/ NeXT Education Software Sampler 1992 Fall / NeXT Education Software Sampler 1992 Fall.iso / Mathematics / Notebooks / CSOMinesCalculus / Chapter3 / ans4.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 33 3a 20 41 6e 73 77 | 65 72 73 20 34 7d 0a 5c | 3: 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 33 3a 20 |.\Bf{Cha|pter 3: |
|00000070| 41 6e 73 77 65 72 73 20 | 34 20 5c 68 66 69 6c 6c |Answers |4 \hfill|
|00000080| 20 4a 61 63 6b 20 4b 2e | 20 43 6f 68 65 6e 20 5c | Jack K.| Cohen \|
|00000090| 68 66 69 6c 6c 20 43 6f | 6c 6f 72 61 64 6f 20 53 |hfill Co|lorado S|
|000000a0| 63 68 6f 6f 6c 20 6f 66 | 20 4d 69 6e 65 73 7d 0a |chool of| Mines}.|
|000000b0| 0a 5c 62 65 67 69 6e 7b | 65 6e 75 6d 65 72 61 74 |.\begin{|enumerat|
|000000c0| 65 7d 0a 5c 69 74 65 6d | 20 20 55 73 65 20 5c 4d |e}.\item|  Use \M|
|000000d0| 6d 61 5c 20 61 6e 64 20 | 69 66 20 79 6f 75 20 73 |ma\ and |if you s|
|000000e0| 74 69 6c 6c 20 64 6f 6e | 27 74 20 73 65 65 20 74 |till don|'t see t|
|000000f0| 68 65 20 6c 69 67 68 74 | 2c 20 61 73 6b 20 79 6f |he light|, ask yo|
|00000100| 75 72 20 66 72 69 65 6e | 64 73 20 6f 72 20 74 68 |ur frien|ds or th|
|00000110| 65 20 69 6e 73 74 72 75 | 63 74 6f 72 20 66 6f 72 |e instru|ctor for|
|00000120| 20 68 65 6c 70 2e 0a 0a | 5c 69 74 65 6d 20 20 55 | help...|\item  U|
|00000130| 73 65 20 74 68 65 20 66 | 61 63 74 20 74 68 61 74 |se the f|act that|
|00000140| 20 20 24 44 20 5c 73 69 | 6e 20 78 20 3d 20 5c 63 |  $D \si|n x = \c|
|00000150| 6f 73 20 78 24 20 74 6f | 20 66 69 6e 64 20 74 68 |os x$ to| find th|
|00000160| 65 20 64 65 72 69 76 61 | 74 69 76 65 73 20 6f 66 |e deriva|tives of|
|00000170| 20 74 68 65 20 66 6f 6c | 6c 6f 77 69 6e 67 20 66 | the fol|lowing f|
|00000180| 75 6e 63 74 69 6f 6e 73 | 3a 0a 09 5c 62 65 67 69 |unctions|:..\begi|
|00000190| 6e 7b 65 6e 75 6d 65 72 | 61 74 65 7d 0a 09 5c 69 |n{enumer|ate}..\i|
|000001a0| 74 65 6d 20 24 44 20 5c | 73 69 6e 20 28 78 5e 33 |tem $D \|sin (x^3|
|000001b0| 29 20 3d 20 5c 63 6f 73 | 20 28 20 5c 63 64 6f 74 |) = \cos| ( \cdot|
|000001c0| 73 20 29 20 5c 63 64 6f | 74 20 44 28 20 5c 63 64 |s ) \cdo|t D( \cd|
|000001d0| 6f 74 73 20 29 20 3d 20 | 5c 63 6f 73 20 28 78 5e |ots ) = |\cos (x^|
|000001e0| 33 29 20 5c 63 64 6f 74 | 20 44 78 5e 33 20 3d 20 |3) \cdot| Dx^3 = |
|000001f0| 33 20 78 5e 32 20 5c 63 | 6f 73 20 78 5e 33 24 2e |3 x^2 \c|os x^3$.|
|00000200| 0a 09 5c 69 74 65 6d 20 | 24 44 20 28 5c 73 69 6e |..\item |$D (\sin|
|00000210| 20 78 29 5e 33 20 3d 20 | 33 20 28 20 5c 63 64 6f | x)^3 = |3 ( \cdo|
|00000220| 74 73 20 29 5e 32 20 5c | 63 64 6f 74 20 44 28 20 |ts )^2 \|cdot D( |
