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\begin{document}
\begin{center}
\bf Math 119\\
Final Exam\\
December 16, 1998
\end{center}


\begin{problems}
\item
 Find the domain of $f(x) =
%\Ds{\sqrt[ {\bf 3}]{\frac{(x-1)^2}{x-2}}}$.
\Ds{\left({\frac{(x-1)^2}{x-2}}\right)^{1/3}}$.


\begin{itemize}
\item[{\bf (a)}]
         \quad $(1, 2)$ 
\item[{\bf (b)}]
         \quad $(1, \infty) $ 
\item[{\bf (c)}]
         \quad $[2, \infty)$
\item[{\bf (d)}]
        \quad $(- \infty, 2) \cup (2, \infty) $
\item[{\bf (e)}]
         \quad $(-\infty, 1) \cup (2, \infty)$
\end{itemize}



\item
Find $\displaystyle{\lim_{x\to 0} \frac{\sqrt{x^2+3} -
\sqrt{3}}{x}  }$.


\begin{itemize}
\item[{\bf (a)}] 
  \quad $\Ds{\frac{1}{\sqrt{3}}} $
\item[{\bf (b)}] 
  \quad $\Ds{\frac{1}{3\sqrt{3}}} $
\item[{\bf (c)}] 
  \quad $\sqrt{3} $
\item[{\bf (d)}] 
  \quad $\Ds{\frac{1}{\sqrt{6}}} $
\item[{\bf (e)}] 
  \quad 0
\end{itemize}







\item
Find $\displaystyle{\lim_{x\to -1} \frac{x+1}{x^2-x-2} }$.

\begin{itemize}
\item[{\bf (a)}] 
  \quad $- \Ds{\frac{1}{3}} $ 
\item[{\bf (b)}] 
  \quad $- \Ds{\frac{1}{2}} $ 
\item[{\bf (c)}] 
  \quad $-1$ 
\item[{\bf (d)}] 
  \quad $- \Ds{\frac{2}{3} }$ 
\item[{\bf (e)}] 
  \quad $- \Ds{\frac{3}{2}} $ 
\end{itemize}

\newpage
The next 2 problems concern the following graph of the
function $f(x)$.
\begin{center}
\scalebox{.6}{\includegraphics{final_graph.eps} }
\end{center}

\item
 Which of the following is closest to the value of
 $f'(13)$?

\begin{itemize}
\item[{\bf (a)}]  
  \quad 2.5  
\item[{\bf (b)}]  
  \quad  0.9  
\item[{\bf (c)}]  
  \quad 0.4 
\item[{\bf (d)}]  
  \quad -2.5 
\item[{\bf (e)}]   
  \quad The derivative doesn't exist.  
\end{itemize}



\item
Only one of the following assertions is true; find
the true statement.

\begin{itemize}

\item[{\bf (a)}] 
  \quad $f'(x)$ is non-negative for $0 \le x \le 13$. 
\item[{\bf (b)}] 
  \quad $f(12)$ is larger than $f(13)$.
\item[{\bf (c)}] 
  \quad $f'(10)$ is smaller than $f'(13)$. 
\item[{\bf (d)}] 
  \quad $f'(15)$ is larger than $f'(13)$. 
\item[{\bf (e)}]  
  \quad  $f'$ is an increasing function on the interval $[0,13]$.
\end{itemize}

\item
Find the coordinates of the point on the curve
\[
y = x^2 + x + 1
\]
where the tangent line is parallel to the line $y = 5x + 2$.

\begin{itemize}
\item[{\bf (a)}] 
  \quad $(1, 3)$ 
\item[{\bf (b)}] 
  \quad $(2, 7)$ 
\item[{\bf (c)}] 
  \quad $(0, 1)$ 
\item[{\bf (d)}] 
  \quad $(-1, 1)$
\item[{\bf (e)}] 
  \quad $(-2, 3)$
\end{itemize}



\item
Let $f(x) = 2x^2+x+3$ and let $h \not = 0$. Which of the
following represents the quantity:
\[\frac{f(x+h) - f(x)}{h}\]

\emph{Note: This is not a limit.}


\begin{itemize}
\item[{\bf (a)}] 
  \quad $4x+2h +h^2$ 
\item[{\bf (b)}] 
  \quad $4x+1$ 
\item[{\bf (c)}] 
  \quad $4x+2h+1$ 
\item[{\bf (d)}] 
  \quad  $2x^2 +x-2h$ 
\item[{\bf (e)}] 
  \quad  $4x+2h^3+1 $ 
\end{itemize}




