | 74 3b 78 26 6c 74 3b 2f | <I&g|t;x</|
|000007c0| 49 26 67 74 3b 2d 76 61 | 6c 75 65 73 20 28 74 68 |I>-va|lues (th|
|000007d0| 65 20 60 60 64 6f 6d 61 | 69 6e 27 27 20 6f 66 20 |e ``doma|in'' of |
|000007e0| 26 6c 74 3b 49 26 67 74 | 3b 41 26 6c 74 3b 2f 49 |<I>|;A</I|
|000007f0| 26 67 74 3b 29 20 61 72 | 65 20 5b 30 2c 20 35 30 |>) ar|e [0, 50|
|00000800| 5d 20 62 65 63 61 75 73 | 65 20 6f 75 74 73 69 64 |] becaus|e outsid|
|00000810| 65 20 74 68 69 73 20 69 | 6e 74 65 72 76 61 6c 20 |e this i|nterval |
|00000820| 61 20 73 69 64 65 20 77 | 69 6c 6c 20 61 73 73 75 |a side w|ill assu|
|00000830| 6d 65 20 61 20 6e 6f 6e | 2d 70 68 79 73 69 63 61 |me a non|-physica|
|00000840| 6c 20 6e 65 67 61 74 69 | 76 65 20 76 61 6c 75 65 |l negati|ve value|
|00000850| 2e 20 20 57 65 20 6b 6e | 6f 77 20 74 68 61 74 20 |. We kn|ow that |
|00000860| 65 76 65 6e 20 74 68 65 | 20 65 6e 64 70 6f 69 6e |even the| endpoin|
|00000870| 74 20 76 61 6c 75 65 73 | 20 30 20 61 6e 64 20 35 |t values| 0 and 5|
|00000880| 30 20 63 61 6e 27 74 20 | 62 65 20 72 69 67 68 74 |0 can't |be right|
|00000890| 20 73 69 6e 63 65 20 74 | 68 65 6e 20 74 68 65 20 | since t|hen the |
|000008a0| 61 72 65 61 20 69 73 20 | 7a 65 72 6f 2c 20 62 75 |area is |zero, bu|
|000008b0| 74 20 69 74 20 64 6f 65 | 73 20 6e 6f 20 68 61 72 |t it doe|s no har|
|000008c0| 6d 20 74 6f 20 69 6e 63 | 6c 75 64 65 20 74 68 65 |m to inc|lude the|
|000008d0| 73 65 20 61 6e 64 20 77 | 65 27 6c 6c 20 73 65 65 |se and w|e'll see|
|000008e0| 20 6c 61 74 65 72 20 74 | 68 61 74 20 70 6f 73 69 | later t|hat posi|
|000008f0| 6e 67 20 6d 61 78 69 6d | 75 6d 20 6f 72 20 6d 69 |ng maxim|um or mi|
|00000900| 6e 69 6d 75 6d 20 70 72 | 6f 62 6c 65 6d 73 20 6f |nimum pr|oblems o|
|00000910| 6e 20 61 20 63 6c 6f 73 | 65 64 20 69 6e 74 65 72 |n a clos|ed inter|
|00000920| 76 61 6c 20 68 61 73 20 | 73 6f 6d 65 20 74 68 65 |val has |some the|
|00000930| 6f 72 65 74 69 63 61 6c | 20 61 64 76 61 6e 74 61 |oretical| advanta|
|00000940| 67 65 73 2e 0a 26 6c 74 | 3b 2f 4c 49 26 67 74 3b |ges..<|;/LI>|
|00000950| 0a 26 6c 74 3b 4c 49 26 | 67 74 3b 54 68 65 20 20 |.<LI&|gt;The |
|00000960| 76 61 6c 75 65 73 20 69 | 6e 20 74 68 65 20 6d 69 |values i|n the mi|
|00000970| 64 64 6c 65 20 6f 66 20 | 74 68 65 20 6d 6f 73 74 |ddle of |the most|
|00000980| 20 72 65 66 69 6e 65 64 | 20 54 61 62 6c 65 20 61 | refined| Table a|
|00000990| 72 65 3a 0a 20 20 20 20 | 20 20 20 20 26 6c 74 3b |re:. | <|
|000009a0| 50 52 45 26 67 74 3b 20 | 20 20 20 20 20 20 20 0a |PRE> | .|
|000009b0| 20 20 20 20 20 20 20 20 | 32 34 2e 37 20 20 20 36 | |24.7 6|
|000009c0| 32 34 2e 39 31 0a 20 20 | 20 20 20 20 20 20 32 34 |24.91. | 24|
|000009d0| 2e 38 20 20 20 36 32 34 | 2e 39 36 0a 20 20 20 20 |.8 624|.96. |
|000009e0| 20 20 20 20 32 34 2e 39 | 20 20 20 36 32 34 2e 39 | 24.9| 624.9|
|000009f0| 39 0a 20 20 20 20 20 20 | 20 20 32 35 2e 20 20 20 |9. | 25. |
|00000a00| 20 36 32 35 2e 0a 20 20 | 20 20 20 20 20 20 32 35 | 625.. | 25|
|00000a10| 2e 31 20 20 20 36 32 34 | 2e 39 39 0a 20 20 20 20 |.1 624|.99. |
|00000a20| 20 20 20 20 32 35 2e 32 | 20 20 20 36 32 34 2e 39 | 25.2| 624.9|
|00000a30| 36 0a 20 20 20 20 20 20 | 20 20 32 35 2e 33 20 20 |6. | 25.3 |
|00000a40| 20 36 32 34 2e 39 31 0a | 26 6c 74 3b 2f 50 52 45 | 624.91.|</PRE|
|00000a50| 26 67 74 3b 0a 0a 3c 50 | 3e 0a 26 6c 74 3b 50 26 |>..<P|>.<P&|
|00000a60| 67 74 3b 0a 26 6c 74 3b | 2f 4c 49 26 67 74 3b 0a |gt;.<|/LI>.|
|00000a70| 26 6c 74 3b 4c 49 26 67 | 74 3b 53 65 65 20 46 69 |<LI&g|t;See Fi|
|00000a80| 67 75 72 65 20 31 2e 0a | 0a 3c 50 3e 0a 26 6c 74 |gure 1..|.<P>.<|
|00000a90| 3b 44 49 56 20 63 6c 61 | 73 73 3d 22 43 45 4e 54 |;DIV cla|ss="CENT|
|00000aa0| 45 52 22 26 67 74 3b 26 | 6c 74 3b 41 20 49 44 3d |ER">&|lt;A ID=|
|00000ab0| 22 31 33 22 26 67 74 3b | 26 6c 74 3b 2f 41 26 67 |"13">|</A&g|
|00000ac0| 74 3b 0a 26 6c 74 3b 54 | 41 42 4c 45 26 67 74 3b |t;.<T|ABLE>|
|00000ad0| 0a 26 6c 74 3b 43 41 50 | 54 49 4f 4e 20 63 6c 61 |.<CAP|TION cla|
|00000ae0| 73 73 3d 22 42 4f 54 54 | 4f 4d 22 26 67 74 3b 26 |ss="BOTT|OM">&|
|00000af0| 6c 74 3b 53 54 52 4f 4e | 47 26 67 74 3b 46 69 67 |lt;STRON|G>Fig|
|00000b00| 75 72 65 3a 26 6c 74 3b | 2f 53 54 52 4f 4e 47 26 |ure:<|/STRONG&|
|00000b10| 67 74 3b 0a 47 72 61 70 | 68 20 6f 66 20 26 6c 74 |gt;.Grap|h of <|
|00000b20| 3b 49 26 67 74 3b 41 26 | 6c 74 3b 2f 49 26 67 74 |;I>A&|lt;/I>|
|00000b30| 3b 28 26 6c 74 3b 49 26 | 67 74 3b 78 26 6c 74 3b |;(<I&|gt;x<|
|00000b40| 2f 49 26 67 74 3b 29 20 | 69 6e 20 74 68 65 20 63 |/I>) |in the c|
|00000b50| 72 69 74 69 63 61 6c 20 | 72 65 67 69 6f 6e 20 6f |ritical |region o|
|00000b60| 66 20 50 72 6f 62 6c 65 | 6d 20 31 2e 26 6c 74 3b |f Proble|m 1.<|
|00000b70| 2f 43 41 50 54 49 4f 4e | 26 67 74 3b 0a 26 6c 74 |/CAPTION|>.<|
|00000b80| 3b 54 52 26 67 74 3b 26 | 6c 74 3b 54 44 26 67 74 |;TR>&|lt;TD>|
|00000b90| 3b 26 6c 74 3b 49 4d 47 | 0a 20 53 54 59 4c 45 3d |;<IMG|. STYLE=|
|00000ba0| 22 68 65 69 67 68 74 3a | 20 32 38 36 2e 37 36 65 |"height:| 286.76e|
|00000bb0| 78 3b 20 22 20 53 52 43 | 3d 22 69 6d 67 31 2e 70 |x; " SRC|="img1.p|
|00000bc0| 6e 67 22 0a 20 41 4c 54 | 3d 22 0a 3c 44 49 56 20 |ng". ALT|=".<DIV |
|00000bd0| 63 6c 61 73 73 3d 22 43 | 45 4e 54 45 52 22 3e 0a |class="C|ENTER">.|
|00000be0| 3c 49 4d 47 0a 20 53 54 | 59 4c 45 3d 22 68 65 69 |<IMG. ST|YLE="hei|
|00000bf0| 67 68 74 3a 20 33 31 34 | 2e 30 30 65 78 3b 20 22 |ght: 314|.00ex; "|
|00000c00| 20 53 52 43 3d 22 69 6d | 67 31 2e 70 6e 67 22 0a | SRC="im|g1.png".|
|00000c10| 20 41 4c 54 3d 22 5c 62 | 65 67 69 6e 7b 66 69 67 | ALT="\b|egin{fig|
|00000c20| 75 72 65 7d 5c 65 70 73 | 66 79 73 69 7a 65 20 31 |ure}\eps|fysize 1|
|00000c30| 30 30 70 74 0a 5c 63 65 | 6e 74 65 72 6c 69 6e 65 |00pt.\ce|nterline|
|00000c40| 7b 5c 65 70 73 66 66 69 | 6c 65 7b 61 6e 73 33 70 |{\epsffi|le{ans3p|
|00000c50| 31 2e 65 70 73 7d 7d 0a | 5c 65 6e 64 7b 66 69 67 |1.eps}}.|\end{fig|
|00000c60| 75 72 65 7d 22 3e 0a 3c | 2f 44 49 56 3e 22 26 67 |ure}">.<|/DIV>"&g|
|00000c70| 74 3b 26 6c 74 3b 2f 54 | 44 26 67 74 3b 26 6c 74 |t;</T|D><|
|00000c80| 3b 2f 54 52 26 67 74 3b | 0a 26 6c 74 3b 2f 54 41 |;/TR>|.</TA|
|00000c90| 42 4c 45 26 67 74 3b 0a | 26 6c 74 3b 2f 44 49 56 |BLE>.|</DIV|
|00000ca0| 26 67 74 3b 0a 0a 3c 50 | 3e 0a 26 6c 74 3b 50 26 |>..<P|>.<P&|
|00000cb0| 67 74 3b 0a 26 6c 74 3b | 2f 4c 49 26 67 74 3b 0a |gt;.<|/LI>.|
|00000cc0| 26 6c 74 3b 4c 49 26 67 | 74 3b 54 68 65 20 6d 61 |<LI&g|t;The ma|
|00000cd0| 78 69 6d 69 7a 69 6e 67 | 20 26 6c 74 3b 49 26 67 |ximizing| <I&g|
|00000ce0| 74 3b 78 26 6c 74 3b 2f | 49 26 67 74 3b 20 69 73 |t;x</|I> is|
|00000cf0| 20 68 61 6c 66 2d 77 61 | 79 20 62 65 74 77 65 65 | half-wa|y betwee|
|00000d00| 6e 20 74 68 65 20 7a 65 | 72 6f 65 73 2c 20 74 68 |n the ze|roes, th|
|00000d10| 61 74 20 69 73 20 26 6c | 74 3b 49 26 67 74 3b 78 |at is &l|t;I>x|
|00000d20| 26 6c 74 3b 2f 49 26 67 | 74 3b 20 3d 20 32 35 2e |</I&g|t; = 25.|
|00000d30| 20 20 20 4f 6e 65 20 77 | 61 79 20 74 6f 20 6a 75 | One w|ay to ju|
