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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 14\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 14} 
\bigskip

\itemitem{1.\hskip12pt a)}  Graph the function $f(x)=\sin x$ on a set
of axes that is at least half the size of a page of notebook paper.

\medskip
\itemitem{b)}  Find the equation of the tangent line to $f$ at $x=0$.
Graph it on the same set of axes.  Notice that near $x=0$, this gives
a reasonable approximation for $f$.  Most likely you have seen the
approximation $\sin x\approx x$ for $x$ small.  (In physics class?) 

\medskip
\itemitem{c)}  Perhaps we could approximate $f(x)=\sin x$ more
accurately with a parabola than with a line.  Find the equation of the
parabola which goes through $(0,0)$, has the same slope as $\sin x$ at
$x=0$, {\bf and} which has the same second derivative as $\sin x$ at
$x=0$. 

\medskip
\itemitem{d)}  Now find the equation of a cubic which goes through
$(0,0)$ and which also has the same value of first, second, {\bf and}
third derivative as $f(x)=\sin x$ at $x=0$.  

\medskip 
\item{2.}  Let $f\colon\R\to\R$ be a function which is infinitely
differentiable.  (What does that mean?)  Consider the $n^{\rm th}$
degree polynomial 
$$
a_0+a_1x+a_2x^2+\cdots+a_nx^n.
$$
Suppose that the values of each of the first $n$ derivatives of $f$ at $x=0$
$$
f'(0),\> f''(0),\> f'''(0),\ldots,\> f^{(n)}(0)
$$
are the same as the values of the respective derivatives of the above
polynomial.  Find the values of the coefficients $a_i$ in terms of the
derivatives of $f$.  This is called the {\it $n^{th}$ order Taylor
polynomial of $f$ at 0}.

\item{}  (Hint:  Evaluate the polynomial and its first $n$
derivatives at $x=0$.)

\medskip
\item{3.}  {\bf Definition.}  Let $f\colon\R\to\R$ be a function.  The
{\it Taylor series of $f$ at $x=0$} is the series
$$
\sum_{n=0}^\infty {f^{(n)}(0)\over n!}x^n.
$$
\itemitem{a)}  Compute the Taylor series at $x=0$ for the following
functions: 
$$
\pt {i} f(x)=\sin x \qquad
\pt {ii} f(x)=\cos x \qquad
\pt {iii} f(x)=e^x \qquad
\pt {iv} f(x)={1\over1-x}\hskip.2in
$$

\itemitem{b)}  For which values of $x$ do these series converge?

\eject
\itemitem{4.\hskip12pt a)}  Which polynomials are odd functions?

\medskip
\itemitem{b)}  If $f$ is an odd function, are its Taylor polynomials
necessarily odd?  Explain.

\medskip
\itemitem{c)}  Which polynomials are even functions?

\medskip
\itemitem{d)}  If $f$ is an even function, are its Taylor polynomials
necessarily even?  Explain.

\medskip
\item{5.}  For values $x$ where the series converges, define
$$
f(x)=\sum_{n=0}^\infty\left({x+3\over5}\right)^n{\atop.}
$$

\medskip
\itemitem{a)}  Find a closed form expression for $f(x)$.  (i.e., one
which does not involve an infinite sum.) 

\medskip
\itemitem{b)}  For what values of $x$ does the series defining $f(x)$
converge? 

\medskip
\itemitem{c)}  On the interval where the series defining $f(x)$
converges, graph $f(x)$.  Looking at the graph, why do you think the
interval of convergence ends where it does?

\medskip
\item{6.}  A train wheel has two rims as shown below and is designed
to run along a bi-level track.  The outer (smaller) rim has radius 10
inches while the inner (larger) rim has radius 12 inches.

\centerline{
\epsfysize=2truein\epsfbox{/home/oehrtman/m210/slowtrain.eps}}

\itemitem{a)}  How far does the outer rim roll over the course of one
rotation of the wheel?
\itemitem{b)}  How far does the inner rim roll over the course of one
rotation of the wheel?
\itemitem{c)}  How far does the train travel over the course of one
rotation of the wheel?






\bye