Mathematics 210 - Spring term 2003 - First assignment

This assignment requires you to submit several spreadsheets concerned with numerical integration. Be sure to sign all graphs.
  • Go to the MathSheet home page and then to the applet page. Open a running copy of the spreadsheet and return to this page.
  • Log in immediately: File/Log in. Your login id is your Mathematics Department login name, and your password is your student number. This allows you to save and load spreadsheet files. You should save your work frequently.
  • Question 1. In the first column lay out values from 0 to 4 in 64 steps. In the second column lay out values of the function e-x2. Make a graph of this function. Save this sheet as
  • Question 2. Estimate the area under the curve between 0 and 4 very rapidly by the right rectangle rule. You can check by eyeball whether you get a reasonable estimate. Fill in a bar graph to show your calculation visually - that is to say, fillin some bars to show the area given by the right rectangel rule. Save this sheet as
  • Question 3. Delete the part of your spreadsheet used to estimate areas. Calculate and graph with essentially the same data the area A(0, x) under that curve between 0 and x (i.e. the integral of e-x2 from 0 to x) as a function of x for the same range of values. But this time by applying the trapezoid rule. Save this sheet as

    For this question, you will need to figure out how to plot the function A(0,x), the area between 0 and x, by one of the approximation rules. You will need a separate column in the spreadsheet to do this. Suppose you wanted to graph A(0,x) by the left rectangle rule, for example. What you need to know is that the area up to x+h is equal to the area up to x plus the area of the new rectangle between x and x+h, or in other words that A(0,x+h) = A(x) + h f(x) (approximately), and furthermore that A(0,0) = 0. Thus you can get A(0,h), A(0,2h), etc. up to the end. The cell starting the column holds 0, and that holding A(0,x+h) is thus the sum of the cell holding A(0,x) plus something involving the cell holding f(x). Something like this happens for the trapezoid rule, too, except that the area h f(x) of a rectangle is replaced by the area of the trapezoid with ends at x and x+h.

  • Question 4. Calculate the areas, on the same spreadsheet, for 128 and 256 steps as well as 64. Each one of these should take up a separate bunch of columns. Put these all on one spreadsheet, taking up maybe 8 columns all in all. Graph all the area functions on one graph, each with a different colour. Save this sheet as
  • Question 5. Use the same spreadsheet. Delete all the previous graphs. Look at successive differences of the areas. I.e. if An is the area you get for n steps, look at values of A2n-An. Format these values to 16 decimal places. Think. Use these data alone to estimate A(0,4) that you would get with 512 steps, then 1024 and 2048 steps. Put all these differences and the estimates in a single column, next to labels in the spreadsheet. Save this sheet as
  • Question 6. Look at the differences you got in the previous question. Now make absolutely the best estimate you possibly can for A(0,4). (Hint: what is 1 + 1/4 + 1/16 + ... ?) Put this estimate in a labeled cell as on the previous qusetion. Save this sheet as
That's it!

If you find these questions confusing, please write me.