|00000d40| 73 74 69 66 79 20 74 68 | 69 73 20 69 73 20 74 6f |stify th|is is to|
|00000d50| 20 6d 75 6c 74 69 70 6c | 79 20 6f 75 74 20 74 68 | multipl|y out th|
|00000d60| 65 20 70 6f 6c 79 6e 6f | 6d 69 61 6c 20 61 6e 64 |e polyno|mial and|
|00000d70| 20 63 6f 6d 70 6c 65 74 | 65 20 74 68 65 20 73 71 | complet|e the sq|
|00000d80| 75 61 72 65 3a 20 26 6c | 74 3b 21 26 6e 64 61 73 |uare: &l|t;!&ndas|
|00000d90| 68 3b 20 4d 41 54 48 0a | 20 3c 21 2d 2d 20 4d 41 |h; MATH.| <!-- MA|
|00000da0| 54 48 0a 20 24 41 28 78 | 29 20 3d 20 35 30 78 20 |TH. $A(x|) = 50x |
|00000db0| 2d 20 78 5e 32 20 3d 20 | 2d 28 78 20 2d 20 32 35 |- x^2 = |-(x - 25|
|00000dc0| 29 5e 32 20 2b 20 36 32 | 35 24 0a 20 2d 2d 3e 0a |)^2 + 62|5$. -->.|
|00000dd0| 3c 49 3e 41 3c 2f 49 3e | 28 3c 49 3e 78 3c 2f 49 |<I>A</I>|(<I>x</I|
|00000de0| 3e 29 20 3d 20 35 30 3c | 49 3e 78 3c 2f 49 3e 20 |>) = 50<|I>x</I> |
|00000df0| 2d 20 3c 49 3e 78 3c 2f | 49 3e 3c 53 55 50 3e 32 |- <I>x</|I><SUP>2|
|00000e00| 3c 2f 53 55 50 3e 20 3d | 20 2d 20 28 3c 49 3e 78 |</SUP> =| - (<I>x|
|00000e10| 3c 2f 49 3e 20 2d 20 32 | 35 29 3c 53 55 50 3e 32 |</I> - 2|5)<SUP>2|
|00000e20| 3c 2f 53 55 50 3e 20 2b | 20 36 32 35 0a 20 26 6e |</SUP> +| 625. &n|
|00000e30| 64 61 73 68 3b 26 67 74 | 3b 0a 26 6c 74 3b 49 26 |dash;>|;.<I&|
|00000e40| 67 74 3b 41 26 6c 74 3b | 2f 49 26 67 74 3b 28 26 |gt;A<|/I>(&|
|00000e50| 6c 74 3b 49 26 67 74 3b | 78 26 6c 74 3b 2f 49 26 |lt;I>|x</I&|
|00000e60| 67 74 3b 29 20 3d 20 35 | 30 26 6c 74 3b 49 26 67 |gt;) = 5|0<I&g|
|00000e70| 74 3b 78 26 6c 74 3b 2f | 49 26 67 74 3b 20 2d 20 |t;x</|I> - |
|00000e80| 26 6c 74 3b 49 26 67 74 | 3b 78 26 6c 74 3b 2f 49 |<I>|;x</I|
|00000e90| 26 67 74 3b 26 6c 74 3b | 53 55 50 26 67 74 3b 32 |><|SUP>2|
|00000ea0| 26 6c 74 3b 2f 53 55 50 | 26 67 74 3b 20 3d 20 2d |</SUP|> = -|
|00000eb0| 20 28 26 6c 74 3b 49 26 | 67 74 3b 78 26 6c 74 3b | (<I&|gt;x<|
|00000ec0| 2f 49 26 67 74 3b 20 2d | 20 32 35 29 26 6c 74 3b |/I> -| 25)<|
|00000ed0| 53 55 50 26 67 74 3b 32 | 26 6c 74 3b 2f 53 55 50 |SUP>2|</SUP|
|00000ee0| 26 67 74 3b 20 2b 20 36 | 32 35 2e 20 20 54 68 69 |> + 6|25. Thi|
|00000ef0| 73 20 66 6f 72 6d 20 6f | 66 20 74 68 65 20 65 71 |s form o|f the eq|
|00000f00| 75 61 74 69 6f 6e 20 73 | 68 6f 77 73 20 63 6c 65 |uation s|hows cle|
|00000f10| 61 72 6c 79 20 74 68 61 | 74 20 26 6c 74 3b 21 26 |arly tha|t <!&|
|00000f20| 6e 64 61 73 68 3b 20 4d | 41 54 48 0a 20 3c 21 2d |ndash; M|ATH. <!-|
|00000f30| 2d 20 4d 41 54 48 0a 20 | 24 41 20 5c 6c 65 71 20 |- MATH. |$A \leq |
|00000f40| 36 32 35 24 0a 20 2d 2d | 3e 0a 3c 49 3e 41 3c 2f |625$. --|>.<I>A</|
|00000f50| 49 3e 26 23 38 38 30 34 | 3b 36 32 35 0a 20 26 6e |I>≤|;625. &n|
|00000f60| 64 61 73 68 3b 26 67 74 | 3b 0a 26 6c 74 3b 49 26 |dash;>|;.<I&|
|00000f70| 67 74 3b 41 26 6c 74 3b | 2f 49 26 67 74 3b 26 61 |gt;A<|/I>&a|
|00000f80| 6d 70 3b 23 38 38 30 34 | 3b 36 32 35 20 66 6f 72 |mp;#8804|;625 for|
|00000f90| 20 61 6c 6c 20 26 6c 74 | 3b 49 26 67 74 3b 78 26 | all <|;I>x&|
|00000fa0| 6c 74 3b 2f 49 26 67 74 | 3b 20 61 6e 64 20 61 63 |lt;/I>|; and ac|
|00000fb0| 74 75 61 6c 6c 79 20 61 | 63 68 69 65 76 65 73 20 |tually a|chieves |
|00000fc0| 74 68 65 20 6d 61 78 69 | 6d 75 6d 20 76 61 6c 75 |the maxi|mum valu|
|00000fd0| 65 20 26 6c 74 3b 49 26 | 67 74 3b 41 26 6c 74 3b |e <I&|gt;A<|
|00000fe0| 2f 49 26 67 74 3b 20 3d | 20 36 32 35 20 61 74 20 |/I> =| 625 at |
|00000ff0| 26 6c 74 3b 49 26 67 74 | 3b 78 26 6c 74 3b 2f 49 |<I>|;x</I|
|00001000| 26 67 74 3b 20 3d 20 32 | 35 2e 20 20 4e 6f 74 65 |> = 2|5. Note|
|00001010| 3a 20 74 68 65 20 60 60 | 62 65 74 77 65 65 6e 20 |: the ``|between |
|00001020| 74 68 65 20 7a 65 72 6f | 65 73 20 69 64 65 61 27 |the zero|es idea'|
|00001030| 27 20 69 73 20 6c 69 6d | 69 74 65 64 20 74 6f 20 |' is lim|ited to |
|00001040| 71 75 61 64 72 61 74 69 | 63 73 26 61 6d 70 3b 6d |quadrati|cs&m|
|00001050| 64 61 73 68 3b 61 6e 6f | 74 68 65 72 20 61 6e 64 |dash;ano|ther and|
|00001060| 20 66 61 72 20 6d 6f 72 | 65 20 67 65 6e 65 72 61 | far mor|e genera|
|00001070| 6c 20 61 6e 73 77 65 72 | 20 74 6f 20 74 68 69 73 |l answer| to this|
|00001080| 20 71 75 65 73 74 69 6f | 6e 20 69 73 20 74 68 61 | questio|n is tha|
|00001090| 74 20 74 68 65 20 6d 61 | 78 69 6d 75 6d 20 6f 63 |t the ma|ximum oc|
|000010a0| 63 75 72 73 20 77 68 65 | 72 65 20 74 68 65 20 74 |curs whe|re the t|
|000010b0| 61 6e 67 65 6e 74 20 74 | 6f 20 74 68 65 20 66 75 |angent t|o the fu|
|000010c0| 6e 63 74 69 6f 6e 20 69 | 73 20 68 6f 72 69 7a 6f |nction i|s horizo|
|000010d0| 6e 74 61 6c 2e 20 20 54 | 68 65 20 70 75 72 73 75 |ntal. T|he pursu|
|000010e0| 69 74 20 61 6e 64 20 72 | 65 66 69 6e 65 6d 65 6e |it and r|efinemen|
|000010f0| 74 20 6f 66 20 74 68 69 | 73 20 69 6e 73 69 67 68 |t of thi|s insigh|
|00001100| 74 20 69 73 20 61 20 6c | 61 72 67 65 20 70 61 72 |t is a l|arge par|
|00001110| 74 20 6f 66 20 74 68 69 | 73 20 73 65 6d 65 73 74 |t of thi|s semest|
|00001120| 65 72 27 73 20 77 6f 72 | 6b 2e 20 20 41 74 20 74 |er's wor|k. At t|
|00001130| 68 65 20 70 72 65 73 65 | 6e 74 20 74 69 6d 65 2c |he prese|nt time,|
|00001140| 20 74 68 69 73 20 63 72 | 69 74 65 72 69 61 20 69 | this cr|iteria i|
|00001150| 73 20 6e 6f 74 20 70 72 | 61 63 74 69 63 61 6c 20 |s not pr|actical |
|00001160| 66 6f 72 20 75 73 2c 20 | 73 69 6e 63 65 20 77 65 |for us, |since we|
|00001170| 20 64 6f 6e 27 74 20 79 | 65 74 20 6b 6e 6f 77 20 | don't y|et know |
|00001180| 68 6f 77 20 74 6f 20 66 | 69 6e 64 20 74 68 65 20 |how to f|ind the |
|00001190| 26 6c 74 3b 49 26 67 74 | 3b 78 26 6c 74 3b 2f 49 |<I>|;x</I|
|000011a0| 26 67 74 3b 27 73 20 74 | 68 61 74 20 6d 61 6b 65 |>'s t|hat make|
|000011b0| 20 74 68 65 20 74 61 6e | 67 65 6e 74 20 68 6f 72 | the tan|gent hor|
|000011c0| 69 7a 6f 6e 74 61 6c 26 | 61 6d 70 3b 6d 64 61 73 |izontal&|amp;mdas|
|000011d0| 68 3b 62 75 74 20 77 65 | 20 73 6f 6f 6e 20 77 69 |h;but we| soon wi|
|000011e0| 6c 6c 21 20 20 28 4e 6f | 20 63 72 65 64 69 74 20 |ll! (No| credit |
|000011f0| 66 6f 72 20 75 73 69 6e | 67 20 74 68 69 73 20 6d |for usin|g this m|
|00001200| 65 74 68 6f 64 20 6e 6f | 77 2c 20 69 66 20 79 6f |ethod no|w, if yo|
|00001210| 75 20 68 61 70 70 65 6e | 20 74 6f 20 68 61 76 65 |u happen| to have|
|00001220| 20 63 6f 76 65 72 65 64 | 20 74 68 69 73 20 6e 6f | covered| this no|
|00001230| 74 69 6f 6e 20 69 6e 20 | 68 69 67 68 20 73 63 68 |tion in |high sch|
|00001240| 6f 6f 6c 2e 29 0a 0a 3c | 50 3e 0a 26 6c 74 3b 50 |ool.)..<|P>.<P|
|00001250| 26 67 74 3b 0a 26 6c 74 | 3b 2f 4c 49 26 67 74 3b |>.<|;/LI>|
|00001260| 0a 26 6c 74 3b 4c 49 26 | 67 74 3b 49 6e 20 60 60 |.<LI&|gt;In ``|
|00001270| 72 65 61 6c 20 6c 69 66 | 65 27 27 20 61 6e 79 20 |real lif|e'' any |
|00001280| 6e 75 6d 62 65 72 20 6f | 66 20 6f 74 68 65 72 20 |number o|f other |
|00001290| 72 65 73 74 72 69 63 74 | 69 6f 6e 73 20 6d 69 67 |restrict|ions mig|
|000012a0| 68 74 20 61 70 70 6c 79 | 2e 20 20 46 6f 72 20 65 |ht apply|. For e|
|000012b0| 78 61 6d 70 6c 65 2c 20 | 74 68 65 20 6e 65 65 64 |xample, |the need|
