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  MIDTERM 1 &\huge TQS 120  &\normalsize Practice\\
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\vskip .5 in
NAME: \normalsize This is a sample midterm to be used for practice.  This is \emph{not} a template for the midterm that will be given in class.  Many of the questions on the Midterm will look quite different than those appearing here.


\begin{enumerate}

\item TRUE/FALSE: Circle T in each of the following cases if the statement
is \emph{always} true.  Otherwise, circle F.
\newline
\newline
\noindent{Let $x$ and $y$ be positive numbers.}

\begin{tabular}{lll}
~\\
T & F & $\log(x+y) = \log(x) \cdot \log(y)$ \\
~\\
T & F & $\log(\frac{x}{y}) = \log(x) - \log(xy)$ \\
~\\
T & F & $\frac{\log x}{\log y}=\frac{x}{y}$ \\
~\\
T & F & $\log_2 5x^7=7\log_25x$ \\
~\\
T & F &  $\log({\log({10})})=0$.\\
~\\
T & F & For all numbers $z$, $\sqrt{z^2}=z$\\
~\\
T & F & The function $f$ with the rule $f(x)=-2^{-x}$ is increasing. \\
~\\
\end{tabular}

LONG ANSWERS: Show all your work and circle you final answer.  Correct
answers will \emph{not} get credit without supporting work.

\item  Given $-x=\frac{2xy}{2y-1}$, solve for $y$.
\vskip 1.5in


\newpage
\item $[2]$ Define the rule of the function $\log$.
\vskip 1.5in

\item $[4]$ Assume $b$, $x$,$y>0$, simplify the following:

$\displaystyle{\frac{(b^x)^{x-1}}{b^{-x}}}$
\hspace{3in}$\displaystyle{\frac{\sqrt[3]{x^2}(y^2)^{\frac{3}{2}}}{x^{\frac{2}{3}}y^2}}$
\vfill

 $2-\log_5 (25z)$ \hspace{3.5in}  $\log_2\frac{1}{4}$

\vfill

\newpage

\item $[3]$ Find $x$ in the following:

$2^{4x-1}=3^{1-x}$  \hspace{3in} $5^x=2$
\vskip 2.5in

\item Find a formula for the inverse function $f^{-1}$ of the indicated function $f$.

$\displaystyle{f(x)=4x^{\frac{3}{7}}-1}$ \hspace{3.5in}  $f(x)=3\cdot2^x+4$
\vskip 2in
\item Write the given expression as a single logarithm.

$2\ln 2x - 3(\ln x^2 +\ln x)$ \hspace{3in} $3-\log_6(36y)$

\vskip 2in

\newpage 






\item $[4]$ Solve for $x$: 

$\log (x-16)=2-\log (x-1)$ \hspace{3in} $4^x-3*2^x=10$
\vskip 1.5 in 


 \item $[]$ Chani has \$20,000 in student loans at the end of her college education with 8\% interest compounded quarterly.  Chani chooses not to make payments after graduation since she has had trouble securing employment.

\begin{enumerate}
\item How much money will Chani owe after 2 years assuming no extra fees will be incurred?
\vskip 1in
\item How long will it take for Chani's loan to double?
\vfill
\end{enumerate}

\newpage


\item  Let $h(x)=\log_2 x$ and $f(x)=\log_2(x+3)+1$.
\begin{enumerate}
\item  List the transformations needed to transform the graph of $h$ to the graph of $f$. 
\vskip 1in
\item Graph $f$. 

\includegraphics[height=3in]{blank_5}

\item Find the inverse function to $f$.
\vskip 1.5in
\item Graph the inverse function to $f$.
\end{enumerate}

\newpage
 


\item Chemists use the pH scale to measure the acidity or baseness of liquids.  The system measure the concentration of hydromium ions (H$_3$O$^+$), denoted by [H$_3$O$^+$] and then uses the following function to determine the liquids pH level:
$$-\log \text{ [H$_3$O$^+$]}=\text{pH level of liquid}$$

Given that vinegar has a pH level of 3.1, how many times stronger is stomach acid (which a pH level of 1.6)?

\vfill
\item  
\vfill

\newpage
\item Paul Atreides is in this course and would like to know if it is still possible to earn a 2.5 now that he's taken two exams.   He has looked at the gradebook on catalyst and has computed the averages listed below. 

Assuming Paul's work does not drastically change in the remaining 3 weeks and his averages remain about the same, find what grade he needs to get on the final to receive a 2.5 in the course.  In case you don't remember, the weights specified in the syllabus and the function $f$ that takes your class percentage $x$ and returns your score on a 4. scale are also provided below.

\vspace{ .3in} \hspace{3.8in} $f(x)=\begin{cases}
				4.0 &\text{ if } 95<x\\ 
				.1x-5.5 &\text{ if }62\leq x\leq 95\\
				0 &\text{ if } x<62\\
			\end{cases}$

\vspace{-1in}

\begin{center}
\begin{tabular*}{6in}{l@{\extracolsep{.3 
in}}cc@{\extracolsep{1 in}}}
~     & weight &Paul's ave        \\

Mini-Quizzes (top score)   & 5\% &95\%	\\
WeBWork assignments        & 10\% 	&98\%\\
Handwritten assignments &  15\%  &84\%\\	
Quizzes   		& 15\% &61\%\\
2 Midterms     	& 30\% 	&73\%	\\
Final                	& 25\% 		\\		

\end{tabular*}
\end{center}


\end{enumerate}
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