Information Technology Reference
In-Depth Information
(i) You work at Numerical Integrators, Inc. A customer has sent you a mail and
asked whether you can write a program that computes
a.t/
D
Z
1
0
sin.
4
tx
2
/ ln
p
x
C
t
dx
for any value of t in the interval Œ1; 2. She wants a program where t is input,
a.t/ is output, where the relative error is less 1=1; 000 (no percent sign here),
and the computation must take less than 1 s of CPU time on a typical desk-top
computer.
Write a project proposal for the customer. Mention what kinds of methods you
would like to try, how you would ensure the accuracy, what kind of computer
you would use, and how many hours you need to do the job in, e.g., Java.
12
1.7.3
Compute the Length of a Cable
In Fig.
1.9
you see the beautiful Storebælt bridge in Denmark connecting Sjælland
and Fyn. The bridge was opened in 1998 and is now used by more than 20,000 cars
every day. In the year 2000, the Øresund bridge connecting Denmark and Sweden
was also opened. These two bridges, together with the smaller Lillebælt bridge,
now tie Norway and Sweden directly to the European mainland. Building bridges is
a major challenge for engineers and applied mathematicians. Most bridges are safe,
but there have been disastrous designs. Probably the most infamous disaster is the
break-down of the Tacoma bridge.
13
In this project we will consider one of the simpler problems connected to building
bridges. Take a look at Fig.
1.9
. Suppose we want to know the length of the cables
that are needed. For the purpose of illustration, we consider a simplified design, see
Fig.
1.10
.
Our problem is to compute the length of a graph. To solve this problem, we start
by considering the graph of some smooth function
y
D
y.x/
for x ranging from a to b. In particular, we consider a small interval ranging from
x to x
C
x,where0<x
1.InFig.
1.11
we have depicted y on this interval.
We want to estimate the length s of the graph of y from x to x
C
x: If x is
sufficiently small, we see from Fig.
1.11
that s can be approximated by the length of
the straight line connecting the two points .x; y.x// and .x
C
x; y.x
C
x//:
12
You may specify the programming environment but nothing that requires a license for the
customer.
13
See http://www.ifi.uio.no/˜scicomp/tacoma.mpg.