|000012c0| 20 66 6f 72 20 6f 75 72 | 20 69 72 72 69 67 61 74 | for our| irrigat|
|000012d0| 69 6f 6e 20 6c 69 6e 65 | 73 20 74 6f 20 62 65 20 |ion line|s to be |
|000012e0| 61 62 6c 65 20 74 6f 20 | 72 65 61 63 68 20 61 6c |able to |reach al|
|000012f0| 6c 20 70 61 72 74 73 20 | 6f 66 20 74 68 65 20 70 |l parts |of the p|
|00001300| 61 74 63 68 2e 0a 0a 3c | 50 3e 0a 26 6c 74 3b 50 |atch...<|P>.<P|
|00001310| 26 67 74 3b 0a 26 6c 74 | 3b 2f 4c 49 26 67 74 3b |>.<|;/LI>|
|00001320| 0a 26 6c 74 3b 4c 49 26 | 67 74 3b 53 74 69 6c 6c |.<LI&|gt;Still|
|00001330| 20 6c 61 62 65 6c 69 6e | 67 20 6f 6e 65 20 73 69 | labelin|g one si|
|00001340| 64 65 20 61 73 20 26 6c | 74 3b 49 26 67 74 3b 78 |de as &l|t;I>x|
|00001350| 26 6c 74 3b 2f 49 26 67 | 74 3b 2c 20 77 65 20 66 |</I&g|t;, we f|
|00001360| 69 6e 64 20 74 68 61 74 | 20 74 68 65 20 61 64 6a |ind that| the adj|
|00001370| 61 63 65 6e 74 20 73 69 | 64 65 20 68 61 73 20 6c |acent si|de has l|
|00001380| 65 6e 67 74 68 20 26 6c | 74 3b 49 26 67 74 3b 50 |ength &l|t;I>P|
|00001390| 26 6c 74 3b 2f 49 26 67 | 74 3b 2f 32 20 2d 20 26 |</I&g|t;/2 - &|
|000013a0| 6c 74 3b 49 26 67 74 3b | 78 26 6c 74 3b 2f 49 26 |lt;I>|x</I&|
|000013b0| 67 74 3b 2c 20 73 6f 20 | 74 68 61 74 20 74 68 65 |gt;, so |that the|
|000013c0| 20 61 72 65 61 20 69 73 | 20 26 6c 74 3b 21 26 6e | area is| <!&n|
|000013d0| 64 61 73 68 3b 20 4d 41 | 54 48 0a 20 3c 21 2d 2d |dash; MA|TH. <!--|
|000013e0| 20 4d 41 54 48 0a 20 24 | 41 28 78 29 20 3d 20 78 | MATH. $|A(x) = x|
|000013f0| 28 50 2f 32 20 2d 20 78 | 29 24 0a 20 2d 2d 3e 0a |(P/2 - x|)$. -->.|
|00001400| 3c 49 3e 41 3c 2f 49 3e | 28 3c 49 3e 78 3c 2f 49 |<I>A</I>|(<I>x</I|
|00001410| 3e 29 20 3d 20 3c 49 3e | 78 3c 2f 49 3e 28 3c 49 |>) = <I>|x</I>(<I|
|00001420| 3e 50 3c 2f 49 3e 2f 32 | 20 2d 20 3c 49 3e 78 3c |>P</I>/2| - <I>x<|
|00001430| 2f 49 3e 29 0a 20 26 6e | 64 61 73 68 3b 26 67 74 |/I>). &n|dash;>|
|00001440| 3b 0a 26 6c 74 3b 49 26 | 67 74 3b 41 26 6c 74 3b |;.<I&|gt;A<|
|00001450| 2f 49 26 67 74 3b 28 26 | 6c 74 3b 49 26 67 74 3b |/I>(&|lt;I>|
|00001460| 78 26 6c 74 3b 2f 49 26 | 67 74 3b 29 20 3d 20 26 |x</I&|gt;) = &|
|00001470| 6c 74 3b 49 26 67 74 3b | 78 26 6c 74 3b 2f 49 26 |lt;I>|x</I&|
|00001480| 67 74 3b 28 26 6c 74 3b | 49 26 67 74 3b 50 26 6c |gt;(<|I>P&l|
|00001490| 74 3b 2f 49 26 67 74 3b | 2f 32 20 2d 20 26 6c 74 |t;/I>|/2 - <|
|000014a0| 3b 49 26 67 74 3b 78 26 | 6c 74 3b 2f 49 26 67 74 |;I>x&|lt;/I>|
|000014b0| 3b 29 2e 20 20 20 55 73 | 69 6e 67 20 74 68 65 20 |;). Us|ing the |
|000014c0| 60 60 68 61 6c 66 2d 77 | 61 79 20 62 65 74 77 65 |``half-w|ay betwe|
|000014d0| 65 6e 20 74 68 65 20 7a | 65 72 6f 65 73 27 27 20 |en the z|eroes'' |
|000014e0| 69 6e 73 69 67 68 74 2c | 20 77 65 20 69 6e 74 75 |insight,| we intu|
|000014f0| 69 74 20 74 68 61 74 20 | 74 68 65 20 6d 61 78 69 |it that |the maxi|
|00001500| 6d 69 7a 69 6e 67 20 26 | 6c 74 3b 49 26 67 74 3b |mizing &|lt;I>|
|00001510| 78 26 6c 74 3b 2f 49 26 | 67 74 3b 20 3d 20 26 6c |x</I&|gt; = &l|
|00001520| 74 3b 49 26 67 74 3b 50 | 26 6c 74 3b 2f 49 26 67 |t;I>P|</I&g|
|00001530| 74 3b 2f 34 20 61 6e 64 | 20 74 68 61 74 20 74 68 |t;/4 and| that th|
|00001540| 65 20 63 6f 72 72 65 73 | 70 6f 6e 64 69 6e 67 20 |e corres|ponding |
|00001550| 61 72 65 61 20 69 73 20 | 26 6c 74 3b 21 26 6e 64 |area is |<!&nd|
|00001560| 61 73 68 3b 20 4d 41 54 | 48 0a 20 3c 21 2d 2d 20 |ash; MAT|H. <!-- |
|00001570| 4d 41 54 48 0a 20 24 41 | 20 3d 20 50 5e 32 2f 31 |MATH. $A| = P^2/1|
|00001580| 36 24 0a 20 2d 2d 3e 0a | 3c 49 3e 41 3c 2f 49 3e |6$. -->.|<I>A</I>|
|00001590| 20 3d 20 3c 49 3e 50 3c | 2f 49 3e 3c 53 55 50 3e | = <I>P<|/I><SUP>|
|000015a0| 32 3c 2f 53 55 50 3e 2f | 31 36 0a 20 26 6e 64 61 |2</SUP>/|16. &nda|
|000015b0| 73 68 3b 26 67 74 3b 0a | 26 6c 74 3b 49 26 67 74 |sh;>.|<I>|
|000015c0| 3b 41 26 6c 74 3b 2f 49 | 26 67 74 3b 20 3d 20 26 |;A</I|> = &|
|000015d0| 6c 74 3b 49 26 67 74 3b | 50 26 6c 74 3b 2f 49 26 |lt;I>|P</I&|
|000015e0| 67 74 3b 26 6c 74 3b 53 | 55 50 26 67 74 3b 32 26 |gt;<S|UP>2&|
|000015f0| 6c 74 3b 2f 53 55 50 26 | 67 74 3b 2f 31 36 2e 20 |lt;/SUP&|gt;/16. |
|00001600| 20 43 6f 6d 70 6c 65 74 | 69 6e 67 20 74 68 65 20 | Complet|ing the |
|00001610| 73 71 75 61 72 65 20 76 | 65 72 69 66 69 65 73 20 |square v|erifies |
|00001620| 74 68 69 73 20 69 6e 74 | 75 69 74 69 6f 6e 3a 20 |this int|uition: |
|00001630| 26 6c 74 3b 21 26 6e 64 | 61 73 68 3b 20 4d 41 54 |<!&nd|ash; MAT|
|00001640| 48 0a 20 3c 21 2d 2d 20 | 4d 41 54 48 0a 20 24 41 |H. <!-- |MATH. $A|
|00001650| 28 78 29 20 3d 20 2d 28 | 78 20 2d 20 50 2f 34 29 |(x) = -(|x - P/4)|
|00001660| 5e 32 20 2b 20 50 5e 32 | 2f 31 36 24 0a 20 2d 2d |^2 + P^2|/16$. --|
|00001670| 3e 0a 3c 49 3e 41 3c 2f | 49 3e 28 3c 49 3e 78 3c |>.<I>A</|I>(<I>x<|
|00001680| 2f 49 3e 29 20 3d 20 2d | 20 28 3c 49 3e 78 3c 2f |/I>) = -| (<I>x</|
|00001690| 49 3e 20 2d 20 3c 49 3e | 50 3c 2f 49 3e 2f 34 29 |I> - <I>|P</I>/4)|
|000016a0| 3c 53 55 50 3e 32 3c 2f | 53 55 50 3e 20 2b 20 3c |<SUP>2</|SUP> + <|
|000016b0| 49 3e 50 3c 2f 49 3e 3c | 53 55 50 3e 32 3c 2f 53 |I>P</I><|SUP>2</S|
|000016c0| 55 50 3e 2f 31 36 0a 20 | 26 6e 64 61 73 68 3b 26 |UP>/16. |–&|
|000016d0| 67 74 3b 0a 26 6c 74 3b | 49 26 67 74 3b 41 26 6c |gt;.<|I>A&l|
|000016e0| 74 3b 2f 49 26 67 74 3b | 28 26 6c 74 3b 49 26 67 |t;/I>|(<I&g|
|000016f0| 74 3b 78 26 6c 74 3b 2f | 49 26 67 74 3b 29 20 3d |t;x</|I>) =|
|00001700| 20 2d 20 28 26 6c 74 3b | 49 26 67 74 3b 78 26 6c | - (<|I>x&l|
|00001710| 74 3b 2f 49 26 67 74 3b | 20 2d 20 26 6c 74 3b 49 |t;/I>| - <I|
|00001720| 26 67 74 3b 50 26 6c 74 | 3b 2f 49 26 67 74 3b 2f |>P<|;/I>/|
|00001730| 34 29 26 6c 74 3b 53 55 | 50 26 67 74 3b 32 26 6c |4)<SU|P>2&l|
|00001740| 74 3b 2f 53 55 50 26 67 | 74 3b 20 2b 20 26 6c 74 |t;/SUP&g|t; + <|
|00001750| 3b 49 26 67 74 3b 50 26 | 6c 74 3b 2f 49 26 67 74 |;I>P&|lt;/I>|
|00001760| 3b 26 6c 74 3b 53 55 50 | 26 67 74 3b 32 26 6c 74 |;<SUP|>2<|
|00001770| 3b 2f 53 55 50 26 67 74 | 3b 2f 31 36 2e 20 20 49 |;/SUP>|;/16. I|
|00001780| 6e 20 6d 6f 73 74 20 70 | 72 6f 62 6c 65 6d 73 2c |n most p|roblems,|
|00001790| 20 77 65 20 77 6f 75 6c | 64 20 6e 6f 74 20 62 65 | we woul|d not be|
|000017a0| 20 61 62 6c 65 20 74 6f | 20 68 61 6e 64 6c 65 20 | able to| handle |
|000017b0| 74 68 65 20 70 61 72 61 | 6d 65 74 65 72 20 73 6f |the para|meter so|
|000017c0| 20 63 6c 65 61 6e 6c 79 | 2e 20 20 49 6e 20 74 68 | cleanly|. In th|
|000017d0| 65 20 61 62 73 65 6e 63 | 65 20 6f 66 20 73 6f 6d |e absenc|e of som|
|000017e0| 65 20 74 68 65 6f 72 65 | 74 69 63 61 6c 20 72 65 |e theore|tical re|
|000017f0| 73 75 6c 74 20 6f 72 20 | 62 6c 69 6e 64 69 6e 67 |sult or |blinding|
|00001800| 20 66 6c 61 73 68 20 6f | 66 20 69 6e 73 69 67 68 | flash o|f insigh|
|00001810| 74 2c 20 77 65 20 6d 69 | 67 68 74 20 77 65 6c 6c |t, we mi|ght well|
|00001820| 20 70 69 63 6b 20 73 6f | 6d 65 20 60 60 74 79 70 | pick so|me ``typ|
|00001830| 69 63 61 6c 27 27 20 26 | 6c 74 3b 49 26 67 74 3b |ical'' &|lt;I>|
|00001840| 50 26 6c 74 3b 2f 49 26 | 67 74 3b 2d 76 61 6c 75 |P</I&|gt;-valu|
