/ NeXT Education Software Sampler 1992 Fall / NeXT Education Software Sampler 1992 Fall.iso / Mathematics / Notebooks / CSOMinesCalculus / Chapter2 / ans1.tex

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|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 31 7d 0a 5c | 2: Answ|ers 1}.\|
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|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 | 31 20 5c 68 66 69 6c 6c |Answers |1 \hfill|
|00000080| 20 4a 61 63 6b 20 4b 2e | 20 43 6f 68 65 6e 20 5c | Jack K.| Cohen \|
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|00000160| 6d 20 53 65 65 20 46 69 | 67 75 72 65 20 31 2e 0a |m See Fi|gure 1..|
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|000001a0| 6f 70 65 20 69 73 20 70 | 6f 73 69 74 69 76 65 20 |ope is p|ositive |
|000001b0| 69 6e 20 74 68 65 20 69 | 6e 74 65 72 76 61 6c 73 |in the i|ntervals|
|000001c0| 20 24 28 61 2c 41 29 2c | 20 28 62 2c 20 42 29 24 | $(a,A),| (b, B)$|
|000001d0| 20 61 6e 64 20 24 28 63 | 2c 20 43 29 24 2e 20 20 | and $(c|, C)$.  |
|000001e0| 54 68 65 73 65 20 61 72 | 65 20 69 6e 64 65 65 64 |These ar|e indeed|
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|00000230| 74 65 6d 20 5a 65 72 6f | 2e 0a 09 5c 69 74 65 6d |tem Zero|...\item|
|00000240| 20 49 66 20 74 68 65 20 | 73 6c 6f 70 65 20 69 73 | If the |slope is|
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|00000290| 69 6e 20 61 6e 20 69 6e | 74 65 72 76 61 6c 20 74 |in an in|terval t|
|000002a0| 6f 20 74 68 65 20 72 69 | 67 68 74 20 6f 66 20 74 |o the ri|ght of t|
|000002b0| 68 69 73 20 70 6f 69 6e | 74 2c 20 74 68 65 6e 20 |his poin|t, then |
|000002c0| 74 68 65 20 70 6f 69 6e | 74 20 67 69 76 65 73 20 |the poin|t gives |
|000002d0| 61 20 6c 6f 63 61 6c 20 | 6d 61 78 69 6d 75 6d 2e |a local |maximum.|
|000002e0| 0a 09 5c 69 74 65 6d 20 | 47 6c 6f 62 61 6c 20 6d |..\item |Global m|
|000002f0| 61 78 69 6d 75 6d 20 69 | 73 20 61 74 20 24 42 24 |aximum i|s at $B$|
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|00000310| 20 69 73 20 61 74 20 24 | 61 24 2e 0a 09 5c 69 74 | is at $|a$...\it|
|00000320| 65 6d 20 54 68 65 20 76 | 65 6c 6f 63 69 74 79 20 |em The v|elocity |
|00000330| 69 73 20 30 20 61 74 20 | 24 41 2c 20 62 2c 20 42 |is 0 at |$A, b, B|
|00000340| 2c 20 63 24 2e 20 54 68 | 65 20 76 65 6c 6f 63 69 |, c$. Th|e veloci|
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|00000390| 65 6c 6f 63 69 74 79 20 | 68 61 73 20 61 20 6c 6f |elocity |has a lo|
|000003a0| 63 61 6c 20 6d 69 6e 69 | 6d 75 6d 20 77 68 65 72 |cal mini|mum wher|
|000003b0| 65 20 74 68 65 20 73 6c | 6f 70 65 20 69 73 20 73 |e the sl|ope is s|
|000003c0| 6d 61 6c 6c 65 73 74 2e | 20 20 0a 49 74 20 6c 6f |mallest.|  .It lo|
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|000003e0| 67 65 73 74 20 73 6c 6f | 70 65 20 28 72 6f 75 67 |gest slo|pe (roug|
|000003f0| 68 6c 79 20 33 30 29 20 | 6f 63 63 75 72 73 20 61 |hly 30) |occurs a|