\item
 A stone is dropped into a lake, creating a circular ripple
with radius increasing at a rate
 of 20 in/sec. Find the rate at which the
area within the circle is increasing when the radius is 3 in.

\begin{itemize}
\item[{\bf (a)}] 
  \quad $6 \pi$ 
\item[{\bf (b)}] 
  \quad $18\pi$ 
\item[{\bf (c)}] 
  \quad $60 \pi$ 
\item[{\bf (d)}]  
  \quad $100 \pi$ 
\item[{\bf (e)}] 
  \quad $120 \pi$ 
\end{itemize}

\newpage

\item
 Find  $\displaystyle{ \frac {d\ }{dx}\left[\sin(x^2+3)\right]. }$

\begin{itemize}
\item[{\bf (a)}]
  \quad $\cos(2x)$ 
\item[{\bf (b)}] 
  \quad $2x\cos(x^2+3)$
\item[{\bf (c)}] 
  \quad $\cos(2x^3+6x)$
\item[{\bf (d)}] 
  \quad $-\cos(x^2+3)$
\item[{\bf (e)}] 
  \quad $ \cos(x^2+3)+2x$
\end{itemize}




\item
Find  $\displaystyle{\frac{d}{dx} \left [
\frac{x+1}{(x^2+3)^2} \right ] }$.

\begin{itemize}
\item[{\bf (a)}] 
  \quad $\displaystyle{\frac{-3x^2 - 2x+1}{(x^2+3)^3} }$ 
\vspace{2ex}
\item[{\bf (b)}] 
  \quad $\displaystyle{\frac{5x^2-2x+1}{(x^2+3)^3} }$ 
\vspace{2ex}
\item[{\bf (c)}] 
  \quad $\displaystyle{\frac{-x^3 - x^2 + x-3}{(x^2 + 3)^3} }$
\vspace{2ex}
\item[{\bf (d)}] 
  \quad $\displaystyle{\frac{1}{4x(x^2 + 3)} }$ 
\vspace{2ex}
\item[{\bf (e)}]  
  \quad $\displaystyle{\frac{3-4x-3x^2}{(x^2 + 3)^3} }$
\end{itemize}




\item
Find the absolute maximum and absolute minimum values
of the function 
$$f(x) = x^4 - 4x + 8$$
 on the interval $[0,2]$.

\begin{itemize}
\item[{\bf (a)}] 
  \quad  maximum value 16, minimum value 5
\item[{\bf (b)}] 
  \quad  maximum value 16, minimum value 1
\item[{\bf (c)}] 
  \quad  maximum value 2, minimum value 0
\item[{\bf (d)}] 
  \quad  maximum value 8, minimum value 5
\item[{\bf (e)}]  
  \quad maximum value 16, minimum value 8
\end{itemize}

\newpage

\item If $f(x) = x \sin (2x)$, find the second derivative
$f^{\prime \prime} (x).$

\begin{itemize}
\item[{\bf (a)}] 
  \quad $f^{\prime \prime} (x) = \cos (2x) $
\item[{\bf (b)}] 
  \quad  $f^{\prime \prime} (x) = 4 \cos (2x) $
\item[{\bf (c)}] 
  \quad  $f^{\prime \prime} (x) = 4 \cos (2x) - 4x \sin (2x)$
\item[{\bf (d)}]  
  \quad $f^{\prime \prime} (x) = 3 \cos x - 4x \sin x$
\item[{\bf (e)}]  
  \quad $f^{\prime \prime} (x) = 2 \cos (2x) + 4x \sin (2x)$
\end{itemize}
\vspace{1cm}

\item
 Let $x^2 + y^2 \sin x = 2$.  What is
$\displaystyle{\frac{dy}{dx} }$?

\begin{itemize}
\item[{\bf (a)}] 
  \quad $\displaystyle{ \frac{dy}{dx} = - \,\frac{x}{y \sin x}}$
\vspace{2ex}
\item[{\bf (b)}] 
  \quad $\displaystyle{ \frac{dy}{dx} =  2 - \frac{y^2 \cos
x + 2x}{2y \sin x} }$
\vspace{2ex}
\item[{\bf (c)}] 
  \quad $\displaystyle{ \frac{dy}{dx} = 2x + 2y \sin x
    }$
\vspace{2ex}
\item[{\bf (d)}] 
  \quad $\displaystyle{ \frac{dy}{dx} = - \,\frac{y^2 \cos x +
2x}{2y \sin x} }$
\vspace{2ex}
\item[{\bf (e)}] 
  \quad $\displaystyle{ \frac{dy}{dx} = \frac{2-x^2}{y \sin x} }$
\end{itemize}