|00001850| 65 73 20 61 6e 64 20 75 | 73 65 20 67 72 61 70 68 |es and u|se graph|
|00001860| 69 63 61 6c 2f 74 61 62 | 75 6c 61 72 20 6f 72 20 |ical/tab|ular or |
|00001870| 6f 74 68 65 72 20 6d 65 | 74 68 6f 64 73 20 74 6f |other me|thods to|
|00001880| 20 61 74 74 65 6d 70 74 | 20 74 6f 20 66 6f 72 6d | attempt| to form|
|00001890| 20 61 20 68 79 70 6f 74 | 68 65 73 69 73 20 61 62 | a hypot|hesis ab|
|000018a0| 6f 75 74 20 74 68 65 20 | 67 65 6e 65 72 61 6c 20 |out the |general |
|000018b0| 63 61 73 65 2e 0a 0a 3c | 50 3e 0a 26 6c 74 3b 2f |case...<|P>.</|
|000018c0| 4c 49 26 67 74 3b 0a 26 | 6c 74 3b 2f 4f 4c 26 67 |LI>.&|lt;/OL&g|
|000018d0| 74 3b 0a 0a 3c 50 3e 0a | 26 6c 74 3b 50 26 67 74 |t;..<P>.|<P>|
|000018e0| 3b 0a 26 6c 74 3b 2f 4c | 49 26 67 74 3b 0a 26 6c |;.</L|I>.&l|
|000018f0| 74 3b 4c 49 26 67 74 3b | 46 72 6f 6d 20 46 69 67 |t;LI>|From Fig|
|00001900| 75 72 65 20 32 20 28 79 | 6f 75 20 64 69 64 20 64 |ure 2 (y|ou did d|
|00001910| 72 61 77 20 61 20 66 69 | 67 75 72 65 2c 20 64 69 |raw a fi|gure, di|
|00001920| 64 6e 27 74 20 79 6f 75 | 3f 29 2c 20 77 65 20 73 |dn't you|?), we s|
|00001930| 65 65 20 74 68 61 74 20 | 6e 6f 77 20 26 6c 74 3b |ee that |now <|
|00001940| 21 26 6e 64 61 73 68 3b | 20 4d 41 54 48 0a 20 3c |!–| MATH. <|
|00001950| 21 2d 2d 20 4d 41 54 48 | 0a 20 24 41 28 78 29 20 |!-- MATH|. $A(x) |
|00001960| 3d 20 32 78 28 35 30 20 | 2d 20 78 29 24 0a 20 2d |= 2x(50 |- x)$. -|
|00001970| 2d 3e 0a 3c 49 3e 41 3c | 2f 49 3e 28 3c 49 3e 78 |->.<I>A<|/I>(<I>x|
|00001980| 3c 2f 49 3e 29 20 3d 20 | 32 3c 49 3e 78 3c 2f 49 |</I>) = |2<I>x</I|
|00001990| 3e 28 35 30 20 2d 20 3c | 49 3e 78 3c 2f 49 3e 29 |>(50 - <|I>x</I>)|
|000019a0| 0a 20 26 6e 64 61 73 68 | 3b 26 67 74 3b 0a 26 6c |. &ndash|;>.&l|
|000019b0| 74 3b 49 26 67 74 3b 41 | 26 6c 74 3b 2f 49 26 67 |t;I>A|</I&g|
|000019c0| 74 3b 28 26 6c 74 3b 49 | 26 67 74 3b 78 26 6c 74 |t;(<I|>x<|
|000019d0| 3b 2f 49 26 67 74 3b 29 | 20 3d 20 32 26 6c 74 3b |;/I>)| = 2<|
|000019e0| 49 26 67 74 3b 78 26 6c | 74 3b 2f 49 26 67 74 3b |I>x&l|t;/I>|
|000019f0| 28 35 30 20 2d 20 26 6c | 74 3b 49 26 67 74 3b 78 |(50 - &l|t;I>x|
|00001a00| 26 6c 74 3b 2f 49 26 67 | 74 3b 29 2e 20 20 20 55 |</I&g|t;). U|
|00001a10| 73 69 6e 67 20 74 68 65 | 20 60 60 68 61 6c 66 2d |sing the| ``half-|
|00001a20| 77 61 79 20 62 65 74 77 | 65 65 6e 27 27 20 69 6e |way betw|een'' in|
|00001a30| 73 69 67 68 74 20 66 6f | 72 20 70 61 72 61 62 6f |sight fo|r parabo|
|00001a40| 6c 61 73 20 6f 72 20 73 | 6f 6d 65 20 6f 74 68 65 |las or s|ome othe|
|00001a50| 72 20 6d 65 74 68 6f 64 | 2c 20 66 69 6e 64 20 74 |r method|, find t|
|00001a60| 68 61 74 20 74 68 65 20 | 6d 61 78 69 6d 69 7a 69 |hat the |maximizi|
|00001a70| 6e 67 20 26 6c 74 3b 49 | 26 67 74 3b 78 26 6c 74 |ng <I|>x<|
|00001a80| 3b 2f 49 26 67 74 3b 20 | 3d 20 32 35 20 61 6e 64 |;/I> |= 25 and|
|00001a90| 20 74 68 65 20 6d 61 78 | 69 6d 75 6d 20 76 61 6c | the max|imum val|
|00001aa0| 75 65 20 69 73 20 26 6c | 74 3b 49 26 67 74 3b 41 |ue is &l|t;I>A|
|00001ab0| 26 6c 74 3b 2f 49 26 67 | 74 3b 20 3d 20 31 32 35 |</I&g|t; = 125|
|00001ac0| 30 20 6d 26 6c 74 3b 53 | 55 50 26 67 74 3b 32 26 |0 m<S|UP>2&|
|00001ad0| 6c 74 3b 2f 53 55 50 26 | 67 74 3b 20 28 65 78 61 |lt;/SUP&|gt; (exa|
|00001ae0| 63 74 6c 79 20 74 77 69 | 63 65 20 61 73 20 6c 61 |ctly twi|ce as la|
|00001af0| 72 67 65 20 61 73 20 66 | 6f 72 20 74 68 65 20 34 |rge as f|or the 4|
|00001b00| 2d 73 69 64 65 64 20 63 | 61 73 65 29 2e 20 20 20 |-sided c|ase). |
|00001b10| 20 46 6f 72 20 74 68 65 | 20 67 65 6e 65 72 61 6c | For the| general|
|00001b20| 20 63 61 73 65 2c 20 74 | 68 65 20 66 6f 72 6d 75 | case, t|he formu|
|00001b30| 6c 61 20 69 73 20 26 6c | 74 3b 21 26 6e 64 61 73 |la is &l|t;!&ndas|
|00001b40| 68 3b 20 4d 41 54 48 0a | 20 3c 21 2d 2d 20 4d 41 |h; MATH.| <!-- MA|
|00001b50| 54 48 0a 20 24 41 28 78 | 29 20 3d 20 32 78 28 50 |TH. $A(x|) = 2x(P|
|00001b60| 2f 32 20 2d 20 78 29 24 | 0a 20 2d 2d 3e 0a 3c 49 |/2 - x)$|. -->.<I|
|00001b70| 3e 41 3c 2f 49 3e 28 3c | 49 3e 78 3c 2f 49 3e 29 |>A</I>(<|I>x</I>)|
|00001b80| 20 3d 20 32 3c 49 3e 78 | 3c 2f 49 3e 28 3c 49 3e | = 2<I>x|</I>(<I>|
|00001b90| 50 3c 2f 49 3e 2f 32 20 | 2d 20 3c 49 3e 78 3c 2f |P</I>/2 |- <I>x</|
|00001ba0| 49 3e 29 0a 20 26 6e 64 | 61 73 68 3b 26 67 74 3b |I>). &nd|ash;>|
|00001bb0| 0a 26 6c 74 3b 49 26 67 | 74 3b 41 26 6c 74 3b 2f |.<I&g|t;A</|
|00001bc0| 49 26 67 74 3b 28 26 6c | 74 3b 49 26 67 74 3b 78 |I>(&l|t;I>x|
|00001bd0| 26 6c 74 3b 2f 49 26 67 | 74 3b 29 20 3d 20 32 26 |</I&g|t;) = 2&|
|00001be0| 6c 74 3b 49 26 67 74 3b | 78 26 6c 74 3b 2f 49 26 |lt;I>|x</I&|
|00001bf0| 67 74 3b 28 26 6c 74 3b | 49 26 67 74 3b 50 26 6c |gt;(<|I>P&l|
|00001c00| 74 3b 2f 49 26 67 74 3b | 2f 32 20 2d 20 26 6c 74 |t;/I>|/2 - <|
|00001c10| 3b 49 26 67 74 3b 78 26 | 6c 74 3b 2f 49 26 67 74 |;I>x&|lt;/I>|
|00001c20| 3b 29 20 77 69 74 68 20 | 6d 61 78 69 6d 75 6d 20 |;) with |maximum |
|00001c30| 61 74 20 26 6c 74 3b 49 | 26 67 74 3b 78 26 6c 74 |at <I|>x<|
|00001c40| 3b 2f 49 26 67 74 3b 20 | 3d 20 26 6c 74 3b 49 26 |;/I> |= <I&|
|00001c50| 67 74 3b 50 26 6c 74 3b | 2f 49 26 67 74 3b 2f 34 |gt;P<|/I>/4|
|00001c60| 20 6f 66 20 73 69 7a 65 | 20 26 6c 74 3b 49 26 67 | of size| <I&g|
|00001c70| 74 3b 41 26 6c 74 3b 2f | 49 26 67 74 3b 20 3d 20 |t;A</|I> = |
|00001c80| 26 6c 74 3b 49 26 67 74 | 3b 50 26 6c 74 3b 2f 49 |<I>|;P</I|
|00001c90| 26 67 74 3b 26 6c 74 3b | 53 55 50 26 67 74 3b 32 |><|SUP>2|
|00001ca0| 26 6c 74 3b 2f 53 55 50 | 26 67 74 3b 2f 38 2e 0a |</SUP|>/8..|
|00001cb0| 0a 3c 50 3e 0a 26 6c 74 | 3b 50 26 67 74 3b 0a 0a |.<P>.<|;P>..|
|00001cc0| 3c 50 3e 0a 26 6c 74 3b | 44 49 56 20 63 6c 61 73 |<P>.<|DIV clas|
|00001cd0| 73 3d 22 43 45 4e 54 45 | 52 22 26 67 74 3b 26 6c |s="CENTE|R">&l|
|00001ce0| 74 3b 41 20 49 44 3d 22 | 31 39 22 26 67 74 3b 26 |t;A ID="|19">&|
|00001cf0| 6c 74 3b 2f 41 26 67 74 | 3b 0a 26 6c 74 3b 54 41 |lt;/A>|;.<TA|
|00001d00| 42 4c 45 26 67 74 3b 0a | 26 6c 74 3b 43 41 50 54 |BLE>.|<CAPT|
|00001d10| 49 4f 4e 20 63 6c 61 73 | 73 3d 22 42 4f 54 54 4f |ION clas|s="BOTTO|
|00001d20| 4d 22 26 67 74 3b 26 6c | 74 3b 53 54 52 4f 4e 47 |M">&l|t;STRONG|
|00001d30| 26 67 74 3b 46 69 67 75 | 72 65 3a 26 6c 74 3b 2f |>Figu|re:</|
|00001d40| 53 54 52 4f 4e 47 26 67 | 74 3b 0a 44 69 61 67 72 |STRONG&g|t;.Diagr|
|00001d50| 61 6d 20 66 6f 72 20 50 | 72 6f 62 6c 65 6d 20 32 |am for P|roblem 2|
|00001d60| 2e 26 6c 74 3b 2f 43 41 | 50 54 49 4f 4e 26 67 74 |.</CA|PTION>|
|00001d70| 3b 0a 26 6c 74 3b 54 52 | 26 67 74 3b 26 6c 74 3b |;.<TR|><|
|00001d80| 54 44 26 67 74 3b 26 6c | 74 3b 49 4d 47 0a 20 53 |TD>&l|t;IMG. S|
|00001d90| 54 59 4c 45 3d 22 68 65 | 69 67 68 74 3a 20 32 38 |TYLE="he|ight: 28|
|00001da0| 36 2e 37 36 65 78 3b 20 | 22 20 53 52 43 3d 22 69 |6.76ex; |" SRC="i|
|00001db0| 6d 67 32 2e 70 6e 67 22 | 0a 20 41 4c 54 3d 22 0a |mg2.png"|. ALT=".|
|00001dc0| 3c 44 49 56 20 63 6c 61 | 73 73 3d 22 43 45 4e 54 |<DIV cla|ss="CENT|
|00001dd0| 45 52 22 3e 0a 3c 49 4d | 47 0a 20 53 54 59 4c 45 |ER">.<IM|G. STYLE|
