\documentclass[11pt,pages]{exam}
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\title{Take Home Exam 1}
\target{Math 158: Spring 2013}

\begin{document}


\begin{exam}{Due:  Tuesday, March 11}



\bp \textbf{[2 points]} Here's a picture of the elliptic curve
$y^2=x^3-5x+8$.  The coordinates of the black point are $(1,2)$.   What are the coordinates of the red point?

\begin{center}
\includegraphics[scale=0.5]{elliptic.pdf}
\end{center}
\ep



 \bp \textbf{[2 points]}  Factoring the identity function.
\ben
\item Find two functions $f$ and $g$ satisfying $f(0)=0$,
 $g(0)=0$ and $x=f(x)g(x)$.  
\vspace{3in}
\item Prove that there do not exist two differentiable functions $f$
  and $g$ with $f(0)=0$,
 $g(0)=0$ and $x=f(x)g(x)$.  
\een

 \ep

\bp \textbf{[2 points]}  Suppose that $f:[0,1]\to [0,1]$ is differentiable and that
$f'(x)\neq 1$ for any $x$.  Prove that there is one and only one
number $c\in [0,1]$ with $f(c)=c.$
\ep


\bp \textbf{[2 points]}  The standard normal curve is given by $$\ds
y=\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}.$$  Give a good sketch of
this curve.  Be sure to indicate the points at which the curve changes concavity.
\ep



\bp \textbf{[2 points]}  For any $x>0$, let $\ds A(x)=\int_2^x f(t)dt$ where $f$ is the
function whose graph is sketched below:

\begin{center}
\includegraphics{plot_of_f.pdf}
\end{center}

Give a good sketch of the graph of $A$.  Indicate the value of $A$ and
$A'$ at $x=0,1,2,3,4,5,6,7,8$ and indicate on which intervals $A$ is
concave up and concave down.
\ep


\bp \textbf{[2 points]}  Suppose $a<b<c<d<e$.  
\ben
\addtolength{\itemsep}{2in}
\item Find the minimum of $f(x)=(x-a)^2+(x-b)^2+(x-c)^2+(x-d)^2+(x-e)^2$
\item Find the minimum of $g(x)=|x-a|+|x-b|+|x-c|+|x-d|+|x-e|$
\item \textbf{[Bonus point]} Prove that if
  $a+\frac{b}{2}+\frac{c}{3}+\frac{d}{4}+\frac{e}{5}=0$ then
  $a+bx+cx^2+dx^3+ex^4=0$ for some $x\in [0,1]$.
\een
\ep


\bp  Here is the unit circle $x^2+y^2=1$.


\begin{center}
\includegraphics[scale=0.5]{circle.pdf}
\end{center}


\ben
\item \textbf{[1 point]}  Express the area of the sector $OAC$ in terms of $t$.
\vspace{2.5in}
\item \textbf{[1 point]} Suppose the point $A$ is moving counterclockwise around the circle so 
  that the area of the sector $OAC$ is increasing at a constant rate.
  Determine the rate is the area of triangle $OAC$ changing at the moment
  that it is isoceles.
\een
\ep

\addtocounter{problemnum}{-1}


\bp \textbf{Continued.}  Here is the unit
hyperbola $x^2-y^2=1$.

\begin{center}
\includegraphics[scale=0.5]{hyperbola.pdf}
\end{center}

\ben
\addtocounter{enumi}{1}
\item \textbf{[1 point]}  Express the area of the hyperbolic sector
  $OAC$ in terms of $t$.
\vspace{2.5in}
\item \textbf{[Bonus point]}  Suppose the point $A$ is moving away from the
  origin so that the area of the sector $OAC$ is increasing at a constant rate.
  Determine the rate is the area of triangle $OAB$ changing at the moment
  that it is isoceles.
\een
\ep



\bp \textbf{[2 points]}  Compute $\ds \int_0^\pi \sin^n(x)dx$ for $n=0,1,2,\ldots$.
\ep


\bp \textbf{[All correct: 3 points.  Eight correct: 1 points]}
Compute.  
Interpret any integrals above as limits if necessary.  For example, 
$$ \int_0^1
  \frac{dt}{\sqrt{1-t^2}}=\ds \lim_{x \to 1^{-}}\int_0^x
  \frac{dt}{\sqrt{1-t^2}}.$$

\ben 
\addtolength{\itemsep}{1in}
\item $\ds \int_0^1 \frac{t dt}{\sqrt{1+t^2}}$
\item $\ds \int_0^1
  \frac{t dt}{\sqrt{1-t^2}}$ 
\item $\ds \int_0^1 \frac{t dt}{1+t^2}$
\item  $\ds \int_0^1 \frac{t dt}{1-t^2}$
\item $\ds \int_0^1 \frac{dt}{\sqrt{1+t^2}}$
\een
\ep

\addtocounter{problemnum}{-1}


\bp \textbf{Continued.}
\ben
\addtocounter{enumi}{5}
\addtolength{\itemsep}{1in}
\item $\ds \int_0^1  \frac{dt}{\sqrt{1-t^2}}$ 
\item $\ds \int_0^1 \sqrt{1-t^2}dt$ 
\item $\ds \int_0^1 \sqrt{1+t^2}dt$
\item $\ds \int_0^1 \frac{dt}{1+t^2}$
\item $\ds \int_0^1 \frac{dt}{1-t^2}$
\een

\ep




\end{exam}
\end{document}