\item  Suppose that for $x \in [0,2\pi]$, we define $g(x) = \sin^2
 x$. 
 On what intervals is $g$ increasing?


\begin{itemize}
\item[{\bf (a)}] 
  \quad  $[0, 2\pi ]$
\item[{\bf (b)}] 
  \quad  $\displaystyle{ \Big[0, \frac{\pi}{2} \Big] \cup \Big[ \pi, \frac{3\pi}2 \Big ] }$
\item[{\bf (c)}]   
  \quad $\displaystyle{\Big [\frac{\pi}{2},  {\pi}\Big]
     \cup \Big[ \frac{3\pi}{2}, 2 \pi\Big] }$
\item[{\bf (d)}] 
  \quad  $\displaystyle{\left [0, \pi \right ] }$
\item[{\bf (e)}] 
  \quad  $\displaystyle{\left [\pi, 2\pi \right ] }$
\end{itemize}


\item
 A balloon is rising vertically from a point $A$.  It has
constant speed of  5m/sec.  A girl is watching from a position exactly
60m from $A$.  How fast is the distance between the girl and the balloon
increasing when the balloon is 80m high?

\unitlength1cm
\mbox{}
\hspace{5cm}%% maybe you want to adjust here for inserting
\begin{picture}(10,4)
\put(0,1){\line(1,0){6}}
\put(3.5,0.9){\circle*{0.2}}
\put(3.4,0.5){\small{A}}
\put(3.5,2.5){\circle*{0.2}}
\put(3.7,2.4){\small{balloon}}
\put(3.5,1){\vector(0,1){2.3}}
\put(1,0.9){\circle{0.2}}
\put(0.7,0.5){\small{girl}}
\put(1.9,1.1){\small{60}}
\end{picture}
\begin{itemize}
\item[{\bf (a)}] 
  \quad 2m/sec.
\item[{\bf (b)}] 
  \quad 4m/sec.
\item[{\bf (c)}] 
  \quad  6m/sec.
\item[{\bf (d)}] 
  \quad 8m/sec.
\item[{\bf (e)}] 
  \quad  10m/sec.
\end{itemize}




\item
On what intervals is $\displaystyle{f(x) = \frac{3x-2}{4x-3}
}$ concave up?

\begin{itemize}
\item[{\bf (a)}] 
  \quad  $\displaystyle{ \left (-\infty, \frac{3}{4} \right ) }$
\item[{\bf (b)}]  
  \quad $\displaystyle{\left (\frac{3}{4},  \infty\right ) }$
\item[{\bf (c)}]  
  \quad none
\item[{\bf (d)}]  
  \quad $\displaystyle{\left (-\infty, \frac{3}{4}\right ) \cup
\left (\frac{3}{4},\infty\right ) }$
\item[{\bf (e)}] 
  \quad  $(-\infty, \infty)$
\end{itemize}




\item
Compute $\displaystyle{\lim_{x\to \infty}
\frac{3x^3}{\sqrt{3x^3 + 1}} }$

\begin{itemize}
\item[{\bf (a)}]  
  \quad $- \infty$
\item[{\bf (b)}]  
  \quad  0
\item[{\bf (c)}]  
  \quad 1
\item[{\bf (d)}]  
  \quad  $\sqrt{3}$
\item[{\bf (e)}] \quad $\infty$
\end{itemize}


\item
Consider $\displaystyle{f(x) = \frac{4}{x}+x.  }$
Then $f$ has

\begin{itemize}

\item[{\bf (a)}]   
  \quad local minima at $x=2$ and $x= -2.$
\item[{\bf (b)}]   
  \quad  local maxima at $x=2$ and $x = -2$.
\item[{\bf (c)}]  
  \quad  a local minimum at $x= -2$ and a local
  maximum at $x=2$.
\item[{\bf (d)}]  
  \quad a local minimum at $x=2$ and a
  local maximum at $x= -2.$
\item[{\bf (e)}]  
  \quad no local extrema.
\end{itemize}
\vspace{3cm}

\item
 The product of two positive numbers is $400$.  What is the minimum their
{\bf sum} can be?

\begin{itemize}
\item[{\bf (a)}]  
  \quad $-20$ 
\item[{\bf (b)}] 
  \quad  $10$ 
\item[{\bf (c)}] 
  \quad $40$ 
\item[{\bf (d)}] 
  \quad $50$
\item[{\bf (e)}] 
  \quad  $100$ 
\end{itemize}
\vspace{2cm}

