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\newcommand{\solution}[1]{\textbf{\color{red}#1}} \begin{document}
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%**********************TITLE PAGE*********************************
\vspace{2in} \begin{large} 

{\bf Name:} \underline{Bill A. Student}\\ 

{\bf Student Identification Number:} \underline{999-9999}\\ 

{\bf TA's Name:} \underline{Sally T. Assistant}\\ \vspace{1cm} 

\begin{center} {\bf Calculus I\\ SKILLS TEST I} \\ Practice Test\\ (Key)
\end{center} 

\end{large} \vspace{3cm} 

{\bf Directions:} You have two hours to complete this test. READ ALL
DIRECTIONS CAREFULLY! Partial credit will be given, but ONLY IF YOU SHOW
YOUR WORK! \\ \vspace{2cm} 

\noindent Unless otherwise specified, you may assume that each problem
is worth 5 points. Please indicate ALL answers clearly by drawing a box
around the final answer.\\ 

\newpage 
\begin{enumerate} 
\mitem (2 points each) {\bf TRUE OR FALSE:} For each of the following,
indicate whether the statement is true or false. 

\begin{enumerate} \mmitem $\sqrt[3]{x^3}=|x|$ ~~~\textbf{\color{red}False:
when x is negative, so is its cube root.} 

\mmitem $\ln{(A^n+B)}=n \ln{A} + \ln{B}$ ~~~\textbf{\color{red}False: the
log of the sum is not the sum of the logs} 

\mmitem $\cos{(2x)}=1-2 \, \mbox{sin}^2(x)$ ~~~\textbf{\color{red}True:
this follows from the identity $\sin^2(x) + \cos^2(x)=1$ and the
addition formula $cos(x+y)=\sin(x)\sin(y)-\cos(x)\cos(y)$.} 