|00001de0| 3d 22 68 65 69 67 68 74 | 3a 20 33 31 34 2e 30 30 |="height|: 314.00|
|00001df0| 65 78 3b 20 22 20 53 52 | 43 3d 22 69 6d 67 32 2e |ex; " SR|C="img2.|
|00001e00| 70 6e 67 22 0a 20 41 4c | 54 3d 22 5c 62 65 67 69 |png". AL|T="\begi|
|00001e10| 6e 7b 66 69 67 75 72 65 | 7d 5c 65 70 73 66 79 73 |n{figure|}\epsfys|
|00001e20| 69 7a 65 20 31 30 30 70 | 74 0a 5c 63 65 6e 74 65 |ize 100p|t.\cente|
|00001e30| 72 6c 69 6e 65 7b 5c 65 | 70 73 66 66 69 6c 65 7b |rline{\e|psffile{|
|00001e40| 61 6e 73 33 70 32 2e 65 | 70 73 7d 7d 0a 5c 65 6e |ans3p2.e|ps}}.\en|
|00001e50| 64 7b 66 69 67 75 72 65 | 7d 22 3e 0a 3c 2f 44 49 |d{figure|}">.</DI|
|00001e60| 56 3e 22 26 67 74 3b 26 | 6c 74 3b 2f 54 44 26 67 |V>">&|lt;/TD&g|
|00001e70| 74 3b 26 6c 74 3b 2f 54 | 52 26 67 74 3b 0a 26 6c |t;</T|R>.&l|
|00001e80| 74 3b 2f 54 41 42 4c 45 | 26 67 74 3b 0a 26 6c 74 |t;/TABLE|>.<|
|00001e90| 3b 2f 44 49 56 26 67 74 | 3b 0a 0a 3c 50 3e 0a 26 |;/DIV>|;..<P>.&|
|00001ea0| 6c 74 3b 50 26 67 74 3b | 0a 26 6c 74 3b 2f 4c 49 |lt;P>|.</LI|
|00001eb0| 26 67 74 3b 0a 26 6c 74 | 3b 4c 49 26 67 74 3b 4e |>.<|;LI>N|
|00001ec0| 6f 74 65 20 74 68 61 74 | 20 6e 6f 77 2c 20 77 65 |ote that| now, we|
|00001ed0| 20 77 61 6e 74 20 74 6f | 20 6d 69 6e 69 6d 69 7a | want to| minimiz|
|00001ee0| 65 20 74 68 65 20 70 65 | 72 69 6d 65 74 65 72 2e |e the pe|rimeter.|
|00001ef0| 20 20 46 72 6f 6d 20 46 | 69 67 75 72 65 20 33 2c | From F|igure 3,|
|00001f00| 20 77 65 20 73 65 65 20 | 74 68 61 74 20 74 68 65 | we see |that the|
|00001f10| 20 73 75 6d 20 6f 66 20 | 73 69 64 65 20 6c 65 6e | sum of |side len|
|00001f20| 67 74 68 73 20 69 73 20 | 26 6c 74 3b 21 26 6e 64 |gths is |<!&nd|
|00001f30| 61 73 68 3b 20 4d 41 54 | 48 0a 20 3c 21 2d 2d 20 |ash; MAT|H. <!-- |
|00001f40| 4d 41 54 48 0a 20 24 4c | 20 3d 20 78 20 2b 20 31 |MATH. $L| = x + 1|
|00001f50| 30 30 30 2f 78 20 2b 20 | 78 20 3d 20 32 78 20 2b |000/x + |x = 2x +|
|00001f60| 20 31 30 30 30 2f 78 24 | 0a 20 2d 2d 3e 0a 3c 49 | 1000/x$|. -->.<I|
|00001f70| 3e 4c 3c 2f 49 3e 20 3d | 20 3c 49 3e 78 3c 2f 49 |>L</I> =| <I>x</I|
|00001f80| 3e 20 2b 20 31 30 30 30 | 2f 3c 49 3e 78 3c 2f 49 |> + 1000|/<I>x</I|
|00001f90| 3e 20 2b 20 3c 49 3e 78 | 3c 2f 49 3e 20 3d 20 32 |> + <I>x|</I> = 2|
|00001fa0| 3c 49 3e 78 3c 2f 49 3e | 20 2b 20 31 30 30 30 2f |<I>x</I>| + 1000/|
|00001fb0| 3c 49 3e 78 3c 2f 49 3e | 0a 20 26 6e 64 61 73 68 |<I>x</I>|. &ndash|
|00001fc0| 3b 26 67 74 3b 0a 26 6c | 74 3b 49 26 67 74 3b 4c |;>.&l|t;I>L|
|00001fd0| 26 6c 74 3b 2f 49 26 67 | 74 3b 20 3d 20 26 6c 74 |</I&g|t; = <|
|00001fe0| 3b 49 26 67 74 3b 78 26 | 6c 74 3b 2f 49 26 67 74 |;I>x&|lt;/I>|
|00001ff0| 3b 20 2b 20 31 30 30 30 | 2f 26 6c 74 3b 49 26 67 |; + 1000|/<I&g|
|00002000| 74 3b 78 26 6c 74 3b 2f | 49 26 67 74 3b 20 2b 20 |t;x</|I> + |
|00002010| 26 6c 74 3b 49 26 67 74 | 3b 78 26 6c 74 3b 2f 49 |<I>|;x</I|
|00002020| 26 67 74 3b 20 3d 20 32 | 26 6c 74 3b 49 26 67 74 |> = 2|<I>|
|00002030| 3b 78 26 6c 74 3b 2f 49 | 26 67 74 3b 20 2b 20 31 |;x</I|> + 1|
|00002040| 30 30 30 2f 26 6c 74 3b | 49 26 67 74 3b 78 26 6c |000/<|I>x&l|
|00002050| 74 3b 2f 49 26 67 74 3b | 2e 20 0a 0a 3c 50 3e 0a |t;/I>|. ..<P>.|
|00002060| 26 6c 74 3b 50 26 67 74 | 3b 0a 0a 3c 50 3e 0a 26 |<P>|;..<P>.&|
|00002070| 6c 74 3b 44 49 56 20 63 | 6c 61 73 73 3d 22 43 45 |lt;DIV c|lass="CE|
|00002080| 4e 54 45 52 22 26 67 74 | 3b 26 6c 74 3b 41 20 49 |NTER">|;<A I|
|00002090| 44 3d 22 32 34 22 26 67 | 74 3b 26 6c 74 3b 2f 41 |D="24"&g|t;</A|
|000020a0| 26 67 74 3b 0a 26 6c 74 | 3b 54 41 42 4c 45 26 67 |>.<|;TABLE&g|
|000020b0| 74 3b 0a 26 6c 74 3b 43 | 41 50 54 49 4f 4e 20 63 |t;.<C|APTION c|
|000020c0| 6c 61 73 73 3d 22 42 4f | 54 54 4f 4d 22 26 67 74 |lass="BO|TTOM">|
|000020d0| 3b 26 6c 74 3b 53 54 52 | 4f 4e 47 26 67 74 3b 46 |;<STR|ONG>F|
|000020e0| 69 67 75 72 65 3a 26 6c | 74 3b 2f 53 54 52 4f 4e |igure:&l|t;/STRON|
|000020f0| 47 26 67 74 3b 0a 44 69 | 61 67 72 61 6d 20 66 6f |G>.Di|agram fo|
|00002100| 72 20 50 72 6f 62 6c 65 | 6d 20 33 2e 26 6c 74 3b |r Proble|m 3.<|
|00002110| 2f 43 41 50 54 49 4f 4e | 26 67 74 3b 0a 26 6c 74 |/CAPTION|>.<|
|00002120| 3b 54 52 26 67 74 3b 26 | 6c 74 3b 54 44 26 67 74 |;TR>&|lt;TD>|
|00002130| 3b 26 6c 74 3b 49 4d 47 | 0a 20 53 54 59 4c 45 3d |;<IMG|. STYLE=|
|00002140| 22 68 65 69 67 68 74 3a | 20 32 38 36 2e 37 36 65 |"height:| 286.76e|
|00002150| 78 3b 20 22 20 53 52 43 | 3d 22 69 6d 67 33 2e 70 |x; " SRC|="img3.p|
|00002160| 6e 67 22 0a 20 41 4c 54 | 3d 22 0a 3c 44 49 56 20 |ng". ALT|=".<DIV |
|00002170| 63 6c 61 73 73 3d 22 43 | 45 4e 54 45 52 22 3e 0a |class="C|ENTER">.|
|00002180| 3c 49 4d 47 0a 20 53 54 | 59 4c 45 3d 22 68 65 69 |<IMG. ST|YLE="hei|
|00002190| 67 68 74 3a 20 33 31 34 | 2e 30 30 65 78 3b 20 22 |ght: 314|.00ex; "|
|000021a0| 20 53 52 43 3d 22 69 6d | 67 33 2e 70 6e 67 22 0a | SRC="im|g3.png".|
|000021b0| 20 41 4c 54 3d 22 5c 62 | 65 67 69 6e 7b 66 69 67 | ALT="\b|egin{fig|
|000021c0| 75 72 65 7d 5c 65 70 73 | 66 79 73 69 7a 65 20 31 |ure}\eps|fysize 1|
|000021d0| 30 30 70 74 0a 5c 63 65 | 6e 74 65 72 6c 69 6e 65 |00pt.\ce|nterline|
|000021e0| 7b 5c 65 70 73 66 66 69 | 6c 65 7b 61 6e 73 33 70 |{\epsffi|le{ans3p|
|000021f0| 33 61 2e 65 70 73 7d 7d | 0a 5c 65 6e 64 7b 66 69 |3a.eps}}|.\end{fi|
|00002200| 67 75 72 65 7d 22 3e 0a | 3c 2f 44 49 56 3e 22 26 |gure}">.|</DIV>"&|
|00002210| 67 74 3b 26 6c 74 3b 2f | 54 44 26 67 74 3b 26 6c |gt;</|TD>&l|
|00002220| 74 3b 2f 54 52 26 67 74 | 3b 0a 26 6c 74 3b 2f 54 |t;/TR>|;.</T|
|00002230| 41 42 4c 45 26 67 74 3b | 0a 26 6c 74 3b 2f 44 49 |ABLE>|.</DI|
|00002240| 56 26 67 74 3b 0a 4f 6e | 65 20 6d 6f 64 65 20 6f |V>.On|e mode o|
|00002250| 66 20 73 6f 6c 75 74 69 | 6f 6e 20 69 73 20 61 20 |f soluti|on is a |
|00002260| 67 72 61 70 68 2e 20 20 | 46 69 67 75 72 65 20 34 |graph. |Figure 4|
|00002270| 20 73 68 6f 77 73 20 61 | 20 7a 6f 6f 6d 20 6f 66 | shows a| zoom of|
|00002280| 20 74 68 65 20 63 72 69 | 74 69 63 61 6c 20 72 65 | the cri|tical re|
|00002290| 67 69 6f 6e 20 6f 62 74 | 61 69 6e 65 64 20 28 61 |gion obt|ained (a|
|000022a0| 66 74 65 72 20 73 6f 6d | 65 20 65 78 70 65 72 69 |fter som|e experi|
|000022b0| 6d 65 6e 74 61 74 69 6f | 6e 29 20 62 79 20 75 73 |mentatio|n) by us|
|000022c0| 69 6e 67 20 74 68 65 20 | 63 6f 6d 6d 61 6e 64 0a |ing the |command.|
|000022d0| 0a 3c 50 3e 0a 26 6c 74 | 3b 50 26 67 74 3b 0a 50 |.<P>.<|;P>.P|
|000022e0| 6c 6f 74 5b 32 78 20 2b | 20 31 30 30 30 2f 78 2c |lot[2x +| 1000/x,|
|000022f0| 20 78 2c 20 31 30 2c 20 | 35 30 2c 20 50 6c 6f 74 | x, 10, |50, Plot|
|00002300| 52 61 6e 67 65 20 2d 26 | 61 6d 70 3b 67 74 3b 20 |Range -&|amp;gt; |
|00002310| 38 30 2c 20 31 35 30 5d | 20 0a 0a 3c 50 3e 0a 26 |80, 150]| ..<P>.&|
|00002320| 6c 74 3b 50 26 67 74 3b | 0a 50 69 63 6b 69 6e 67 |lt;P>|.Picking|
|00002330| 20 74 68 65 20 70 6f 69 | 6e 74 20 61 74 20 74 68 | the poi|nt at th|
|00002340| 65 20 6d 69 6e 69 6d 75 | 6d 20 67 69 76 65 73 20 |e minimu|m gives |
|00002350| 26 6c 74 3b 21 26 6e 64 | 61 73 68 3b 20 4d 41 54 |<!&nd|ash; MAT|
