# 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.

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)*

h | Integral | Error |
---|---|---|

10^{-1} | 1.495 | 5×10^{-3} |

10^{-2} | 1.49995 | 5×10^{-5} |

10^{-3} | 1.4999995 | 5×10^{-7} |

10^{-4} | 1.499999995 | 5×10^{-9} |

10^{-5} | 1.49999999995 | 4.9998×10^{-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.

- The paper will have a title, author information, an introduction, conclusions, and probably be divided into sections.
- The paper will be carefully written, with lucid description, proper use of grammar, spelling, and punctuation, and few typographical errors.
- The paper will include equations. References to those equations will be by number.
- The paper will include a list of references. If no references appear, the reader will assume you made everything up, which does not inspire confidence.
- There are some hints about mathematical writing here.
- Papers shorter than a couple of pages or longer than five will probably not make a positive impression on the reader.
- You will read this assignment carefully to be sure that you have included everything required.

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 C is posted.

The test solution is available.