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\noindent
{\bf Math 1030 Workshop \#5 (Due: July 27${}^{\mathrm{th}}$, 2016)} \hfill
{\bf Name: \underline{\hspace*{2.5in}}}

\vspace{0.05in}

\noindent
Please turn in a paper copy and {\bf SHOW YOUR WORK!}

\vspace{0.05in}

\begin{enumerate}
   \item Consider the function 
$\displaystyle{ f(x) = \left\{ \begin{array}{cc}
 \displaystyle 3x^3+9x^2+5x+3       & x \leq -1  \Large\strut \\
 \displaystyle (7x^4-6x)e^{-x^2} & -1 < x \leq 1 \Large\strut \\
 \displaystyle  x^2-5x+2\ln(x)  & x > 1  \Large\strut
 \end{array} \right. }$

Be careful! Wolfram Alpha may have trouble with this function. You might want to deal with it one piece at a time.

   \begin{enumerate}
   \item Find all of the critical points of $f(x)$.
             \qquad $x =$ \underline{\hspace{3in}}

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   \item Restricting our attention to the interval $[-2,4]$...
   
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         The maximum value of $f(x)$ is \underline{\hspace{1.25in}}. This occurs when $x=$ \underline{\hspace{1.25in}}.

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         The minimum value of $f(x)$ is \underline{\hspace{1.25in}}. This occurs when $x=$ \underline{\hspace{1.25in}}.
   \end{enumerate}

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  \item Riverdale Highschool has an expensive copying machine. It could cost \$7,000 to replace their machine. They have recorded that they spent \$50 reparing the copier during its first year of operation. The next year it cost \$190 to fix their machine. The third year it cost \$350 to fix it. 

Use Excel to find a power model for the \underline{\bf average} annual repair costs. Then model the average annual cost of operating this copying machine using a function of the form: $\displaystyle{A(t) = \frac{C}{t} + Rt^r}$ where $C$ is the cost of purchasing the copier and $Rt^r$ models the repair costs. 
            
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\hspace{2in}            $A(t) =$ \underline{\hspace{4in}}
            
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             When $t=$ \underline{\hspace{1in}}, $A(t)$ is minimized. [Keep 5 decimal places.]
             
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            Riverdale should replace its copying machine after \underline{\hspace{1in}} years and \underline{\hspace{1in}} months of opertion. \\
            {}[Round up to the next whole month.]
            
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            If they do this, their average annual cost (for the copier) should be \$ \underline{\hspace{1.5in}}.                       

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   \item Stew's business uses specialty print cartridges for their photo ID printer each year. These cartridges cast \$150 a piece. Placing an order costs \$10. Stew's inventory cost is rather low. He estimates that it costs him \$0.10 per year per cartridge (based on average inventory with all of the standard assumptions). Let $C(x)$ be Stew's annual cost function for these cartridges.

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   \begin{enumerate}
      \item If Stew needs 200 cartridges each year, \quad $C(x)=$ \underline{\hspace{3in}}

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             His {\bf ideal} EOQ is \underline{\hspace{1in}} and his {\bf ideal} minimum annual cost is \underline{\hspace{2in}}.
   
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      \item If Stew needs 1,000 cartridges each year, \quad $C(x)=$ \underline{\hspace{3in}}

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             His {\bf ideal} EOQ is \underline{\hspace{1in}} and his {\bf ideal} minimum annual cost is \underline{\hspace{2in}}.
   
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   \item Suppose that Stew needs 2,000 cartridges each year. In addition, he found out that he gets a discount if he places a large order. For orders of 300 or more, he pay \$135 per cartridge. However, there is an additional delivery fee for these large orders. Instead of \$10, it costs \$50 to place a large order. His inventory costs stay the same.

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             $\displaystyle{C(x) = \left\{ \Huge\strut \right.}$ 

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             His {\bf ideal} EOQ is \underline{\hspace{1in}}. His {\bf ideal} minimum annual cost is \underline{\hspace{2in}}.

   \end{enumerate}

\end{enumerate}

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