|00002360| 48 0a 20 3c 21 2d 2d 20 | 4d 41 54 48 0a 20 24 78 |H. <!-- |MATH. $x|
|00002370| 20 5c 61 70 70 72 6f 78 | 20 32 32 24 0a 20 2d 2d | \approx| 22$. --|
|00002380| 3e 0a 3c 49 3e 78 3c 2f | 49 3e 20 3c 49 4d 47 0a |>.<I>x</|I> <IMG.|
|00002390| 20 53 54 59 4c 45 3d 22 | 68 65 69 67 68 74 3a 20 | STYLE="|height: |
|000023a0| 32 2e 31 30 65 78 3b 20 | 76 65 72 74 69 63 61 6c |2.10ex; |vertical|
|000023b0| 2d 61 6c 69 67 6e 3a 20 | 31 37 37 2e 32 37 65 78 |-align: |177.27ex|
|000023c0| 3b 20 22 20 53 52 43 3d | 22 69 6d 67 34 2e 70 6e |; " SRC=|"img4.pn|
|000023d0| 67 22 0a 20 41 4c 54 3d | 22 24 5c 61 70 70 72 6f |g". ALT=|"$\appro|
|000023e0| 78 24 22 3e 20 32 32 0a | 20 26 6e 64 61 73 68 3b |x$"> 22.| –|
|000023f0| 26 67 74 3b 0a 26 6c 74 | 3b 49 26 67 74 3b 78 26 |>.<|;I>x&|
|00002400| 6c 74 3b 2f 49 26 67 74 | 3b 20 26 6c 74 3b 49 4d |lt;/I>|; <IM|
|00002410| 47 0a 20 53 54 59 4c 45 | 3d 22 68 65 69 67 68 74 |G. STYLE|="height|
|00002420| 3a 20 31 37 39 2e 32 32 | 65 78 3b 20 76 65 72 74 |: 179.22|ex; vert|
|00002430| 69 63 61 6c 2d 61 6c 69 | 67 6e 3a 20 2d 30 2e 31 |ical-ali|gn: -0.1|
|00002440| 31 65 78 3b 20 22 20 53 | 52 43 3d 22 69 6d 67 34 |1ex; " S|RC="img4|
|00002450| 2e 70 6e 67 22 0a 20 41 | 4c 54 3d 22 3c 49 4d 47 |.png". A|LT="<IMG|
|00002460| 0a 20 53 54 59 4c 45 3d | 22 68 65 69 67 68 74 3a |. STYLE=|"height:|
|00002470| 20 32 2e 31 30 65 78 3b | 20 76 65 72 74 69 63 61 | 2.10ex;| vertica|
|00002480| 6c 2d 61 6c 69 67 6e 3a | 20 31 37 37 2e 32 37 65 |l-align:| 177.27e|
|00002490| 78 3b 20 22 20 53 52 43 | 3d 22 69 6d 67 34 2e 70 |x; " SRC|="img4.p|
|000024a0| 6e 67 22 0a 20 41 4c 54 | 3d 22 24 5c 61 70 70 72 |ng". ALT|="$\appr|
|000024b0| 6f 78 24 22 3e 22 26 67 | 74 3b 20 32 32 2e 0a 54 |ox$">"&g|t; 22..T|
|000024c0| 68 65 6e 20 74 68 65 20 | 63 6f 6d 6d 61 6e 64 0a |hen the |command.|
|000024d0| 0a 3c 50 3e 0a 26 6c 74 | 3b 50 26 67 74 3b 0a 54 |.<P>.<|;P>.T|
|000024e0| 61 62 6c 65 5b 78 2c 20 | 32 78 20 2b 20 31 30 30 |able[x, |2x + 100|
|000024f0| 30 2f 78 2c 20 78 2c 20 | 32 32 2e 32 2c 20 32 32 |0/x, x, |22.2, 22|
|00002500| 2e 34 2c 20 30 2e 30 35 | 5d 20 2f 2f 54 61 62 6c |.4, 0.05|] //Tabl|
|00002510| 65 46 6f 72 6d 0a 0a 3c | 50 3e 0a 26 6c 74 3b 50 |eForm..<|P>.<P|
|00002520| 26 67 74 3b 0a 6c 65 74 | 73 20 75 73 20 68 6f 6d |>.let|s us hom|
|00002530| 65 20 69 6e 20 6f 6e 20 | 74 68 65 20 61 6e 73 77 |e in on |the answ|
|00002540| 65 72 3a 0a 0a 3c 50 3e | 0a 26 6c 74 3b 50 26 67 |er:..<P>|.<P&g|
|00002550| 74 3b 0a 26 6c 74 3b 50 | 52 45 26 67 74 3b 0a 32 |t;.<P|RE>.2|
|00002560| 32 2e 32 20 20 20 20 38 | 39 2e 34 34 35 0a 32 32 |2.2 8|9.445.22|
|00002570| 2e 32 35 20 20 20 38 39 | 2e 34 34 33 38 0a 32 32 |.25 89|.4438.22|
|00002580| 2e 33 20 20 20 20 38 39 | 2e 34 34 33 0a 32 32 2e |.3 89|.443.22.|
|00002590| 33 35 20 20 20 38 39 2e | 34 34 32 37 0a 32 32 2e |35 89.|4427.22.|
|000025a0| 34 20 20 20 20 38 39 2e | 34 34 32 39 0a 26 6c 74 |4 89.|4429.<|
|000025b0| 3b 2f 50 52 45 26 67 74 | 3b 0a 54 68 75 73 20 26 |;/PRE>|;.Thus &|
|000025c0| 6c 74 3b 21 26 6e 64 61 | 73 68 3b 20 4d 41 54 48 |lt;!&nda|sh; MATH|
|000025d0| 0a 20 3c 21 2d 2d 20 4d | 41 54 48 0a 20 24 78 20 |. <!-- M|ATH. $x |
|000025e0| 5c 61 70 70 72 6f 78 20 | 32 32 2e 33 35 24 0a 20 |\approx |22.35$. |
|000025f0| 2d 2d 3e 0a 3c 49 3e 78 | 3c 2f 49 3e 20 3c 49 4d |-->.<I>x|</I> <IM|
|00002600| 47 0a 20 53 54 59 4c 45 | 3d 22 68 65 69 67 68 74 |G. STYLE|="height|
|00002610| 3a 20 32 2e 31 30 65 78 | 3b 20 76 65 72 74 69 63 |: 2.10ex|; vertic|
|00002620| 61 6c 2d 61 6c 69 67 6e | 3a 20 31 37 37 2e 32 37 |al-align|: 177.27|
|00002630| 65 78 3b 20 22 20 53 52 | 43 3d 22 69 6d 67 34 2e |ex; " SR|C="img4.|
|00002640| 70 6e 67 22 0a 20 41 4c | 54 3d 22 24 5c 61 70 70 |png". AL|T="$\app|
|00002650| 72 6f 78 24 22 3e 20 32 | 32 2e 33 35 0a 20 26 6e |rox$"> 2|2.35. &n|
|00002660| 64 61 73 68 3b 26 67 74 | 3b 0a 26 6c 74 3b 49 26 |dash;>|;.<I&|
|00002670| 67 74 3b 78 26 6c 74 3b | 2f 49 26 67 74 3b 20 26 |gt;x<|/I> &|
|00002680| 6c 74 3b 49 4d 47 0a 20 | 53 54 59 4c 45 3d 22 68 |lt;IMG. |STYLE="h|
|00002690| 65 69 67 68 74 3a 20 31 | 37 39 2e 32 32 65 78 3b |eight: 1|79.22ex;|
|000026a0| 20 76 65 72 74 69 63 61 | 6c 2d 61 6c 69 67 6e 3a | vertica|l-align:|
|000026b0| 20 2d 30 2e 31 31 65 78 | 3b 20 22 20 53 52 43 3d | -0.11ex|; " SRC=|
|000026c0| 22 69 6d 67 34 2e 70 6e | 67 22 0a 20 41 4c 54 3d |"img4.pn|g". ALT=|
|000026d0| 22 3c 49 4d 47 0a 20 53 | 54 59 4c 45 3d 22 68 65 |"<IMG. S|TYLE="he|
|000026e0| 69 67 68 74 3a 20 32 2e | 31 30 65 78 3b 20 76 65 |ight: 2.|10ex; ve|
|000026f0| 72 74 69 63 61 6c 2d 61 | 6c 69 67 6e 3a 20 31 37 |rtical-a|lign: 17|
|00002700| 37 2e 32 37 65 78 3b 20 | 22 20 53 52 43 3d 22 69 |7.27ex; |" SRC="i|
|00002710| 6d 67 34 2e 70 6e 67 22 | 0a 20 41 4c 54 3d 22 24 |mg4.png"|. ALT="$|
|00002720| 5c 61 70 70 72 6f 78 24 | 22 3e 22 26 67 74 3b 20 |\approx$|">"> |
|00002730| 32 32 2e 33 35 20 61 6e | 64 20 26 6c 74 3b 21 26 |22.35 an|d <!&|
|00002740| 6e 64 61 73 68 3b 20 4d | 41 54 48 0a 20 3c 21 2d |ndash; M|ATH. <!-|
|00002750| 2d 20 4d 41 54 48 0a 20 | 24 41 20 5c 61 70 70 72 |- MATH. |$A \appr|
|00002760| 6f 78 20 38 39 2e 34 34 | 32 37 24 0a 20 2d 2d 3e |ox 89.44|27$. -->|
|00002770| 0a 3c 49 3e 41 3c 2f 49 | 3e 20 3c 49 4d 47 0a 20 |.<I>A</I|> <IMG. |
|00002780| 53 54 59 4c 45 3d 22 68 | 65 69 67 68 74 3a 20 32 |STYLE="h|eight: 2|
|00002790| 2e 31 30 65 78 3b 20 76 | 65 72 74 69 63 61 6c 2d |.10ex; v|ertical-|
|000027a0| 61 6c 69 67 6e 3a 20 31 | 37 37 2e 32 37 65 78 3b |align: 1|77.27ex;|
|000027b0| 20 22 20 53 52 43 3d 22 | 69 6d 67 34 2e 70 6e 67 | " SRC="|img4.png|
|000027c0| 22 0a 20 41 4c 54 3d 22 | 24 5c 61 70 70 72 6f 78 |". ALT="|$\approx|
|000027d0| 24 22 3e 20 38 39 2e 34 | 34 32 37 0a 20 26 6e 64 |$"> 89.4|427. &nd|
|000027e0| 61 73 68 3b 26 67 74 3b | 0a 26 6c 74 3b 49 26 67 |ash;>|.<I&g|
|000027f0| 74 3b 41 26 6c 74 3b 2f | 49 26 67 74 3b 20 26 6c |t;A</|I> &l|
|00002800| 74 3b 49 4d 47 0a 20 53 | 54 59 4c 45 3d 22 68 65 |t;IMG. S|TYLE="he|
|00002810| 69 67 68 74 3a 20 31 37 | 39 2e 32 32 65 78 3b 20 |ight: 17|9.22ex; |
|00002820| 76 65 72 74 69 63 61 6c | 2d 61 6c 69 67 6e 3a 20 |vertical|-align: |
|00002830| 2d 30 2e 31 31 65 78 3b | 20 22 20 53 52 43 3d 22 |-0.11ex;| " SRC="|
|00002840| 69 6d 67 34 2e 70 6e 67 | 22 0a 20 41 4c 54 3d 22 |img4.png|". ALT="|
|00002850| 3c 49 4d 47 0a 20 53 54 | 59 4c 45 3d 22 68 65 69 |<IMG. ST|YLE="hei|
|00002860| 67 68 74 3a 20 32 2e 31 | 30 65 78 3b 20 76 65 72 |ght: 2.1|0ex; ver|
|00002870| 74 69 63 61 6c 2d 61 6c | 69 67 6e 3a 20 31 37 37 |tical-al|ign: 177|
|00002880| 2e 32 37 65 78 3b 20 22 | 20 53 52 43 3d 22 69 6d |.27ex; "| SRC="im|
|00002890| 67 34 2e 70 6e 67 22 0a | 20 41 4c 54 3d 22 24 5c |g4.png".| ALT="$\|
|000028a0| 61 70 70 72 6f 78 24 22 | 3e 22 26 67 74 3b 20 38 |approx$"|>"> 8|
|000028b0| 39 2e 34 34 32 37 20 61 | 74 20 74 68 65 20 6d 69 |9.4427 a|t the mi|
|000028c0| 6e 69 6d 75 6d 2e 20 20 | 43 61 6c 63 75 6c 75 73 |nimum. |Calculus|
|000028d0| 20 6d 65 74 68 6f 64 73 | 20 6c 65 74 20 75 73 65 | methods| let use|