|00000400| 74 20 24 61 24 20 6f 72 | 20 24 43 24 2c 20 74 68 |t $a$ or| $C$, th|
|00000410| 65 20 73 6d 61 6c 6c 65 | 73 74 20 73 6c 6f 70 65 |e smalle|st slope|
|00000420| 20 28 61 62 6f 75 74 20 | 2d 31 35 29 0a 6f 63 63 | (about |-15).occ|
|00000430| 75 72 73 20 62 65 74 77 | 65 65 6e 20 24 42 24 20 |urs betw|een $B$ |
|00000440| 61 6e 64 20 24 63 24 2e | 0a 09 5c 65 6e 64 7b 65 |and $c$.|..\end{e|
|00000450| 6e 75 6d 65 72 61 74 65 | 7d 0a 0a 5c 69 74 65 6d |numerate|}..\item|
|00000460| 20 28 32 2e 31 2e 33 36 | 29 20 20 53 65 65 20 46 | (2.1.36|)  See F|
|00000470| 69 67 75 72 65 20 32 2e | 20 20 57 65 20 6b 6e 6f |igure 2.|  We kno|
|00000480| 77 20 74 68 61 74 20 74 | 68 65 20 73 6c 6f 70 65 |w that t|he slope|
|00000490| 20 66 75 6e 63 74 69 6f | 6e 20 28 64 65 72 69 76 | functio|n (deriv|
|000004a0| 61 74 69 76 65 29 20 6f | 66 20 24 79 20 3d 20 78 |ative) o|f $y = x|
|000004b0| 5e 32 24 20 69 73 20 24 | 32 78 24 2c 20 73 6f 20 |^2$ is $|2x$, so |
|000004c0| 74 68 65 20 73 6c 6f 70 | 65 20 61 74 20 24 78 20 |the slop|e at $x |
|000004d0| 3d 20 78 5f 30 24 20 69 | 73 20 24 6d 20 3d 20 32 |= x_0$ i|s $m = 2|
|000004e0| 20 78 5f 30 24 2e 20 20 | 53 69 6e 63 65 20 24 28 | x_0$.  |Since $(|
|000004f0| 78 5f 30 2c 20 78 5f 30 | 5e 32 29 24 20 69 73 20 |x_0, x_0|^2)$ is |
|00000500| 61 20 70 6f 69 6e 74 20 | 6f 6e 20 74 68 65 20 74 |a point |on the t|
|00000510| 61 6e 67 65 6e 74 2c 20 | 74 68 65 20 70 6f 69 6e |angent, |the poin|
|00000520| 74 2d 73 6c 6f 70 65 20 | 66 6f 72 6d 75 6c 61 20 |t-slope |formula |
|00000530| 67 69 76 65 73 20 74 68 | 65 20 74 61 6e 67 65 6e |gives th|e tangen|
|00000540| 74 20 6c 69 6e 65 20 61 | 73 20 24 79 20 20 2d 20 |t line a|s $y  - |
|00000550| 78 5f 30 5e 32 20 3d 20 | 32 20 78 5f 30 20 28 20 |x_0^2 = |2 x_0 ( |
|00000560| 78 20 2d 20 78 5f 30 29 | 24 2e 20 20 54 68 69 73 |x - x_0)|$.  This|
|00000570| 20 6c 69 6e 65 20 69 6e | 74 65 72 73 65 63 74 73 | line in|tersects|
|00000580| 20 74 68 65 20 24 78 24 | 2d 61 78 69 73 20 77 68 | the $x$|-axis wh|
|00000590| 65 6e 20 24 78 20 3d 20 | 78 5f 31 2c 20 79 20 3d |en $x = |x_1, y =|
|000005a0| 20 30 24 2c 20 73 6f 20 | 24 30 20 20 2d 20 78 5f | 0$, so |$0  - x_|
|000005b0| 30 5e 32 20 3d 20 32 20 | 78 5f 30 20 28 20 78 5f |0^2 = 2 |x_0 ( x_|
|000005c0| 31 20 2d 20 78 5f 30 29 | 24 2e 20 20 53 6f 6c 76 |1 - x_0)|$.  Solv|
|000005d0| 69 6e 67 20 66 6f 72 20 | 24 78 5f 31 24 20 67 69 |ing for |$x_1$ gi|
|000005e0| 76 65 73 20 24 78 5f 31 | 20 3d 20 78 5f 30 20 2f |ves $x_1| = x_0 /|
|000005f0| 20 32 24 20 61 73 20 77 | 61 73 20 74 6f 20 62 65 | 2$ as w|as to be|
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|00000660| 6d 20 66 6f 72 20 70 72 | 6f 62 6c 65 6d 20 32 2e |m for pr|oblem 2.|
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|00000680| 0a 5c 69 74 65 6d 20 28 | 32 2e 31 2e 34 32 29 20 |.\item (|2.1.42) |
|00000690| 20 53 65 65 20 46 69 67 | 75 72 65 20 33 2e 20 20 | See Fig|ure 3.  |
