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\lhead{Math 1100-003, Business Calculus}
\rhead{Instructor : Sabine Lang}
\begin{document}

\section*{Worksheet}

\textbf{Here are some examples of what you can do with \LaTeX. You can type text, use formulas, play with alignments and (numbered) lists, tables, include pictures, etc...}


\begin{enumerate}
\item Calculate the area bounded by $f(x)=x^2-1$ and $g(x)=2x-2$, between $x=0$ and $x=3$.
\item Let $\displaystyle f(x)=\frac{x^2-6x+5}{x-1}$. Find the horizontal and vertical asymptote(s) of $f(x)$, if they exist.
\item Sketch the graph of $\displaystyle f(x)=\sqrt{-x+3}$ and $g(x)=2x^3-4$ using transformations of graphs.
\item Calculate the following limits, if they exist. If they do not exist, explain why.
\begin{enumerate}
\item $\displaystyle{\lim_{x\to -1}} \frac{3x^2-6x-9}{x+1}$,
\item $\displaystyle{\lim_{x\to 0}} \frac{-2x^2-7x+10}{-x+4}$,
\item $\displaystyle{\lim_{x\to 3}} \frac{x^2+2}{2x-6}$.
\end{enumerate}
\item Calculate $\displaystyle\frac{\partial^2f}{\partial x \partial y}$ and $\displaystyle\frac{\partial^2f}{\partial y \partial x}$ and check that they are equal in the following cases :
\begin{enumerate}
 \item $f(x,y)=6x^2y^3-8x+12y^2$,
 \item $\displaystyle f(x,y)=\frac{7y-5x}{x}+e^x$,
 \item $f(x,y)=\ln(x)e^y$.
\end{enumerate}
\item What is the average value of $f(x)=e^{2x}-3x^2+x$ on the interval $[-1,2]$?
\item Use geometry to calculate the shaded area in the picture.

\begin{center}
\includegraphics[scale=1]{Function.pdf}
\end{center}

\pagebreak

\item Calculate the following integrals 
\begin{enumerate}
 \item $\displaystyle\int \ln(x)x^2 dx$,
 \item $\displaystyle\int 2e^xx^3 dx$,
 \item $\displaystyle\int (x^4-1)\ln(x) dx$.
\end{enumerate}

\item The table shows the variable cost and profit of producing and selling $q$ items.

\renewcommand{\arraystretch}{2}
\begin{tabular}{|c||c|c|}
\hline
Number of items ($q$) & Variable cost ($TVC(q)$) & Profit ($P(q)$) \\
\hline
\hline
$2$ & \$$1,300$ & \$$1,000$\\
\hline
$4$ & \$$1,800$ & \$$1,500$\\
\hline
$6$ & \$$2,200$ & \$$1,800$\\
\hline
$8$ & \$$4,000$ & \$$1,900$\\
\hline
$10$ & \$$4,200$ & \$$2,500$\\ 
\hline
\end{tabular}

Using this table, and assuming that the fixed cost is $\$2,500$, calculate or approximate :
\begin{enumerate}
 \item the total cost for producing $6$ items,
 \item the revenue for selling $8$ items,
 \item the marginal profit for selling $10$ items,
 \item the average revenue for selling $4$ items.
\end{enumerate}
\item Let $\displaystyle f(x)=\frac{5-x}{3x-4}$ and $\displaystyle g(x)=\frac{x}{3}$. Calculate and simplify if necessary 
\begin{enumerate}
\item $(f \circ g)(x)$, 
\item $(g \circ f)(x)$, 
\item $(f \circ f)(x)$, 
\item $(f + g)(x)$, 
\item $\displaystyle (\frac{f}{g})(x)$.
\end{enumerate}
\item Solve the inequality $(x-3)(x+2)(x-3)(x-5)<0$.
\end{enumerate}
\end{document}          