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\headline={ESP Math 408D - AP\hfil Fall 1996}
\footline={\ifnum\pageno>1 \hfil Worksheet 2\hskip.3in Page \folio \fi}
\footnote{}{Mike Oehrtman}

\centerline
{\bf WORKSHEET 2} 

\bigskip 

\itemitem{1.\hskip12pt a)}  Give the definition of a definite integral.

\medskip
\itemitem{b)}  State the Fundamental Theorem of Calculus.

\medskip
\itemitem{c)}  What is wrong with the following argument?
$$
\int_{-1}^2{dt\over t^2}=\left.{-1\over t}\right|_{-1}^2
=-{1\over2}-\left(-{1\over-1}\right)=-{1\over2}-1=-{3\over2}
$$



\medskip 
\item{2.}  Determine whether or not the definite integral of the following
functions over the corresponding intervals exist.  If not, state
why not.  If so, evaluate the integral.
$$
\eqalign{
\pt a& f(x)=e^x \qquad [0,2]\cr
\pt b& f(x)=e^x \qquad (-\infty,0]\cr
\pt c& f(x)={1\over x^2} \qquad [-1,1]\cr
\pt d& f(x)=5x^5-2\sin x \qquad [-1,1]\cr
\pt e& f(x)=\cases{1&if $x$ is in $[0,1]$, $[2.3,2.5]$, or
$[3,3.6]$;\cr {1\over2}& otherwise.\cr} \qquad [-1,5]\cr
\pt f& f(x)=\cases{0&if $x$ is rational;\cr 1&if $x$ is irrational.\cr}
        \qquad [0,1]\cr
\pt g& f(x)=\cases{x^2+1& $x\neq0$;\cr 0 & x=0.\cr} \qquad [-2,2]\cr
\pt h& f(x)=\ln x \qquad [0,1]\cr}\hskip1.3in
$$


\medskip
\item{3.}  We are given a differentiable, odd function $f(x)$ defined
on $[-3,3]$ which has zeros at $x=-2$, $x=0$, and $x=2$ (nowhere else)
and critical points at $x=-1$ and $x=1$ (nowhere else).  Also, we know
that $f(-1)=1$. Define a new function $F$ on $[-3,3]$ by the formula
$$
F(x)=\int_{-2}^x f(t)\,dt.
$$

\medskip
\itemitem{a)}  Sketch a rough graph of $f(x)$.

\medskip
\itemitem{b)}  Find the values of $F(-2)$, $F(2)$, and an upper and
lower bound on $F(0)$.

\medskip
\itemitem{c)}  Find the critical points and inflection points of
$F(x)$ on $[-3,3]$.

\medskip
\itemitem{d)}  Sketch a rough graph of $F(x)$.

\medskip
\itemitem{e)}  Interpret the points found in (c) in terms of the
graphs of $f(x)$ and $F(x)$.


\medskip
\item{4.}  A cake has dimensions $15''\times15''\times3''$.  It is
frosted on the sides and top.  How can it be divided into $5$ pieces
so that each piece has the same amount of cake and the same amount of
frosting?  What if the dimensions are changed?  What about $6$ pieces?
$7$ pieces? $n$ pieces?




\bye