|000006a0| 53 6f 6d 65 20 73 74 65 | 70 73 20 61 6c 6f 6e 67 |Some ste|ps along|
|000006b0| 20 74 68 65 20 77 61 79 | 3a 20 24 6d 20 3d 20 34 | the way|: $m = 4|
|000006c0| 20 2d 20 32 61 24 2c 20 | 74 61 6e 67 65 6e 74 20 | - 2a$, |tangent |
|000006d0| 6c 69 6e 65 20 69 73 20 | 24 79 20 2d 20 28 34 61 |line is |$y - (4a|
|000006e0| 20 2d 20 61 5e 32 29 20 | 3d 20 28 34 20 2d 20 32 | - a^2) |= (4 - 2|
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|00000710| 20 28 32 2c 20 35 29 20 | 66 6f 72 20 24 28 78 2c | (2, 5) |for $(x,|
|00000720| 20 79 29 24 20 74 6f 20 | 67 65 74 20 65 71 75 61 | y)$ to |get equa|
|00000730| 74 69 6f 6e 20 66 6f 72 | 20 24 61 24 20 61 6e 64 |tion for| $a$ and|
|00000740| 20 73 69 6d 70 6c 69 66 | 79 3a 20 24 28 61 20 2d | simplif|y: $(a -|
|00000750| 20 31 29 28 61 20 2d 20 | 33 29 20 3d 20 30 24 2e | 1)(a - |3) = 0$.|
|00000760| 20 20 41 6e 73 77 65 72 | 73 3a 20 24 79 20 3d 20 |  Answer|s: $y = |
|00000770| 32 78 20 2b 20 31 2c 20 | 79 20 3d 20 2d 32 78 20 |2x + 1, |y = -2x |
|00000780| 2b 20 39 24 2e 0a 0a 5c | 62 65 67 69 6e 7b 66 69 |+ 9$...\|begin{fi|
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|000007a0| 79 73 69 7a 65 20 31 31 | 30 70 74 0a 5c 63 65 6e |ysize 11|0pt.\cen|
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|000007d0| 63 61 70 74 69 6f 6e 7b | 44 69 61 67 72 61 6d 20 |caption{|Diagram |
|000007e0| 66 6f 72 20 70 72 6f 62 | 6c 65 6d 20 33 2e 7d 20 |for prob|lem 3.} |
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|00000800| 69 74 65 6d 20 28 32 2e | 31 2e 34 33 29 20 20 53 |item (2.|1.43)  S|
|00000810| 65 65 20 46 69 67 75 72 | 65 20 34 2e 20 20 54 68 |ee Figur|e 4.  Th|
|00000820| 65 20 73 6c 6f 70 65 20 | 6f 66 20 74 68 65 20 74 |e slope |of the t|
|00000830| 61 6e 67 65 6e 74 20 61 | 74 20 24 78 20 3d 20 61 |angent a|t $x = a|
|00000840| 24 20 77 6f 75 6c 64 20 | 62 65 20 24 32 61 24 2c |$ would |be $2a$,|
|00000850| 20 73 6f 20 74 68 65 20 | 73 6c 6f 70 65 20 6f 66 | so the |slope of|
|00000860| 20 74 68 65 20 6e 6f 72 | 6d 61 6c 20 69 73 20 24 | the nor|mal is $|
|00000870| 2d 31 2f 32 61 24 2e 20 | 20 55 73 65 20 70 6f 69 |-1/2a$. | Use poi|
|00000880| 6e 74 2d 73 6c 6f 70 65 | 20 74 6f 20 67 65 74 20 |nt-slope| to get |
|00000890| 74 68 65 20 65 71 75 61 | 74 69 6f 6e 20 6f 66 20 |the equa|tion of |
|000008a0| 74 68 65 20 6e 6f 72 6d | 61 6c 3a 20 24 79 20 2d |the norm|al: $y -|
|000008b0| 20 61 5e 32 20 3d 28 20 | 2d 31 2f 32 61 29 20 28 | a^2 =( |-1/2a) (|
|000008c0| 20 78 20 2d 20 61 29 24 | 2e 20 20 53 75 62 73 74 | x - a)$|.  Subst|
|000008d0| 69 74 75 74 65 20 74 68 | 65 20 67 69 76 65 6e 20 |itute th|e given |
|000008e0| 70 6f 69 6e 74 20 28 33 | 2c 20 30 29 20 6f 6e 20 |point (3|, 0) on |
|000008f0| 74 68 65 20 6e 6f 72 6d | 61 6c 20 74 6f 20 67 65 |the norm|al to ge|
|00000900| 74 20 61 6e 20 65 71 75 | 61 74 69 6f 6e 20 66 6f |t an equ|ation fo|
|00000910| 72 20 24 61 24 3a 20 24 | 32 61 5e 33 20 2b 20 61 |r $a$: $|2a^3 + a|
