Answer to Quiz 1 Fri 26 Aug 2005

Consider f(x) = x^2 - x - 2 = 0. Apply three steps with the bisection method to find an approximation for a root of f(x) = 0 inside the interval [a,b] = [0,3].

Illustrate on the plot of f(x) = 0 below how the bisection method works by indicating the end points of the intervals [a1,b1] (=[a,b]), [a2,b2], and [a3,b3].

Answer:
```
   Use the values for [a,b] in the table below.
```

Write the numerical intermediate results of three steps of the bisection method in scientific notation with four significant decimal places after rounding:

Answer:

+------+-----------+-----------+-----------+-----------+
|      |           |           |   a + b   |  ( a + b )|
| step |     a     |     b     |   -----   | f( ----- )|
|      |           |           |     2     |  (   2   )|
+======+===========+===========+===========+===========+
|   1  |  0.000E+0 |  3.000E+0 |  1.500E+0 | -1.250E+0 |
|   2  |  1.500E+0 |  3.000E+0 |  2.250E+0 |  8.125E-1 |
|   3  |  1.500E+0 |  2.250E+0 |  1.875E+0 | -3.594E-1 |
+------+-----------+-----------+-----------+-----------+

Estimate how many steps of the bisection method it will take to get four decimal places of the root correct. Justify your estimate!

Answer:

  3-0
 ----- = 10^(-4) => 3 . 2^(-N) = 10^(-4)
  2^N                    
                 => N = 4 log[2](10) + log[2](3)

                      ~= 4 ( 3.4 ) + 1. 6 = 14.9
      
  N = 15 steps