Department of Mathematics

Math 300: Mathematical Computing

Assignment 5

You will write a paper about the composite trapezoidal rule. This will discuss the fact that not all integrable functions have antiderivatives that we can expres simply, so that we cannot always use the fundamental theorem to evaluate the integral. It will describe the two-point trapezoidal rule, and then describe how that can be used on a partition of the interval of integration to write a sum that can be used to evaluate the integral with arbitrary precision. The subintervals of the partition of \([a,b]\) traditionally have width \(h=(b-a)/n\), where \(n\) is the number of subintervals of the partition. You will discuss the fact that the error in the trapezoidal rule is \(O(h^2)\), and so the error in the approximation to the integral goes down proportionally to the square of the inverse of the number of subintervals used. Since your reader might not have seen "big O" notation before, you will write a definition for that and describe its significance.

Trapezoidal Rule
Figure 1: Trapezoidal Rule on 4 subintervals

The exciting part of your paper will involve applying the trapezoidal rule to the function \(p(x)=-x^2+2x+\frac12\) on the interval \([0,3]\). You need not do this yourself, but you should discuss how it is done, and you must include Figure 1 and Table 1 as shown on this page. You can save the figure directly from this page to include in your document, but you will need to type the table yourself. Note that both the figure and table will need captions, and you will refer to them by label; not using words such as "the figure below." Be sure to tell the reader that h in the table represents the uniform spacing of the partition for the interval of integration, and include a formula for that.

Table 1: Trapezoidal rule for p(x)
hIntegralError
10-11.4955×10-3
10-21.499955×10-5
10-31.49999955×10-7
10-41.4999999955×10-9
10-51.499999999954.9998×10-11

The paper will be single-spaced in a 10pt font, with 1" margins on both left and right sides. The following things are obvious, but we note them in passing, nonetheless.

Equally obviously, the typesetting for this paper will be done using LaTeχ. The assignment is worth 50 points and is due at 9AM on Thursday, 4 October. It is turned in when the .tex file appears as an attachment to an email message in the instructors inbox. You do not need to send the image - the instructor has that. For emphasis: send the .tex file, not the .pdf.


Assignment 7 is posted.




A solution to the test is available.







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