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

\centerline
{\bf WORKSHEET 1} 
\medskip

\centerline{Math 408C Review}


\bigskip 

\item{1.}  Let $g\colon\R\to\R$ be a function.  What does it mean for
$g$ to be differentiable?  

\medskip 
\item{2.}  Using the definition of the derivative, compute $g'(x)$ if
$g(x)=2x^2-2$. 

\medskip
\item{3.}  Give an example of a function which is continuous but not
differentiable. 

\medskip
\item{4.}  Let $h\colon[a,b]\to\R$ be a real-valued function on the
closed interval $[a,b]$.  Define the definite integral
$$
\int_a^b h(x)\,dx.
$$

\medskip
\itemitem{5.\hskip12pt a)}  State the Fundamental Theorem of Calculus
(either version). 

\medskip
\itemitem{b)}  What makes this theorem beautiful, powerful, or useful?

\medskip
\itemitem{c)}  Outline reasoning to believe the Fundamental Theorem of
Calculus.  A complete proof is not necessary.

\medskip
\itemitem{6.\hskip12pt a)}  Find 
$$
{\partial\over\partial x}\int_{-5}^{x^2}{1\over \sec^2t+t}\,dt.
$$

\itemitem{b)}  Give another function with the same derivative as the
one computed in part a).

\medskip
\item{7.}  Give an example of a function that is not integrable.

\eject


\item{8.}  In each part below, you are given the graph of the
derivative $f'$ of some function $f$.  Sketch a graph of $f$
satisfying the condition listed.
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\+\hskip.3in a) $f(0)=0$ &b) $f(0)=0$ &c) $f(1)=1$\cr
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\+\hskip.3in d) $\min(f(x))=0$ &e) $\lim_{x\to\infty}f(x)=2$ 
     &f) $\lim_{x\to0}f(x)=-1$\cr 
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\bye