Cantor Set The Cantor set (Georg Cantor, 1845-1918) is a subset of the unit interval [0, 1]. To construct the Cantor set, first remove the middle third ( 1 3 , 2 3 ) of the interval, leaving two line segments. For the second step, remove the middle third of each of the two remaining segments, leaving four line segments. Continue this procedure indefinitely, as shown in the figure. The Cantor set consists of all numbers in the unit interval [0, 1] that still remain. (a) Find the total length of all the line segments that are removed. (b) Write down three numbers that are in the Cantor set. (c) Let C n denote the total length of the remaining line segments after n steps. Find lim n → ∞ C n
Cantor Set The Cantor set (Georg Cantor, 1845-1918) is a subset of the unit interval [0, 1]. To construct the Cantor set, first remove the middle third ( 1 3 , 2 3 ) of the interval, leaving two line segments. For the second step, remove the middle third of each of the two remaining segments, leaving four line segments. Continue this procedure indefinitely, as shown in the figure. The Cantor set consists of all numbers in the unit interval [0, 1] that still remain. (a) Find the total length of all the line segments that are removed. (b) Write down three numbers that are in the Cantor set. (c) Let C n denote the total length of the remaining line segments after n steps. Find lim n → ∞ C n
Solution Summary: The author explains that the Cantor set consists of all numbers in the unit interval left[0,1right] and first remove the open middle third.
Cantor Set The Cantor set (Georg Cantor, 1845-1918) is a subset of the unit interval [0, 1]. To construct the Cantor set, first remove the middle third
(
1
3
,
2
3
)
of the interval, leaving two line segments. For the second step, remove the middle third of each of the two remaining segments, leaving four line segments. Continue this procedure indefinitely, as shown in the figure. The Cantor set consists of all numbers in the unit interval [0, 1] that still remain.
(a) Find the total length of all the line segments that are removed.
(b) Write down three numbers that are in the Cantor set.
(c) Let
C
n
denote the total length of the remaining line segments after n steps. Find
lim
n
→
∞
C
n
Calculus III
May I please have some elaborations on Example 2 part a? Thank you.
1. A bicyclist is riding their bike along the Chicago Lakefront Trail. The velocity (in
feet per second) of the bicyclist is recorded below. Use (a) Simpson's Rule, and (b)
the Trapezoidal Rule to estimate the total distance the bicyclist traveled during the
8-second period.
t
0 2
4 6 8
V
10 15
12 10 16
2. Find the midpoint rule approximation for
(a) n = 4
+5
x²dx using n subintervals.
1° 2
(b) n = 8
36
32
28
36
32
28
24
24
20
20
16
16
12
8-
4
1
2
3
4
5
6
12
8
4
1
2
3
4
5
6
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