\item
$\displaystyle{\sum _{i=1}^{100}(3i+1) = }$

\begin{itemize}
\item[{\bf (a)}] 
  \quad $15150$    
\item[{\bf (b)}] 
  \quad $15250$ 
\item[{\bf (c)}] 
  \quad $100200$ 
\item[{\bf (d)}] 
  \quad $151250$ 
\item[{\bf (e)}] 
  \quad $300100$
\end{itemize}


\newpage

\item
The graph below is the graph of the {\bf derivative} of  a function
\begin{math}
  f(x).
\end{math}


\begin{center}
\resizebox{1.5in}{!}{\rotatebox{270}{\includegraphics{grapht01.eps}}}\qquad
\end{center}

\bigskip
\noindent
Which of the following graphs best represents the graph of
\begin{math}
f(x)
\end{math}?

\medskip
\noindent
{\bf (a)}~\resizebox{1.5in}{!}{\rotatebox{270}{\includegraphics{grapht03.eps}}}\qquad
{\bf
  (b)}~\resizebox{1.5in}{!}{\rotatebox{270}{\includegraphics{grapht04.eps}}}\qquad
{\bf (c)}~\resizebox{1.5in}{!}{\rotatebox{270}{\includegraphics{grapht02.eps}}}\qquad
\newline
{\bf (d)}~\resizebox{1.5in}{!}{\rotatebox{270}{\includegraphics{grapht05.eps}}}\qquad
{\bf (e)}~\resizebox{1.5in}{!}{\rotatebox{270}{\includegraphics{grapht06.eps}}}




\item
Find  the area under the graph of $y=x^3+1$ above  the
interval
from $0$ to $2$ on the $x$-axis.  The region is shown in the picture
below,
which is not to scale.

\resizebox{2in}{!}{\rotatebox{270}{\includegraphics{area01.eps}}}\qquad

\begin{itemize}
\item[{\bf (a)}] 
  \quad $ 1$
\item[{\bf (b)}] 
  \quad $4.5$
\item[{\bf (c)}] 
  \quad $6$
\item[{\bf (d)}] 
  \quad $7$
\item[{\bf (e)}] 
  \quad $9$
\end{itemize}

\newpage

\item
Consider the function 
\begin{math}
  f(x) = x^2 + 1
\end{math}
on the interval 
\begin{math}
  [1,2].
\end{math}
Use
the partition 
\begin{math}
 \displaystyle{\{1,\frac 32, 2\}}.
\end{math}
Take
\begin{math}
  x_i^*
\end{math}
to be
the right endpoint of the subinterval 
\begin{math}
  [x_{i-1},x_i].
\end{math}
Calculate the Riemann sum 
\begin{math}
  \displaystyle{\sum _{i=1}^n f(x_i^*)\Delta x_i}
\end{math}
where 
\begin{math}
  \displaystyle{\Delta x_i=x_i-x_{i-1}}.
\end{math}

\begin{itemize}
\item[{\bf (a)}] 
  \quad $\displaystyle{\frac {33}8}$
\item[{\bf (b)}] 
  \quad $\displaystyle{\frac {23}4}$
\item[{\bf (c)}] 
  \quad $\displaystyle{\frac {21}8}$
\item[{\bf (d)}] 
  \quad $\displaystyle{\frac {10}3}$
\item[{\bf (e)}] 
  \quad\  $\displaystyle{4}$
\end{itemize}
\vspace{1cm}

\item 
Calculate 
\begin{math}
  \displaystyle{\int (x^2-5x)\,dx}.
\end{math}

\begin{itemize}
\item[{\bf (a)}] 
  \quad $x-5$
\item[{\bf (b)}] 
  \quad $\displaystyle{-\frac {13}6}$
\item[{\bf (c)}] 
  \quad $\displaystyle{\frac {x^3}3 -\frac 52 x^2 +C}$
\item[{\bf (d)}] 
  \quad ${x^2-5x+C}$
\item[{\bf (e)}] 
  \quad $\displaystyle{\frac{x^3}2 -5x^2 +C}$
\end{itemize}



\item
Calculate 
\begin{math}
  \displaystyle{\int _{-1}^2(2x-1)\,dx}.
\end{math}

\begin{itemize}
\item[{\bf (a)}]  
  \quad 4
\item[{\bf (b)}]  
  \quad $\displaystyle{\frac 32}$
\item[{\bf (c)}]  
  \quad 0
\item[{\bf (d)}]  
  \quad 6
\item[{\bf (e)}]  
  \quad 2 
\end{itemize}


\end{problems}

\end{document}





\end{document}



