Answer to Quiz 1 Fri 29 Aug 2003

Consider f(x) = e^x - 8x = 0.
Apply three steps with the bisection method to find an approximation
for a root of f(x) = 0 inside the interval [a,b] = [-2,3].

1. 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:

Mark on the horizontal axes the following begin points of
the intervals: a1 = -2, a2 = -2, and a3 = -0.75. 
The end point of the intervals are:
b1 = 3, b2 = 0.5, and b3 = 0.5.

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

   +--------+-----------+-----------+-----------+-------------+
   |  step  |     a     |     b     |  (a+b)/2  | ln((a+b)/2) |
   +========+===========+===========+===========+=============+
   |    1   | -2.000E+0 |  3.000E+0 |  5.000E-1 |  -2.351E+0  |
   |    2   | -2.000E+0 |  5.000E-1 | -7.500E-1 |   6.472E+0  |
   |    3   | -7.500E-1 |  5.000E-1 | -1.250E-1 |   1.882E+0  |
   +--------+-----------+-----------+-----------+-------------+