Ceng 375 Numerical Computing
Final
Jan 11, 2008 15.00-17.00
Good Luck!

Each question is 25 pts. Solve only 4 of them.

1. The following table and figure are given as

   Figure 1: The population growth of Turkey between years of 1965 and 2000

   i

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

   ii

   Estimate the population at the years of 1995, 2007,2010 with least squares method.

   iii

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

   iv

   Estimate the population at the years of 1995, 2007,2010 with fitted polynomial.

   v

   Compare your results for both least squares and interpolated polynomial methods.

2. Consider the difference approximation
   
   $$f'_n=\frac{-f_{n+2}+4f_{n+1}-3f_n}{2h}$$
   
   
   where $f_n$ means $f(x)$ and $f_{n+1}$ means $f(x+h)$

   vi

   Use this formula to approximate the derivative of $f(x) = cos(x)$ at $x = 0$ using step sizes of $h = 0.10$ and $0.20$.

   vii

   Make an error analysis. Estimate the order of error $(O(h^?))$.
   Hints: The ratio of errors and the difference with the exact value.

3. Find the power fit $y=Ax^2$ for the following data,

   $x_k$	$y_k$
   2.0	5.1
   2.3	7.5
   2.6	10.6
   2.9	14.4
   3.2	10.0

   Hint: Use the least-squares method and find only the value of ``A''.
4. Write the expression to economize the the Maclaurin series for $e^{3x}$ with the precision 4.0 by using Chebyshev polynomials.
   Hint: The two-term recursion formula
   
   $$\begin{array}{l} T_{n+1}(x)=2xT_n(x)-T_{n-1}(x)\\ T_0(x)=1\\ T_1(x)=x\\ \end{array}$$
   
   

5. Consider the following table of data

   $x_i$	$f_i$
   0.0000	0.0000
   0.2000	0.5879
   0.4000	1.0637
   0.6000	1.3927
   0.8000	1.5573
   1.0000	1.5575
   1.2000	1.4091

   viii

   Approximate $\int_0^{1.2} f(x)$dx using the Trapezoidal Rule and a step size of $h = 0.4$.

   ix

   Approximate $\int_0^{1.2} f(x)$dx using the Trapezoidal Rule and a step size of $h = 0.2$.

   x

   Estimate the error in your answer to previous item. Hint: Use the procedure to estimate the proportionality factor, $C$.