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\course \hfill Russell Brown \\
Review 4  \hfill \today

The final will be cumulative with an emphasis on Chapter 4. Below are
review problems for Chapter 4. You will also want to study the review
problems for the first three exams. 

\begin{enumerate}

\item Consider  the function $f(x) = x^4-x^2$. Find all critical
numbers. Determine if each critical number is a local maximum, local
minimum or neither. Find local maximum values and local minimum
values. Find intervals of increase or decrease. Find intervals of
concavity. Find inflection points. Sketch a graph using this
information. Check using TI-82. 

\item Suppose a light is revolving at two revolutions/min. The light
is three meters from a wall. a)  Find the speed with which the light
is 
moving along the wall when the beam forms a right angle with wall. 
b) Find the speed with which the light is  moving along the wall  if
the light forms an 
angle of $45^\circ$ 
with the wall.  

\item A rectangle is inscribed in the ellipse $ 2x^2 + y^2 =1$. Find
the largest possible area of the rectangle.  

\vspace{2in}

\item Find the global  maximum and minimum values of $ f(x) = \sin x -
\frac 1 2 x$ on $[0,\pi]$. 

\item Suppose that a prism whose base is an equilateral triangle has
volume 100 cm${}^3$. Find the small possible surface area.

\vspace{2in}

\newpage

\item If we apply Newton's method to the following function starting a
$D$, which root will Newton's method find?  Same question for $E$ and
$F$.

\vspace{2in}

\item Explain Newton's method and derive the formula 
$$
 x_{n+1} = x_n - f(x_n)/ f'(x_n).
$$



\item Show how use Newton's method to find $ \sqrt[3]{10}$.
Give $x_1$, $x_2$ and $x_3$ if $x_0= 4$.

\item State the first derivative test for a global minimum. Use this
test to show that $f(x)= x+ 1/x$ has a global minimum on
$(0,\infty)$. 

\item State the mean value theorem. State the definition of increasing
function. Use the mean value theorem to prove that if $ f'(x) > 0$ for
$ x\in (0,1)$, then $f$ is increasing on $(0,1)$. 

\item Let $ f(x) = x^3-x$  for $ x$ in $[-2,2]$.  Find the global
maximum and minimum. What are the global maximum and minimum values.


\end{enumerate}
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