Wrestling with the Fundamental Theorem of Calculus
Wrestling with the Fundamental Theorem of Calculus
Wrestling with the Fundamental Theorem of Calculus
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
1 1 1 1 1 1 1 1<br />
[ 0, 0. 1)( , . 99, 1]<br />
,<br />
"<br />
! 3 , +<br />
$<br />
3 ,<br />
"<br />
! 4 , +<br />
$<br />
4 ,<br />
# 2 10 2 10 % # 3 10 3 10 %<br />
"<br />
#<br />
2<br />
3<br />
" 1<br />
# 5<br />
1 2 1 1 1 1 1 3 1 3 1<br />
! 5, +<br />
$<br />
5 ,<br />
"<br />
! 6 , +<br />
$<br />
6 ,<br />
"<br />
! 7 , +<br />
$<br />
7 ,<br />
10 3 10 % # 4 10 4 10 % # 4 10 4 10 %<br />
1 1 1<br />
! 8 , +<br />
10 5 10<br />
$ 2 1 2 1 3 1 3 1<br />
,<br />
"<br />
! , +<br />
$<br />
,<br />
"<br />
! , +<br />
$<br />
,!<br />
% # 5 10 5 10 % # 5 10 5 10 %<br />
8 9 9 10 10<br />
Every rational number in [0,1] is in at least one <strong>of</strong> <strong>the</strong>se<br />
intervals. The sum <strong>of</strong> <strong>the</strong> lengths <strong>of</strong> <strong>the</strong>se intervals is less than<br />
1 1 2 1 1<br />
+ +<br />
!<br />
3 1+<br />
+<br />
10 100 10 "<br />
2<br />
10 10<br />
11 2 10 101<br />
= + % =<br />
100 1000 9 900<br />
1<br />
+ 3<br />
10<br />
+ !<br />
#<br />
$