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\centerline{\bigrm Polynomials and The Exponential Function}
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\noindent
{\bf Purpose:} In this project you will investigate the feasibility of
approximating the
function $e^x$ using polynomials.
Although there is a general method using
calculus to determine how closely a function can be approximated using
polynomials you do not need to know the method to do this project.
You will investigate how well polynomials you
derive approximate $e^x$.
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\noindent
{\bf Procedure:}  You are to follow the steps below.  You do not need to
do the steps in exactly the order indicated as long as you cover
everything you are asked to do.
After completing the outline below you should write your
results and calculations {\bf neatly}.  Be sure to include relevant
graphs
in what you turn in. Your grade will be determined
in part by your presentation.
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\noindent
{\bf Project Outline:}
Elmer is writing a computer program for a small business.  The program
must compute interest compounded continuously, so he needs to use the
exponential function.
Elmer is using C++ to write the
program.  In his version of C++, if he includes the standard library of
math functions it will increase his code size beyond what would be
feasible for the market.  He therefore wishes to write his own function
in C++ which would give him a good approximation for $e^x$.  He wishes to
find a polynomial $p(x)$ so that $|p(x)-e^x|<.000005$
for any value of
$x$ between $-1$ and $1$.  Elmer thinks he remembers
from his calculus course several years ago that it is possible to find
polynomials to approximate $e^x$ to any
desired degree of accuracy for {\bf every} real value of $x$.
His daughter Anna, who
is taking precalculus says that would be impossible.
Who is right? Is it possible for Elmer to find a polynomial $p$
with the property given above?  If so what is the polynomial?
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\item{1.}Use the limit $(1+{x\over n})^n$ and Mathematica to
         approximate $e^x$ with a polynomial of {\bf very high degree}.
\item{2.}Determine what the coefficient of $x^k$ gets close to for each
         fixed $k$ as $n$ gets very large. Justify your answer.
\item{3.}Use the first few limiting coefficients to approxiamte the
         function $e^x$ with polynomials of fairly low degree. Use
         Mathematica to find a polynomial of degree as low as you can
         which will meet the stated condition.
\item{4.}Who is right, Elmer or Anna?  Either give a polynomial to
         support Elmer's statement or else give a proof of Anna's
         statement (assuming the facts covered in class about
         polynomials).
\item{5.}Try to approximate $e$ using what you have done above.
\item{6.}Describe how Elmer can use the polynomials you found to
         approximate $e^x$ accurately for values of $x$ which are not
         between $-1$ and $1$. (Remember that Elmer can only use your
         polynomial, addition, subtraction, multiplication, and division
         along with the standard conditionals and loops used in computer
         programs.)
\bye