Information Technology Reference
In-Depth Information
Fig. 1.6
The fi
gu
re shows
1
the graphs of
p
x (
upper
), x
4
(
middle
), and x
20
(
lower
)on
the unit interval
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Tabl e 1. 2
The table shows how accurate the trapezoidal method is for three definite integrals
where the exact solutions are known. Note that convergence seems to be obtained for all three
cases. For the two first cases, we note that E
h
=h
2
is constant as h is reduced. This is, however, not
the case for the third function
h
R
0
x
4
dx
R
0
x
20
dx
R
0
p
xdx
1
5
1
21
2
3
D
D
D
10
5
E
h
E
h
=h
2
10
5
E
h
E
h
=h
2
10
5
E
h
E
h
=h
2
0:01
3:33
0:33
16:66
1:67
20:37
2:04
0:005
0:83
0:33
4:17
1:67
7:25
2:90
0:0025
0:21
0:33
1:04
1:67
2:57
4:17
0:00125
0:05
0:33
0:26
1:67
0:91
5:84
1.5
Back to the Bagels
Let us recall that our problem is to compute
C
Z
b
300
1
2
p
D
p.b/
D
f.x/dx;
(1.22)
where
1
p
220
e
.x300/
2
f.x/
D
:
(1.23)
220
2
See the discussion leading to (1.6). Recall also that for a given value of b,the
function p
D
p.b/ denotes the probability that the number of bagels sold on one
particular day is less than or equal to b. In particular, we want to find the smallest
possible value of b
D
b
such that p
D
p.b
/
0:95. When the owner orders b
bagels, she knows that this will be sufficient on 95% of the days.
In the section above, we derived a method for computing integrals of the form
encountered in (1.22). Here we will apply this method to compute p.b/ for integer
values of b ranging from 300 to 370. Note that since
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