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\begin{center}
  Math 108\\
  Exam I\vspace{3mm}\\
 January ??, 1998
\end{center}

\bigskip
\noindent
\textbf{1.} 
A factory installs new and more efficient machines that are expected
to generate savings at the rate of $600,000 e^{-0.3t}$ dollars per year,
after $t$ years in operation.  If the cost of the machines is
$\$1,000,000,$ find the time needed for the machines to pay for their
cost.

\vspace*{1cm}
\textbf{Solution.} Let $S = S(t)$ be the savings after $t$ years.  Then
$S^\prime(t) = 600,000 e^{-0.3t}$ and $S(0) = 0$.  Thus $S(t) = \int?
600,000 e^{-0.3t} dt$ or $s(t) = \frac{600,000}{-0.3} \ e^{-0.3t} + c = -
2,000,000 e^{-0.3t} + c$.

Since $0 = S(0) = -2,000,000 + c$ we have $c = 2,000,000$.  Therefore
$S(t) = 2,000,000(1 - e^{-0.3t})$.  To find the time needed for
the machines to pay for their cost, we solve the equation $2,000,000 (1
- e^{-0.3t}) = 1,000,000,$ or $1 - e^{-0.3t} = \frac{1}{2}$ or $e^{-0.3t}
= 0.5$ or $-0.3t = \ell n 0.5$ or $t = - \frac{\ell n 0.5}{0.3} \simeq 2.3$
years.


\vspace*{1cm}
\textbf{2.} 
The marginal profit function of a company for producing and selling $x$
units of a product is given by  $MP = -0.6x + 1,200 $ in dollars.  When
500 units are produced and sold, the company's profit is 25,000.  Find:

\begin{itemize}
\item[a.]  The profit function
\item[b.]  The production level which maximixes the profit.
\end{itemize}



\vspace*{1cm}
\textbf{Solution.} 

\begin{itemize}
\item[a.]  Let $P(x)$ be the profit function.  Then $P^\prime = -0.6x +
1,200$, and $P(500) = 25,000$.  Thus $P = \int (-0.6x + 1,200) dx$ or
$P(x) = -0.3x^2 + 1,200x + c$.  Since $25,000 = P(500) = -0.3 \cdot
250,000 + 1,200 \cdot 500 + c$ or $c = -500,000$.  Thus
\[
P(x) = -0.3x^2 + 1,200x - 500,000.
\]
\item[b.]  The profit is maximized when $MP = 0$ or $-0.6x + 1,200 =
0$ or $x =-1,200/0.6$ or $x = 2,000$.
\end{itemize}

\vspace*{1cm}
\textbf{3.}   Consider the following situation.  You are hired to advise
a company for maximizing its profit.  You find that it produces and
sells daily 3,000 units of a certain product for the profit of
$\$100,000$.  You also estimate the marginal profit function to be
$MP=-0.2x + 800$, where $x$ is the number of units produced and sold. 
Compute the amount of units which the company needs to produce and
sell to maximize its profit.  Also compute the maximum profit.







\end{document}

\vspace*{1cm}
\item[2.]  Let $(X, \mathcal{M}, \mu)$ be a measure space
and $f_n: X \longrightarrow \mathbb{C} $ be a sequence of
measurable functions such that $f_n \longrightarrow f$ in
measure, and $|f_n| \le g$ for some integrable function $g$
show that
\[
\int f_n \longrightarrow \int f.
\]
\item[3.]  If $f \in L(\mathbb{R})$ and $f (x) \ge 0 $ for all
$x \in \mathbb{R}$, then show that
\[
\lim_{K \to \infty} k \int_{\mathbb{R}} \ell n \left [ 1 +
\frac{f(x)}{k}\right ] dx = \int_{\mathbb{R}} f(x) dx.
\]
\item[4.]   Let $\{f_k\}$ and $\{g_k\}$ be two sequences of
functions such that $f_k \longrightarrow f$ in
$L^p(\mathbb{R}^n), \ 1 \le p < \infty, \ g_k \longrightarrow g$ a.e.
in $\mathbb{R}$, and $\|g_k\| \le M$ for all $k$, for some $M
> 0$.  Then show that
\[ f_k g_k \longrightarrow fg \mbox{ in } L^p(\mathbb{R}^n).
\]
\item[5.]  If $f \in L^1 (\mathbb{R})$ has compact support then show
that its Fourier transform
\[
\widehat f(\xi) = \int e^{- ix\xi} f(x) dx, \ \xi \in \mathbb{R}
\]
extends as a holomorphic function on the whole $\mathbb{C}$.