|000028e0| 20 64 65 74 65 72 6d 69 | 6e 65 20 74 68 61 74 20 | determi|ne that |
|000028f0| 74 68 65 20 65 78 61 63 | 74 20 61 6e 73 77 65 72 |the exac|t answer|
|00002900| 20 69 73 20 26 6c 74 3b | 21 26 6e 64 61 73 68 3b | is <|!–|
|00002910| 20 4d 41 54 48 0a 20 3c | 21 2d 2d 20 4d 41 54 48 | MATH. <|!-- MATH|
|00002920| 0a 20 24 78 20 3d 20 32 | 20 5c 63 64 6f 74 20 35 |. $x = 2| \cdot 5|
|00002930| 5e 7b 33 2f 32 7d 20 5c | 61 70 70 72 6f 78 20 20 |^{3/2} \|approx |
|00002940| 32 32 2e 33 36 30 36 37 | 39 37 37 34 39 39 37 39 |22.36067|97749979|
|00002950| 24 0a 20 2d 2d 3e 0a 3c | 49 3e 78 3c 2f 49 3e 20 |$. -->.<|I>x</I> |
|00002960| 3d 20 32 26 23 38 39 30 | 31 3b 35 3c 53 55 50 3e |= 2ͺ|1;5<SUP>|
|00002970| 33 2f 32 3c 2f 53 55 50 | 3e 20 3c 49 4d 47 0a 20 |3/2</SUP|> <IMG. |
|00002980| 53 54 59 4c 45 3d 22 68 | 65 69 67 68 74 3a 20 32 |STYLE="h|eight: 2|
|00002990| 2e 31 30 65 78 3b 20 76 | 65 72 74 69 63 61 6c 2d |.10ex; v|ertical-|
|000029a0| 61 6c 69 67 6e 3a 20 31 | 37 37 2e 32 37 65 78 3b |align: 1|77.27ex;|
|000029b0| 20 22 20 53 52 43 3d 22 | 69 6d 67 34 2e 70 6e 67 | " SRC="|img4.png|
|000029c0| 22 0a 20 41 4c 54 3d 22 | 24 5c 61 70 70 72 6f 78 |". ALT="|$\approx|
|000029d0| 24 22 3e 20 32 32 2e 33 | 36 30 36 37 39 37 37 34 |$"> 22.3|60679774|
|000029e0| 39 39 37 39 0a 20 26 6e | 64 61 73 68 3b 26 67 74 |9979. &n|dash;>|
|000029f0| 3b 0a 26 6c 74 3b 49 26 | 67 74 3b 78 26 6c 74 3b |;.<I&|gt;x<|
|00002a00| 2f 49 26 67 74 3b 20 3d | 20 32 26 61 6d 70 3b 23 |/I> =| 2&#|
|00002a10| 38 39 30 31 3b 35 26 6c | 74 3b 53 55 50 26 67 74 |8901;5&l|t;SUP>|
|00002a20| 3b 33 2f 32 26 6c 74 3b | 2f 53 55 50 26 67 74 3b |;3/2<|/SUP>|
|00002a30| 20 26 6c 74 3b 49 4d 47 | 0a 20 53 54 59 4c 45 3d | <IMG|. STYLE=|
|00002a40| 22 68 65 69 67 68 74 3a | 20 31 37 39 2e 32 32 65 |"height:| 179.22e|
|00002a50| 78 3b 20 76 65 72 74 69 | 63 61 6c 2d 61 6c 69 67 |x; verti|cal-alig|
|00002a60| 6e 3a 20 2d 30 2e 31 31 | 65 78 3b 20 22 20 53 52 |n: -0.11|ex; " SR|
|00002a70| 43 3d 22 69 6d 67 34 2e | 70 6e 67 22 0a 20 41 4c |C="img4.|png". AL|
|00002a80| 54 3d 22 3c 49 4d 47 0a | 20 53 54 59 4c 45 3d 22 |T="<IMG.| STYLE="|
|00002a90| 68 65 69 67 68 74 3a 20 | 32 2e 31 30 65 78 3b 20 |height: |2.10ex; |
|00002aa0| 76 65 72 74 69 63 61 6c | 2d 61 6c 69 67 6e 3a 20 |vertical|-align: |
|00002ab0| 31 37 37 2e 32 37 65 78 | 3b 20 22 20 53 52 43 3d |177.27ex|; " SRC=|
|00002ac0| 22 69 6d 67 34 2e 70 6e | 67 22 0a 20 41 4c 54 3d |"img4.pn|g". ALT=|
|00002ad0| 22 24 5c 61 70 70 72 6f | 78 24 22 3e 22 26 67 74 |"$\appro|x$">">|
|00002ae0| 3b 20 32 32 2e 33 36 30 | 36 37 39 37 37 34 39 39 |; 22.360|67977499|
|00002af0| 37 39 2c 20 67 69 76 69 | 6e 67 20 61 20 6d 69 6e |79, givi|ng a min|
|00002b00| 69 6d 75 6d 20 6f 66 20 | 26 6c 74 3b 21 26 6e 64 |imum of |<!&nd|
|00002b10| 61 73 68 3b 20 4d 41 54 | 48 0a 20 3c 21 2d 2d 20 |ash; MAT|H. <!-- |
|00002b20| 4d 41 54 48 0a 20 24 41 | 20 5c 61 70 70 72 6f 78 |MATH. $A| \approx|
|00002b30| 20 38 39 2e 34 34 32 37 | 31 39 30 39 39 39 39 31 | 89.4427|19099991|
|00002b40| 36 24 0a 20 2d 2d 3e 0a | 3c 49 3e 41 3c 2f 49 3e |6$. -->.|<I>A</I>|
|00002b50| 20 3c 49 4d 47 0a 20 53 | 54 59 4c 45 3d 22 68 65 | <IMG. S|TYLE="he|
|00002b60| 69 67 68 74 3a 20 32 2e | 31 30 65 78 3b 20 76 65 |ight: 2.|10ex; ve|
|00002b70| 72 74 69 63 61 6c 2d 61 | 6c 69 67 6e 3a 20 31 37 |rtical-a|lign: 17|
|00002b80| 37 2e 32 37 65 78 3b 20 | 22 20 53 52 43 3d 22 69 |7.27ex; |" SRC="i|
|00002b90| 6d 67 34 2e 70 6e 67 22 | 0a 20 41 4c 54 3d 22 24 |mg4.png"|. ALT="$|
|00002ba0| 5c 61 70 70 72 6f 78 24 | 22 3e 20 38 39 2e 34 34 |\approx$|"> 89.44|
|00002bb0| 32 37 31 39 30 39 39 39 | 39 31 36 0a 20 26 6e 64 |27190999|916. &nd|
|00002bc0| 61 73 68 3b 26 67 74 3b | 0a 26 6c 74 3b 49 26 67 |ash;>|.<I&g|
|00002bd0| 74 3b 41 26 6c 74 3b 2f | 49 26 67 74 3b 20 26 6c |t;A</|I> &l|
|00002be0| 74 3b 49 4d 47 0a 20 53 | 54 59 4c 45 3d 22 68 65 |t;IMG. S|TYLE="he|
|00002bf0| 69 67 68 74 3a 20 31 37 | 39 2e 32 32 65 78 3b 20 |ight: 17|9.22ex; |
|00002c00| 76 65 72 74 69 63 61 6c | 2d 61 6c 69 67 6e 3a 20 |vertical|-align: |
|00002c10| 2d 30 2e 31 31 65 78 3b | 20 22 20 53 52 43 3d 22 |-0.11ex;| " SRC="|
|00002c20| 69 6d 67 34 2e 70 6e 67 | 22 0a 20 41 4c 54 3d 22 |img4.png|". ALT="|
|00002c30| 3c 49 4d 47 0a 20 53 54 | 59 4c 45 3d 22 68 65 69 |<IMG. ST|YLE="hei|
|00002c40| 67 68 74 3a 20 32 2e 31 | 30 65 78 3b 20 76 65 72 |ght: 2.1|0ex; ver|
|00002c50| 74 69 63 61 6c 2d 61 6c | 69 67 6e 3a 20 31 37 37 |tical-al|ign: 177|
|00002c60| 2e 32 37 65 78 3b 20 22 | 20 53 52 43 3d 22 69 6d |.27ex; "| SRC="im|
|00002c70| 67 34 2e 70 6e 67 22 0a | 20 41 4c 54 3d 22 24 5c |g4.png".| ALT="$\|
|00002c80| 61 70 70 72 6f 78 24 22 | 3e 22 26 67 74 3b 20 38 |approx$"|>"> 8|
|00002c90| 39 2e 34 34 32 37 31 39 | 30 39 39 39 39 31 36 2e |9.442719|0999916.|
|00002ca0| 0a 0a 3c 50 3e 0a 26 6c | 74 3b 50 26 67 74 3b 0a |..<P>.&l|t;P>.|
|00002cb0| 0a 3c 50 3e 0a 26 6c 74 | 3b 44 49 56 20 63 6c 61 |.<P>.<|;DIV cla|
|00002cc0| 73 73 3d 22 43 45 4e 54 | 45 52 22 26 67 74 3b 26 |ss="CENT|ER">&|
|00002cd0| 6c 74 3b 41 20 49 44 3d | 22 33 33 22 26 67 74 3b |lt;A ID=|"33">|
|00002ce0| 26 6c 74 3b 2f 41 26 67 | 74 3b 0a 26 6c 74 3b 54 |</A&g|t;.<T|
|00002cf0| 41 42 4c 45 26 67 74 3b | 0a 26 6c 74 3b 43 41 50 |ABLE>|.<CAP|
|00002d00| 54 49 4f 4e 20 63 6c 61 | 73 73 3d 22 42 4f 54 54 |TION cla|ss="BOTT|
|00002d10| 4f 4d 22 26 67 74 3b 26 | 6c 74 3b 53 54 52 4f 4e |OM">&|lt;STRON|
|00002d20| 47 26 67 74 3b 46 69 67 | 75 72 65 3a 26 6c 74 3b |G>Fig|ure:<|
|00002d30| 2f 53 54 52 4f 4e 47 26 | 67 74 3b 0a 50 6c 6f 74 |/STRONG&|gt;.Plot|
|00002d40| 20 6f 66 20 32 78 20 2b | 20 31 30 30 30 2f 78 20 | of 2x +| 1000/x |
|00002d50| 69 6e 20 74 68 65 20 63 | 72 69 74 69 63 61 6c 20 |in the c|ritical |
|00002d60| 20 72 65 67 69 6f 6e 2e | 26 6c 74 3b 2f 43 41 50 | region.|</CAP|
|00002d70| 54 49 4f 4e 26 67 74 3b | 0a 26 6c 74 3b 54 52 26 |TION>|.<TR&|
|00002d80| 67 74 3b 26 6c 74 3b 54 | 44 26 67 74 3b 26 6c 74 |gt;<T|D><|
|00002d90| 3b 49 4d 47 0a 20 53 54 | 59 4c 45 3d 22 68 65 69 |;IMG. ST|YLE="hei|
|00002da0| 67 68 74 3a 20 32 38 36 | 2e 37 36 65 78 3b 20 22 |ght: 286|.76ex; "|
|00002db0| 20 53 52 43 3d 22 69 6d | 67 35 2e 70 6e 67 22 0a | SRC="im|g5.png".|
|00002dc0| 20 41 4c 54 3d 22 0a 3c | 44 49 56 20 63 6c 61 73 | ALT=".<|DIV clas|
|00002dd0| 73 3d 22 43 45 4e 54 45 | 52 22 3e 0a 3c 49 4d 47 |s="CENTE|R">.<IMG|
|00002de0| 0a 20 53 54 59 4c 45 3d | 22 68 65 69 67 68 74 3a |. STYLE=|"height:|
|00002df0| 20 33 31 34 2e 30 30 65 | 78 3b 20 22 20 53 52 43 | 314.00e|x; " SRC|
|00002e00| 3d 22 69 6d 67 35 2e 70 | 6e 67 22 0a 20 41 4c 54 |="img5.p|ng". ALT|
|00002e10| 3d 22 5c 62 65 67 69 6e | 7b 66 69 67 75 72 65 7d |="\begin|{figure}|
|00002e20| 5c 65 70 73 66 79 73 69 | 7a 65 20 39 30 70 74 0a |\epsfysi|ze 90pt.|
|00002e30| 5c 63 65 6e 74 65 72 6c | 69 6e 65 7b 5c 65 70 73 |\centerl|ine{\eps|
|00002e40| 66 66 69 6c 65 7b 61 6e | 73 33 70 33 62 2e 65 70 |ffile{an|s3p3b.ep|