|00000230| 5c 63 64 6f 74 73 20 29 | 20 3d 20 33 20 28 5c 73 |\cdots )| = 3 (\s|
|00000240| 69 6e 20 78 29 5e 32 20 | 5c 63 64 6f 74 20 44 20 |in x)^2 |\cdot D |
|00000250| 5c 73 69 6e 20 78 20 3d | 20 33 20 5c 73 69 6e 5e |\sin x =| 3 \sin^|
|00000260| 32 20 78 20 5c 63 6f 73 | 20 78 24 2e 0a 09 5c 69 |2 x \cos| x$...\i|
|00000270| 74 65 6d 20 24 44 5c 73 | 69 6e 28 20 5c 73 69 6e |tem $D\s|in( \sin|
|00000280| 20 78 29 20 3d 20 5c 63 | 6f 73 20 28 20 5c 63 64 | x) = \c|os ( \cd|
|00000290| 6f 74 73 20 29 20 5c 63 | 64 6f 74 20 44 28 20 5c |ots ) \c|dot D( \|
|000002a0| 63 64 6f 74 73 20 29 20 | 3d 20 5c 63 6f 73 20 28 |cdots ) |= \cos (|
|000002b0| 20 5c 73 69 6e 20 78 29 | 20 5c 63 64 6f 74 20 44 | \sin x)| \cdot D|
|000002c0| 20 5c 73 69 6e 20 78 20 | 3d 20 5c 63 6f 73 20 28 | \sin x |= \cos (|
|000002d0| 5c 73 69 6e 20 78 29 20 | 5c 63 6f 73 20 78 24 2e |\sin x) |\cos x$.|
|000002e0| 0a 09 5c 69 74 65 6d 20 | 24 44 5c 73 69 6e 28 20 |..\item |$D\sin( |
|000002f0| 5c 73 69 6e 20 28 20 5c | 73 69 6e 20 78 29 29 20 |\sin ( \|sin x)) |
|00000300| 3d 20 5c 63 6f 73 20 28 | 20 5c 63 64 6f 74 73 20 |= \cos (| \cdots |
|00000310| 29 20 5c 63 64 6f 74 20 | 44 28 20 5c 63 64 6f 74 |) \cdot |D( \cdot|
|00000320| 73 20 29 20 3d 20 5c 63 | 6f 73 20 28 5c 73 69 6e |s ) = \c|os (\sin|
|00000330| 20 28 5c 73 69 6e 20 78 | 29 29 20 5c 63 64 6f 74 | (\sin x|)) \cdot|
|00000340| 20 44 20 5c 73 69 6e 20 | 28 5c 73 69 6e 20 78 29 | D \sin |(\sin x)|
|00000350| 20 3d 20 5c 63 6f 73 20 | 28 5c 73 69 6e 20 28 5c | = \cos |(\sin (\|
|00000360| 73 69 6e 20 78 29 29 20 | 5c 63 6f 73 20 28 20 5c |sin x)) |\cos ( \|
|00000370| 63 64 6f 74 73 20 29 20 | 5c 63 64 6f 74 20 44 28 |cdots ) |\cdot D(|
|00000380| 20 5c 63 64 6f 74 73 20 | 29 20 3d 20 5c 63 6f 73 | \cdots |) = \cos|
|00000390| 20 28 5c 73 69 6e 20 28 | 5c 73 69 6e 20 78 29 29 | (\sin (|\sin x))|
|000003a0| 20 5c 63 6f 73 20 28 5c | 73 69 6e 20 78 29 20 5c | \cos (\|sin x) \|
|000003b0| 63 64 6f 74 20 44 20 5c | 73 69 6e 20 78 20 3d 20 |cdot D \|sin x = |
|000003c0| 5c 63 6f 73 20 28 5c 73 | 69 6e 20 28 5c 73 69 6e |\cos (\s|in (\sin|
|000003d0| 20 78 29 29 20 5c 63 6f | 73 20 28 5c 73 69 6e 20 | x)) \co|s (\sin |
|000003e0| 78 29 20 5c 63 6f 73 20 | 78 24 2e 20 20 28 57 68 |x) \cos |x$.  (Wh|
|000003f0| 65 77 21 29 0a 09 5c 65 | 6e 64 7b 65 6e 75 6d 65 |ew!)..\e|nd{enume|
|00000400| 72 61 74 65 7d 0a 5c 69 | 74 65 6d 20 54 68 65 20 |rate}.\i|tem The |
|00000410| 61 72 65 61 20 6f 66 20 | 61 20 63 69 72 63 6c 65 |area of |a circle|
|00000420| 20 69 6e 20 74 65 72 6d | 73 20 6f 66 20 74 68 65 | in term|s of the|
