\item 
% Section 3.1:
% Study the error formula (identity) for Lagrange interpolation
% and how to obtain an error bound (inequality).
% Take for granted that $|(x-x_0)\cdots (x-x_n)| \le n! h^{n+1}$ 
% for all $0 \le x \le 1$.
Let $P_n$ be the degree $n$ interpolating polynomial of $\cos(2x)$
on the uniformly spaced nodes $x_0, \cdots, x_n$ on $[0,1]$
with $x_j = j h$, $h = 1/n$. Is it true that
$$
\max_{0\le x \le 1} | \cos(2 x) - P_n(x) | \to 0 \quad \mbox{as } n \to \infty ?
$$
Give all details and explain.

\item 
% Section 3.2:
% Study how to obtain $P_{0,1,\cdots,k}$
% from
% $P_{0,1,\cdots,j-1,j+1,\cdots,k}$
% and
% $P_{0,1,\cdots,i-1,i+1,\cdots,k}$.
Denote by 
$P_{0,1,\cdots,k}(x)$ the Lagrange interpolating polynomial
on the data set \newline $(x_0,f(x_0)), (x_1,f(x_1), \cdots
(x_k,f(x_k))$.
Express $P_{0,1,\cdots,k}$
in terms of 
$P_{0,1,\cdots,j-1,j+1,\cdots,k}$
and
$P_{0,1,\cdots,i-1,i+1,\cdots,k}$.
Then \underline{verify} that your answer is indeed 
the Lagrange interpolating polynomial.

\item  
% Section 3.5:
% Study the meaning of cubic spline and how to match the
% coefficients at $x_j$ (such as problems 12, 13, 14).
% Memorize the meaning of natural and clamped boundary conditions.
A natural cubic spline $S$ on $[0,2]$ is defined by 
$$
S(x) = \left\{ \begin{array}{ll}
                S_0(x) = 1 + 2x - x^3, & 0\le x \le 1                 \\
                S_1(x) = 2 + b(x-1) + c(x-1)^2 + d(x-1)^3, & 1\le x \le 2 
               \end{array} \right .
$$
Find $b, c, d$.

%% \item  Section 3.5:
%% Study how to obtain the degree of the piecewise polynomial
%% and number of boundary condition needed for a $C^k$ spline. 
\item 
Suppose that we are to construct a piecewise polynomial interpolation $S(x)$
on the data
$(x_0, f(x_0)),
(x_1, f(x_1)),
\cdots,
(x_n, f(x_n))$,
with additional continuity conditions for $S'$, $S''$ and $S'''$
on the interior nodes $x_1, \cdots, x_{n-1}$.
If we use polynomials of the same degree on
each of the interval $[x_0,x_1], \cdots, [x_{n-1}, x_n]$,
what is the minimal degree needed in each interval?
How many additional end conditions are needed? 
Count carefully and explain (give details).

\item 
% Section 3.2:
% Study how to solve nonlinear equations with Inverse Interpolation.
Given four data $(x_i,\exp(-2 x_i))$:
$(0.3, 0.5488)$,
$(0.4, 0.4493)$,
$(0.5, 0.3679)$ and
$(0.6, 0.3012)$
(you should generate the data yourself to avoid typo in inputting data).
Use Inverse Interpolation to find the root of $x=\exp(-2x)$.
You can use any algorithm for Lagrange interpolation.
% Extra credits for Neville's method (or divided difference, if anyone knows it).
After finding $x$, check yourself that 
$x=\exp(-2x)$ is indeed satisfied in case of a bug in your code.
Just check it yourself and need not show this last part.

Hand in code, put all data within the code so that it can be
executed immediately.

\end{enumerate}

{\noindent Name your codes in the same format as 
${\rm your_student_id_number}.m$ or
${\rm your_student_id_number}.c$ 
and make sure it is executable/compilable.}



\end{document}