|00002e50| 73 7d 7d 0a 5c 65 6e 64 | 7b 66 69 67 75 72 65 7d |s}}.\end|{figure}|
|00002e60| 22 3e 0a 3c 2f 44 49 56 | 3e 22 26 67 74 3b 26 6c |">.</DIV|>">&l|
|00002e70| 74 3b 2f 54 44 26 67 74 | 3b 26 6c 74 3b 2f 54 52 |t;/TD>|;</TR|
|00002e80| 26 67 74 3b 0a 26 6c 74 | 3b 2f 54 41 42 4c 45 26 |>.<|;/TABLE&|
|00002e90| 67 74 3b 0a 26 6c 74 3b | 2f 44 49 56 26 67 74 3b |gt;.<|/DIV>|
|00002ea0| 0a 0a 3c 50 3e 0a 26 6c | 74 3b 50 26 67 74 3b 0a |..<P>.&l|t;P>.|
|00002eb0| 26 6c 74 3b 2f 4c 49 26 | 67 74 3b 0a 26 6c 74 3b |</LI&|gt;.<|
|00002ec0| 4c 49 26 67 74 3b 41 20 | 64 69 61 67 72 61 6d 20 |LI>A |diagram |
|00002ed0| 69 73 20 73 68 6f 77 6e | 20 69 6e 20 46 69 67 75 |is shown| in Figu|
|00002ee0| 72 65 20 35 2e 20 20 54 | 68 65 20 75 6e 6b 6e 6f |re 5. T|he unkno|
|00002ef0| 77 6e 20 69 73 20 74 68 | 65 20 72 61 64 69 75 73 |wn is th|e radius|
|00002f00| 20 26 6c 74 3b 49 26 67 | 74 3b 72 26 6c 74 3b 2f | <I&g|t;r</|
|00002f10| 49 26 67 74 3b 2e 20 20 | 41 73 20 73 68 6f 77 6e |I>. |As shown|
|00002f20| 2c 20 74 68 65 20 71 75 | 61 6e 74 69 74 69 65 73 |, the qu|antities|
|00002f30| 20 26 6c 74 3b 49 26 67 | 74 3b 68 26 6c 74 3b 2f | <I&g|t;h</|
|00002f40| 49 26 67 74 3b 20 3d 20 | 31 30 20 61 6e 64 20 26 |I> = |10 and &|
|00002f50| 6c 74 3b 49 26 67 74 3b | 73 26 6c 74 3b 2f 49 26 |lt;I>|s</I&|
|00002f60| 67 74 3b 20 3d 20 31 30 | 30 30 20 61 72 65 20 67 |gt; = 10|00 are g|
|00002f70| 69 76 65 6e 2e 0a 0a 3c | 50 3e 0a 26 6c 74 3b 4f |iven...<|P>.<O|
|00002f80| 4c 26 67 74 3b 0a 26 6c | 74 3b 4c 49 26 67 74 3b |L>.&l|t;LI>|
|00002f90| 53 65 65 20 46 69 67 75 | 72 65 20 35 2e 0a 0a 3c |See Figu|re 5...<|
|00002fa0| 50 3e 0a 26 6c 74 3b 50 | 26 67 74 3b 0a 0a 3c 50 |P>.<P|>..<P|
|00002fb0| 3e 0a 26 6c 74 3b 44 49 | 56 20 63 6c 61 73 73 3d |>.<DI|V class=|
|00002fc0| 22 43 45 4e 54 45 52 22 | 26 67 74 3b 26 6c 74 3b |"CENTER"|><|
|00002fd0| 41 20 49 44 3d 22 33 38 | 22 26 67 74 3b 26 6c 74 |A ID="38|"><|
|00002fe0| 3b 2f 41 26 67 74 3b 0a | 26 6c 74 3b 54 41 42 4c |;/A>.|<TABL|
|00002ff0| 45 26 67 74 3b 0a 26 6c | 74 3b 43 41 50 54 49 4f |E>.&l|t;CAPTIO|
|00003000| 4e 20 63 6c 61 73 73 3d | 22 42 4f 54 54 4f 4d 22 |N class=|"BOTTOM"|
|00003010| 26 67 74 3b 26 6c 74 3b | 53 54 52 4f 4e 47 26 67 |><|STRONG&g|
|00003020| 74 3b 46 69 67 75 72 65 | 3a 26 6c 74 3b 2f 53 54 |t;Figure|:</ST|
|00003030| 52 4f 4e 47 26 67 74 3b | 0a 44 69 61 67 72 61 6d |RONG>|.Diagram|
|00003040| 20 66 6f 72 20 50 72 6f | 62 6c 65 6d 20 34 2e 26 | for Pro|blem 4.&|
|00003050| 6c 74 3b 2f 43 41 50 54 | 49 4f 4e 26 67 74 3b 0a |lt;/CAPT|ION>.|
|00003060| 26 6c 74 3b 54 52 26 67 | 74 3b 26 6c 74 3b 54 44 |<TR&g|t;<TD|
|00003070| 26 67 74 3b 26 6c 74 3b | 49 4d 47 0a 20 53 54 59 |><|IMG. STY|
|00003080| 4c 45 3d 22 68 65 69 67 | 68 74 3a 20 32 38 36 2e |LE="heig|ht: 286.|
|00003090| 37 36 65 78 3b 20 22 20 | 53 52 43 3d 22 69 6d 67 |76ex; " |SRC="img|
|000030a0| 36 2e 70 6e 67 22 0a 20 | 41 4c 54 3d 22 0a 3c 44 |6.png". |ALT=".<D|
|000030b0| 49 56 20 63 6c 61 73 73 | 3d 22 43 45 4e 54 45 52 |IV class|="CENTER|
|000030c0| 22 3e 0a 3c 49 4d 47 0a | 20 53 54 59 4c 45 3d 22 |">.<IMG.| STYLE="|
|000030d0| 68 65 69 67 68 74 3a 20 | 33 31 34 2e 30 30 65 78 |height: |314.00ex|
|000030e0| 3b 20 22 20 53 52 43 3d | 22 69 6d 67 36 2e 70 6e |; " SRC=|"img6.pn|
|000030f0| 67 22 0a 20 41 4c 54 3d | 22 5c 62 65 67 69 6e 7b |g". ALT=|"\begin{|
|00003100| 66 69 67 75 72 65 7d 5c | 65 70 73 66 79 73 69 7a |figure}\|epsfysiz|
|00003110| 65 20 39 30 70 74 0a 5c | 63 65 6e 74 65 72 6c 69 |e 90pt.\|centerli|
|00003120| 6e 65 7b 5c 65 70 73 66 | 66 69 6c 65 7b 61 6e 73 |ne{\epsf|file{ans|
|00003130| 33 70 34 2e 65 70 73 7d | 7d 0a 5c 65 6e 64 7b 66 |3p4.eps}|}.\end{f|
|00003140| 69 67 75 72 65 7d 22 3e | 0a 3c 2f 44 49 56 3e 22 |igure}">|.</DIV>"|
|00003150| 26 67 74 3b 26 6c 74 3b | 2f 54 44 26 67 74 3b 26 |><|/TD>&|
|00003160| 6c 74 3b 2f 54 52 26 67 | 74 3b 0a 26 6c 74 3b 2f |lt;/TR&g|t;.</|
|00003170| 54 41 42 4c 45 26 67 74 | 3b 0a 26 6c 74 3b 2f 44 |TABLE>|;.</D|
|00003180| 49 56 26 67 74 3b 0a 0a | 3c 50 3e 0a 26 6c 74 3b |IV>..|<P>.<|
|00003190| 2f 4c 49 26 67 74 3b 0a | 26 6c 74 3b 4c 49 26 67 |/LI>.|<LI&g|
|000031a0| 74 3b 55 73 65 20 74 68 | 65 20 72 69 67 68 74 20 |t;Use th|e right |
|000031b0| 74 72 69 61 6e 67 6c 65 | 20 74 6f 20 64 65 72 69 |triangle| to deri|
|000031c0| 76 65 3a 20 26 6c 74 3b | 21 26 6e 64 61 73 68 3b |ve: <|!–|
|000031d0| 20 4d 41 54 48 0a 20 3c | 21 2d 2d 20 4d 41 54 48 | MATH. <|!-- MATH|
|000031e0| 0a 20 24 5c 63 6f 73 20 | 28 73 2f 72 29 20 3d 20 |. $\cos |(s/r) = |
|000031f0| 72 20 2f 20 28 72 20 2b | 20 68 29 24 0a 20 2d 2d |r / (r +| h)$. --|
|00003200| 3e 0a 63 6f 73 28 3c 49 | 3e 73 3c 2f 49 3e 2f 3c |>.cos(<I|>s</I>/<|
|00003210| 49 3e 72 3c 2f 49 3e 29 | 20 3d 20 3c 49 3e 72 3c |I>r</I>)| = <I>r<|
|00003220| 2f 49 3e 2f 28 3c 49 3e | 72 3c 2f 49 3e 20 2b 20 |/I>/(<I>|r</I> + |
|00003230| 3c 49 3e 68 3c 2f 49 3e | 29 0a 20 26 6e 64 61 73 |<I>h</I>|). &ndas|
|00003240| 68 3b 26 67 74 3b 0a 63 | 6f 73 28 26 6c 74 3b 49 |h;>.c|os(<I|
|00003250| 26 67 74 3b 73 26 6c 74 | 3b 2f 49 26 67 74 3b 2f |>s<|;/I>/|
|00003260| 26 6c 74 3b 49 26 67 74 | 3b 72 26 6c 74 3b 2f 49 |<I>|;r</I|
|00003270| 26 67 74 3b 29 20 3d 20 | 26 6c 74 3b 49 26 67 74 |>) = |<I>|
|00003280| 3b 72 26 6c 74 3b 2f 49 | 26 67 74 3b 2f 28 26 6c |;r</I|>/(&l|
|00003290| 74 3b 49 26 67 74 3b 72 | 26 6c 74 3b 2f 49 26 67 |t;I>r|</I&g|
|000032a0| 74 3b 20 2b 20 26 6c 74 | 3b 49 26 67 74 3b 68 26 |t; + <|;I>h&|
|000032b0| 6c 74 3b 2f 49 26 67 74 | 3b 29 2c 20 0a 68 65 72 |lt;/I>|;), .her|
|000032c0| 65 20 26 6c 74 3b 21 26 | 6e 64 61 73 68 3b 20 4d |e <!&|ndash; M|
|000032d0| 41 54 48 0a 20 3c 21 2d | 2d 20 4d 41 54 48 0a 20 |ATH. <!-|- MATH. |
|000032e0| 24 68 20 3d 20 32 30 2c | 20 73 20 3d 20 31 30 30 |$h = 20,| s = 100|
|000032f0| 30 24 0a 20 2d 2d 3e 0a | 3c 49 3e 68 3c 2f 49 3e |0$. -->.|<I>h</I>|
|00003300| 20 3d 20 32 30 2c 20 3c | 49 3e 73 3c 2f 49 3e 20 | = 20, <|I>s</I> |
|00003310| 3d 20 31 30 30 30 0a 20 | 26 6e 64 61 73 68 3b 26 |= 1000. |–&|
|00003320| 67 74 3b 0a 26 6c 74 3b | 49 26 67 74 3b 68 26 6c |gt;.<|I>h&l|
|00003330| 74 3b 2f 49 26 67 74 3b | 20 3d 20 32 30 2c 20 26 |t;/I>| = 20, &|
|00003340| 6c 74 3b 49 26 67 74 3b | 73 26 6c 74 3b 2f 49 26 |lt;I>|s</I&|
|00003350| 67 74 3b 20 3d 20 31 30 | 30 30 2e 0a 0a 3c 50 3e |gt; = 10|00...<P>|
|00003360| 0a 26 6c 74 3b 2f 4c 49 | 26 67 74 3b 0a 26 6c 74 |.</LI|>.<|
|00003370| 3b 2f 4f 4c 26 67 74 3b | 0a 26 6c 74 3b 2f 4c 49 |;/OL>|.</LI|
|00003380| 26 67 74 3b 0a 26 6c 74 | 3b 2f 4f 4c 26 67 74 3b |>.<|;/OL>|
|00003390| 0a 26 6c 74 3b 48 52 26 | 67 74 3b 0a 0a 3c 50 3e |.<HR&|gt;..<P>|
|000033a0| 0a 26 6c 74 3b 2f 42 4f | 44 59 26 67 74 3b 0a 26 |.</BO|DY>.&|
|000033b0| 6c 74 3b 2f 48 54 4d 4c | 26 67 74 3b 0a 0a 3c 48 |lt;/HTML|>..<H|
|000033c0| 52 3e 0a 0a 3c 2f 42 4f | 44 59 3e 0a 3c 2f 48 54 |R>..</BO|DY>.</HT|
|000033d0| 4d 4c 3e 0a | |ML>. | |
+--------+-------------------------+-------------------------+--------+--------+
