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\noindent{\bf MATH 5365\hfill Homework 8: Excel\hfill\today}\medskip\\
{\bf The homework is due on April 17.}
\begin{pro}{5} 
Use \textit{Excel} to implement Newton's algorithm: Compute approximations of $\sqrt{2}$ and $\sqrt[5]{17}$.
\end{pro}
\begin{pro}{5}
Let $f(x)$ be a polynomial. Given the values $f(k)$ for $k\in\intg$, the first difference values $d_1(k)$ are defined for $k\in\intg$ by setting
\[d_1(k)=f(k)-f(k-1).\] 
The second differences $d_2(k)$ are similarly defined as consecutive differences of the first difference values, etc:
\[d_2(k)=d_1(k)-d_1(k-1).\]

Using \textit{Excel}, illustrate for $n=2,3,4$ that for a polynomial with degree $n$ the $n$th differences are constant. 

(This is a generalization of the fact that a linear function has constant slope. Actually, the converse of this is also true: If a polynomial has constant $n$th differences, then its degree is at most $n$. This ``pre-calculus" characterization of polynomials is due to Isaac Newton.) 
\end{pro}
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