Ceng 375 Numerical Computing
Midterm
Nov 30, 2007 12.40-14.30
Good Luck!

1. (40 pts) Consider the function:
   
   $$f(x) = 2x-6log(x)$$
   
   
   Plot of the function is given at the following figure;

   

   i

   (20 pts) Use three iterations of Newton's method to estimate only one of the roots of this function. Hint: $\int \frac{1}{x}dx=log(x)$

   ii

   (10 pts) Estimate the error in your answer to part i.

   iii

   (10 pts) Approximately how many iterations of the bisection method would have been required to achieve the same error?
   Hint: Take the interval as ((initial+1)-initial)

2. (40 pts) The following table and figure are given as the population growth of Turkey between years of 1927 and 2000

   iv

   (10 pts) What is the relationship that the graph suggests? Use least squares method to find out the necessary parameters of this suggested formula.

   v

   (5 pts) Estimate the population at the years of 1995, 2007 and 2010 with least squares method.

   vi

   (10 pts) Fit a cubic ($P_3$) polynomial to the given data.

   vii

   (5 pts) Estimate the population at the years of 1995, 2007 and 2010 with fitted polynomial.

   viii

   (10 pts) Compare your results for both least squares and interpolated polynomial methods.

3. (40 pts) For the given data points;
   
   $$\begin{array}{rr} x & y \\ \hline 2.1 & -12.4 \\ 4.1 & 7.3 \\ 7.1 & 10.1 \\ \end{array}$$
   
   
   1. (20 pts) Write out the Lagrangian polynomial from this table

      ix

      confirm that it reproduces the $y$'s for each $x$-value.

      x

      interpolate with it to estimate $y$ at $x=3$.

      xi

      extrapolate with it to estimate $y$ at $x=8$.

   2. (10 pts) Suppose in previous item that the $y$-value for $x=4.1$ is mistakenly entered as $7.2$ rather than $7.3$. Repeat the previous item with this incorrect value. How much difference does this make?
   3. (10 pts) Expand the Lagrangian polynomials in the previous items to get the quadratics in the form $ax^2+bx+c$. How different are the values for $a$,$b$, and $c$?