Let c be the Champernowne's constant,
or c=0.123456789101112131415161718192021....
Show, without calculator aid, that
sin(c) + cos(c) + tan(c) > 10c
(In reply to
re(4): solution. An idea by Jer)
Let f(x) be sin(x) + cos(x) + tan(x)
x f(x) f(x)/x
0.100 1.19517 11.95172
0.101 1.19708 11.85225
0.102 1.19898 11.75472
0.103 1.20088 11.65907
0.104 1.20279 11.56525
0.105 1.20469 11.47321
0.106 1.20659 11.38290
0.107 1.20849 11.29427
0.108 1.21039 11.20728
0.109 1.21228 11.12187
0.110 1.21418 11.03800
0.111 1.21608 10.95564
0.112 1.21797 10.87474
0.113 1.21987 10.79527
0.114 1.22176 10.71718
0.115 1.22365 10.64044
0.116 1.22554 10.56502
0.117 1.22743 10.49088
0.118 1.22932 10.41799
0.119 1.23121 10.34632
0.120 1.23310 10.27583
0.121 1.23499 10.20651
0.122 1.23687 10.13831
0.123 1.23876 10.07121
0.124 1.24064 10.00519
0.125 1.24253 9.94022
0.126 1.24441 9.87627
0.127 1.24629 9.81333
0.128 1.24817 9.75136
0.129 1.25005 9.69034
0.130 1.25193 9.63026
0.131 1.25381 9.57108
0.132 1.25569 9.51280
0.133 1.25757 9.45539
0.134 1.25944 9.39883
0.135 1.26132 9.34309
0.136 1.26319 9.28818
0.137 1.26507 9.23406
0.138 1.26694 9.18071
0.139 1.26881 9.12813
When x >= 0.121 (at this level of precision) f(x) exceeds c (Champernowne's number).
When x <= 0.124 f(x) exceeds 10*x.
So there are two ways of coming at it (at least).
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Posted by Charlie
on 2014-12-14 08:13:24 |
re(5): solution. Useful (it's to be hoped) table
