## Mathematics 210 - Spring term 2005 - Third assignment

This assignment requires you to submit several spreadsheets concerned with some limitations of computers. It is due before class time Monday, January 31.
• Question 1. This exercise is to build a spreadsheet that displays the base B expansion of a number.

Allot some fixed cell, say e0, to the integer B, which must be greater than 1.

Then build a spread sheet with this feature: if you put a real number y in a0 it displays in a1 the exponent e (an integer) for base B of y (i.e. with respect to scientific notation). Let x = y/Be. Use column b for working space. In b0 goes x. In column c for 60 cells starting at c0 goes the expression for x in base B. That is to say x = c0.c1 c2 c3 c4 ... in base B. Explain somewhere easily visible, and briefly, and in your own words, what happens for the last several cells, at least for B = 10 and y = 1/3.

The grader will put various values for B and y into the sheet.

Save as m210.3.1.ms.

• Question 2. This exercise is to show one of the limitations of computers. Read first the documentation on conditionals in the spreadsheet, on the page
`http://ugrad.math.ubc.ca:8099/mathsheet/docs/manpages/formulas.html`
.

Put in column a three numbers A, B, C. These are to be the coefficients of a quadratic polynomial Ax2 + Bx + C. Set A = 1, B = 3, C=2 for a start, to check your work easily.

Put in cells b0 and b1 the formulas for the roots of the polynomial if they are real, and if they are complex put the real and imaginary parts in the next lower cells in b. Put in c0 and c1 the sum and product of the real roots, in the next lower rows the same if they are complex. Set the format of all cells to 16 figures.

Set A = 1, B = 1,000,000, C=1. Explain what odd behaviour you see. Then explain how to adjust your formulas for roots to take this behaviour into account, using a spreadsheet conditional.

Save as m210.3.2.ms.

• Question 3. Set up the spreadsheet so it displays in colums a and b the values of x and f(x) = e-x2/2 over the range x=0 to 5 in steps of 0.05. Graph the function f(x).

Then lay out columns that will let you compute values of this function by summing the Taylor series at 0. The way this should work is that the value of x being used is in one fixed cell, and the value of f(x) appears way down in the sheet, around row 60. By changing he valoue of x, compute the different values of f(x). Store in a column the values of f(x) over the same range. You will have to do some things by hand here, using the Edit/Convert function to make the task possible. Graph these data also on the same graph. Explain what you see in the graphs. You might have to zoom in to see what is happening.

Let me summarize: the goal of this exercise is to compare two ways of calculating and displaying a range of values of f(x), one by formula and one by series.

Save as m210.3.3.ms.

If you find these questions confusing, please write me.