|00000920| 20 2d 20 33 20 3d 20 30 | 24 2e 20 20 54 68 65 20 | - 3 = 0|$.  The |
|00000930| 73 6f 6c 75 74 69 6f 6e | 20 24 61 20 3d 20 31 24 |solution| $a = 1$|
|00000940| 20 69 73 20 60 60 6f 62 | 76 69 6f 75 73 27 27 2e | is ``ob|vious''.|
|00000950| 20 20 20 42 75 74 20 6d | 6f 76 65 20 74 68 65 20 |   But m|ove the |
|00000960| 70 6f 69 6e 74 20 28 33 | 2c 20 30 29 20 74 6f 20 |point (3|, 0) to |
|00000970| 61 6e 6f 74 68 65 72 20 | 70 6f 73 69 74 69 6f 6e |another |position|
|00000980| 20 61 6e 64 20 69 74 20 | 69 73 20 61 6c 6d 6f 73 | and it |is almos|
|00000990| 74 20 63 65 72 74 61 69 | 6e 20 74 68 61 74 20 74 |t certai|n that t|
|000009a0| 68 65 20 63 6f 72 72 65 | 73 70 6f 6e 64 69 6e 67 |he corre|sponding|
|000009b0| 20 63 75 62 69 63 20 77 | 69 6c 6c 20 6e 6f 74 20 | cubic w|ill not |
|000009c0| 68 61 76 65 20 61 6e 79 | 20 60 60 6e 69 63 65 27 |have any| ``nice'|
|000009d0| 27 20 72 6f 6f 74 73 2d | 2d 2d 79 6f 75 20 63 6f |' roots-|--you co|
|000009e0| 75 6c 64 20 73 74 69 6c | 6c 20 75 73 65 20 5c 54 |uld stil|l use \T|
|000009f0| 74 7b 50 6c 6f 74 7d 20 | 61 6e 64 2f 6f 72 20 5c |t{Plot} |and/or \|
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|00000a60| 69 6f 6e 7b 44 69 61 67 | 72 61 6d 20 66 6f 72 20 |ion{Diag|ram for |
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|00000a90| 6d 20 46 6f 6c 6c 6f 77 | 69 6e 67 20 75 70 20 6f |m Follow|ing up o|
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|00000ad0| 46 72 6f 6d 20 61 20 70 | 6f 69 6e 74 20 5c 45 6d |From a p|oint \Em|
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|00000b60| 68 65 20 63 6f 72 72 65 | 73 70 6f 6e 64 69 6e 67 |he corre|sponding|
|00000b70| 20 71 75 61 64 72 61 74 | 69 63 20 65 71 75 61 74 | quadrat|ic equat|
|00000b80| 69 6f 6e 20 77 6f 75 6c | 64 20 68 61 76 65 20 63 |ion woul|d have c|
|00000b90| 6f 6d 70 6c 65 78 20 69 | 6e 73 74 65 61 64 20 6f |omplex i|nstead o|
|00000ba0| 66 20 72 65 61 6c 20 72 | 6f 6f 74 73 29 2e 0a 09 |f real r|oots)...|
|00000bb0| 5c 69 74 65 6d 20 20 41 | 67 61 69 6e 2c 20 74 68 |\item  A|gain, th|
|00000bc0| 65 20 61 6e 73 77 65 72 | 20 69 73 20 79 65 73 20 |e answer| is yes |
|00000bd0| 66 6f 72 20 65 78 74 65 | 72 69 6f 72 20 70 6f 69 |for exte|rior poi|
|00000be0| 6e 74 73 2c 20 62 75 74 | 20 66 6f 72 20 69 6e 74 |nts, but| for int|
|00000bf0| 65 72 69 6f 72 20 70 6f | 69 6e 74 73 20 74 68 65 |erior po|ints the|
|00000c00| 72 65 20 77 6f 75 6c 64 | 20 62 65 20 5c 45 6d 7b |re would| be \Em{|
|00000c10| 74 68 72 65 65 7d 20 6e | 6f 72 6d 61 6c 73 21 0a |three} n|ormals!.|
|00000c20| 09 5c 65 6e 64 7b 65 6e | 75 6d 65 72 61 74 65 7d |.\end{en|umerate}|
|00000c30| 0a 0a 5c 65 6e 64 7b 65 | 6e 75 6d 65 72 61 74 65 |..\end{e|numerate|
|00000c40| 7d 0a 5c 65 6e 64 7b 64 | 6f 63 75 6d 65 6e 74 7d |}.\end{d|ocument}|
|00000c50| 0a                      |                         |.       |        |
+--------+-------------------------+-------------------------+--------+--------+