\item[6.a.]  Show that
\[
\left \{ \frac{e^{inx}}{\sqrt{2\pi}} \right \}^\infty_{n = - \infty}
\] is an orthonormal family in $L^2 [-\pi, \pi].$

\vspace*{.5cm}
\textbf{(b)}  If $f$ is a continuously differentiable function of period
$2\pi$ then show that its Fourier series
\[
\sum^\infty_{n = - \infty} \left ( f, \frac{e^{inx}}{\sqrt{2\pi}} 
\right ) \frac{e^{inx}}{\sqrt{2\pi}}
\]
converges uniformly to $f$.  You may use the fact that the Fourier
series of $f$ converges pointwise to $f$.

\medskip
\item[7.]   If $f: [0, 1] \longrightarrow \mathbb{R}$ is continuously
differentiable then show that
\[
\int^1_{-1} |f(x)|^2 dx \le \int^1_{-1} |f^\prime(x)|^2 dx.
\]
\item[8.] A bounded linear function $\lambda: L^2 [-1, 1]
\longrightarrow \mathbb{C}$ satisfies
\[
\lambda (f) = - \lambda (f^\prime)
\]
when applied to continuously differentiable functions $f \in L^2[-1,
1]$ satisfying $f(-1) = f(1) = 0$.  Find $g_\lambda$ for representing
$\lambda$, as in the Riesz theorem, and compute $\|\lambda\|$.

\medskip
\item[9.]  Consider $C[-1, 1]$ as a subspace of $L^2 [-1, 1]$ and
define
$T:C[-1, 1] \longrightarrow \mathbb{C}$ by $T(f) = f(0)$.  Do the
Hahn-Banach and Riesz theorem imply the existence of $g_T \in L^2
[-1, 1]$ such that
\[
T(f) = (f, g_T) \mbox{ for all } f \in L^2 [-1, 1]?
\]
Explain fully your answer.
\item[10.]  Define a sequence of integral operators $Tn: [0,1]
\longrightarrow C[0, 1]$ by
\[
T_n f(x) = \int_1^{1 + n} \int^1_0 e^{-t^2|x-y|} f(y) dy dt.
\]

\vspace*{.5cm}
\textbf{(a)}  Show that $\|T_n\| < \infty $ for all $n = 1, 2, \dots$

\vspace*{.5cm}
\textbf{(b)}  Show that $\{T_n\}$ is a Cauchy sequence in
$\mathcal{L} (C[0,1], C[0, 1]).$
\end{description}

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 DESCRIPTION OF INVENTION--If chemical compounds, describe
variations  in substituents and groups and synthetic route from known
starting materials.  If composition or new use, describe proportions and
applications.  For new process, describe all critical conditions; attach
additional sheets if necessary;  also attach copies of all available
experimental data.  If only in concept state, and not yet reduced to
practice, so state.
\vspace{6cm}
\item  UTILITY OF INVENTION
\vspace{6cm}
\item  OBJECTIVE OF INVENTION--Describe how invention solves problem of
the prior art.  What is believed inventive about the process, product or
apparatus?
\vspace{6cm}
\item WORKABLE EXTENT OF INVENTION--Describe any variations of a broader
scope than those described in Part 1.  Do you have plans for future work?
\vspace{6cm}
\item   RELEVANT PRIOR ART--Include patents, scientific articules,
technical bulletins, and the like insofar as they are known to you. 
Specifically mention whether publication or outside disclosure has or
will be made and if so when.
\vspace{6cm}
\item JOINT INVENTION--If a joint invention, list both University and
non-University inventors.  Explain relationship with joint inventor.
\vspace{5cm}
\item  SPECIAL CIRCUMSTANCES WHICH MIGHT AFFECT COMMITTEE DELIBERATIONS
AND ACTION.
\vspace{5cm}
\item[8] COMMENTS
\vspace{6cm}
\item[9] INVENTOR(S)
\vspace{2cm}
\item[SIGNATURE] \hrulefill\vspace{-1mm}
Date\hrulefill\vspace{-1mm}
 \medskip\\
\hrulefill\vspace{-1mm} \ \ \  \hrulefill\\
\item[Date of Conception]\hrulefill\\
\item[Date of First Written Record] \hrulefill Notebook Ref.
\hrulefill\vspace{-1mm} \\
\item[Date of First Preparation] \hrulefill\vspace{-1mm} Notebook Ref.
\hrulefill\\
\item[Date of First Test] \hrulefill\vspace{-1mm} Notebook Ref.
\hrulefill\\
\end{enumerate}

\end{document}