|00000430| 20 72 61 64 69 75 73 20 | 69 73 20 24 41 20 3d 20 | radius |is $A = |
|00000440| 5c 70 69 20 72 5e 32 24 | 2e 0a 09 5c 62 65 67 69 |\pi r^2$|...\begi|
|00000450| 6e 7b 65 6e 75 6d 65 72 | 61 74 65 7d 0a 09 5c 69 |n{enumer|ate}..\i|
|00000460| 74 65 6d 20 24 5c 66 72 | 61 63 7b 64 41 7d 7b 64 |tem $\fr|ac{dA}{d|
|00000470| 74 7d 20 3d 20 5c 66 72 | 61 63 7b 64 41 7d 7b 64 |t} = \fr|ac{dA}{d|
|00000480| 72 7d 20 5c 66 72 61 63 | 7b 64 72 7d 7b 64 74 7d |r} \frac|{dr}{dt}|
|00000490| 20 3d 20 32 20 5c 70 69 | 20 72 20 5c 66 72 61 63 | = 2 \pi| r \frac|
|000004a0| 7b 64 72 7d 7b 64 74 7d | 24 2e 0a 09 5c 69 74 65 |{dr}{dt}|$...\ite|
|000004b0| 6d 20 28 33 2e 33 2e 34 | 39 29 20 20 24 5c 66 72 |m (3.3.4|9)  $\fr|
|000004c0| 61 63 7b 64 41 7d 7b 64 | 74 7d 20 3d 20 32 20 5c |ac{dA}{d|t} = 2 \|
|000004d0| 70 69 20 72 20 5c 66 72 | 61 63 7b 64 72 7d 7b 64 |pi r \fr|ac{dr}{d|
|000004e0| 74 7d 20 3d 20 32 20 5c | 70 69 20 20 5c 63 64 6f |t} = 2 \|pi  \cdo|
|000004f0| 74 20 31 30 20 5c 63 64 | 6f 74 20 32 20 3d 20 34 |t 10 \cd|ot 2 = 4|
|00000500| 30 20 5c 70 69 20 24 20 | 63 6d 24 5e 32 24 2f 73 |0 \pi $ |cm$^2$/s|
|00000510| 65 63 2e 0a 09 5c 69 74 | 65 6d 20 46 72 6f 6d 20 |ec...\it|em From |
|00000520| 70 61 72 74 20 28 61 29 | 2c 20 24 5c 66 72 61 63 |part (a)|, $\frac|
|00000530| 7b 64 72 7d 7b 64 74 7d | 20 3d 20 5c 66 72 61 63 |{dr}{dt}| = \frac|
|00000540| 7b 31 7d 7b 32 20 5c 70 | 69 20 72 7d 20 5c 66 72 |{1}{2 \p|i r} \fr|
|00000550| 61 63 7b 64 41 7d 7b 64 | 74 7d 24 2e 20 20 55 73 |ac{dA}{d|t}$.  Us|
|00000560| 65 20 24 72 20 3d 20 5c | 73 71 72 74 7b 41 2f 20 |e $r = \|sqrt{A/ |
|00000570| 5c 70 69 7d 24 20 61 6e | 64 20 73 69 6d 70 6c 69 |\pi}$ an|d simpli|
|00000580| 66 79 20 74 6f 20 67 65 | 74 20 20 24 5c 66 72 61 |fy to ge|t  $\fra|
|00000590| 63 7b 64 72 7d 7b 64 74 | 7d 20 3d 20 5c 66 72 61 |c{dr}{dt|} = \fra|
|000005a0| 63 7b 31 7d 7b 32 20 5c | 73 71 72 74 7b 5c 70 69 |c{1}{2 \|sqrt{\pi|
|000005b0| 20 41 7d 7d 20 5c 66 72 | 61 63 7b 64 41 7d 7b 64 | A}} \fr|ac{dA}{d|
|000005c0| 74 7d 24 2e 0a 09 5c 69 | 74 65 6d 20 28 33 2e 33 |t}$...\i|tem (3.3|
|000005d0| 2e 35 30 29 20 24 2d 20 | 5c 73 71 72 74 7b 33 7d |.50) $- |\sqrt{3}|
|000005e0| 2f 20 31 35 24 20 63 6d | 2f 73 65 63 2e 0a 09 5c |/ 15$ cm|/sec...\|
|000005f0| 65 6e 64 7b 65 6e 75 6d | 65 72 61 74 65 7d 0a 0a |end{enum|erate}..|
|00000600| 0a 5c 65 6e 64 7b 65 6e | 75 6d 65 72 61 74 65 7d |.\end{en|umerate}|
|00000610| 0a 5c 65 6e 64 7b 64 6f | 63 75 6d 65 6e 74 7d 0a |.\end{do|cument}.|
+--------+-------------------------+-------------------------+--------+--------+