\mmitem Any rational function has a domain of all real numbers.
~~~\textbf{\color{red}False: the domain does not include numbers $x$
where the function takes on the value 0.} 

\mmitem If $f(x)$ is a polynomial such that $f(a)=0,$ then $x-a$ is a
factor of $f(x).$ \textbf{\color{red}True: This is called the remainder
theorem.} \end{enumerate} 

\mitem What is the {\bf Domain} of $f(x)=\tan(x).$ 

\solution{Since the tangent is the sine over the cosine, its domain is
the set of all numbers $x$ where $\cos(x) \neq 0$.} 

\mitem What is the {\bf Range} of the function $f(x)=x^2 - 4.$ 

\solution{By inspection, the minimum value is $f(0)=-4$, and the range
is the closed ray $[0, \infty)$.} 

\mitem Which of the following functions has a {\bf Range} which is equal
to its {\bf Domain}? 

\begin{enumerate} 

\item $f(x)=\bfrac{1}{x^2}$ \solution{~the domain is all $x\neq 0$, the
range is all $x>0$} 

\item $f(x)=\sin{x}$ \solution{~domain all real $x$, range $[-1,1]$.} 

\item $f(x)=\bfrac{1}{x}$ \solution{~domain all $x \neq 0$, range all $x
\neq 0$} 

\item $f(x)=5$ \solution{~dom all $x$, range $\{5\}$.} 

\item None of the above\\ 

\end{enumerate} 

\solution{The function $\bfrac{1}{x}$ has domain and range the same.} 

\mitem Do the lines given by the equations $7y+3x-17=0$ and $3y-2x+15=0$
ever intersect? If so, where? If not, why not? 

\solution{The first line has slope $\bfrac{-3}{7}$ and the second has
slope $\bfrac{2}{3}$, so they will intersect. The point of intersection
is $\{y = \bfrac{-11}{23}, x = \bfrac{156}{23}\}$, obtained by solving
simultaneously.} 

\mitem Find the equation of the line whose $x$-intercept is 10 and whose
$y$-intercept is 5. 

\solution{An equation for this line is $\bfrac{x}{10 }+ \bfrac{y}{5} =
1$, as is easily checked by substituting in the point $(10,0)$ and
$(0,5)$.} 

\mitem Which of the following functions is {\bf even}? 

\begin{enumerate} 

\item $f(x)=\sin{x}$ ~\solution{ the sin function is odd, since
$\sin(-x)=-\sin(x)$.} 

\item $f(x)=\bfrac{\sin^2(x)}{\cos(x)}$ \solution{ ~This function is
even: It is the product of two even functions.} 

\item $f(x)=\bfrac{1}{x}$ 

\item $f(x)=\ln{(x^2-2)}$ 

\item None of the above 

\end{enumerate} 

\solution{The first even function in the list is item b.} 

\mitem (2 pts each) Given that $\ln{A}=f,$ $\ln{B}=g,$ and $\ln{C}=h,$
evaluate the following expressions in terms of $f,~g,$ and $h.$ 

\begin{enumerate} 

\item $\ln {ABC} =$ \solution{$\ln{A} + \ln{B} + \ln{ C} = f + g + h$}. 

\item $\ln{((AB)^2)}=$ \solution{$2 \ln(AB) = 2 (\ln(A) + \ln(B)) =
2(f+g) $}. 

\item $\ln(\bfrac{1}{\sqrt{C}}) = $ \solution{$- \ln(\sqrt{ C}) = -
\bfrac{1}{2} h$. }

\end{enumerate} 

\item If $f(g(x))=(x^2+4x+4) + 2x + 4$ and $g(x)=x+2$, what is $f(x)$? 

\begin{enumerate} 

\item $f(x)=(x^2+4x+4) + 2x + 2$ 

\item $f(x)=x^2+4$ 

\item $f(x)=x^2+2x$ 

\item $f(x)=(x+2)+2$ 

\item None of the above\\ 

\end{enumerate} 

\solution{By inspection, the most likely candidate is item c
$f(x)=x^2+2x$. We check by composing $$f(g(x)) = (g(x))^2 + 2 g(x) =
(x+2)^2 + 2(x+2) = (x^2+4x + 4) + 2x + 4$$ which checks out.} 

\item If $h(x)=\sqrt{x}$, $k(x)=x^2,$ and $l(x)=x+1$, what composition
of the functions $h,~k$ and $l$ would give us $|x+1|$? 

\solution{We know that $\sqrt{x^2} =|x|$, so the answer is 
$h(k(l(x+1)))=|x+1|$.}

\item Given the graph of {\bf any} function $f(x)$, which of the
following choices would produce a graph that is identical in shape to
$f(x)$ but that is moved 2 units up and 2 units to the left? 


\begin{enumerate} 

\item $f(x)+2$ \solution{~Here the graph of $f$ is raised up 2 units.}

\item $f(x)-2$ \solution{~Here it's lowered 2 units.}

\item $f(x+2)+2$ \solution{~Here the graph   is shifted 2 units
to the right and    up 2 units.}

\item $f(x-2)+2$ \solution{~Here the graph  is shifted 2 units
to the left and  raised up 2 units.}

\item None of the above  
\end{enumerate} 

\solution{item d}

\item Find the following limits.   
\begin{enumerate}  

\item $\limit{x}{2} \bfrac{x^{2}-4}{x-2}= $ 
\solution{ $\lim_{x \to 2} \bfrac{(x-2)(x+2)}{x-2}= \lim_{x\to 2}
x +2 = 4$.}

\item $\limit{x}{2} \bfrac{x^{2}-4}{x^{2} + 8}=$
\solution{$\bfrac{2^2 - 4}{2^2 + 8} = \bfrac{0}{12} =0$.}  
\item $\limit{x}{0} 12\pi$ \\ \vspace{.5in} \end{enumerate} 

\item Find all points where the graphs of the functions $f(x)=x^2+2$ and
$g(x)=2x+1$ intersect.\\ 

\solution{Solve  $f(x)=g(x)$ or $x^2 - 2x +1=0$.  This factors into
$(x-1)^2=0$ and so the solution is $x=1$. The graphs intersect in the 
point $(1,3)$.}

\item (2 points each)  From their general shapes match each of the 
following functions with one of the graphs.  Write down the color
of the correct graph.


\begin{enumerate}

\item \underline{\solution{red}}   $f(x) = x^3$\\
\item \underline{\solution{green}}   $f(x) = \sin(x)$\\
\item \underline{\solution{yellow}}   $f(x) = \bfrac{1}{x} $\\
\item \underline{\solution{blue}}   $f(x) = |x| $\\

\end{enumerate}

\includegraphics[height=5in,width=4in,angle=-90]{funcs.eps}

\end{enumerate} 
\end{document}
