\magnification=\magstep1
\centerline{Mathematics 133 -- Intensive Calculus for Science 1}
\centerline{Discussion 6 -- Using $f'(x)$ and $f''(x)$}
\centerline{November 1, 2000}
\bigskip
\noindent
{\it Background}
\bigskip
Now that we have learned how to differentiate all of the
elementary functions (the power, exponential, and logarithm functions,
trigonometric functions and their inverses, and functions built 
up from them by sums, products, quotients, and compositions), we
will look next at several applications of differential 
calculus.  The idea will be to use all the information that
the first and second derivatives of a function can tell us
to get information about the original function.  We already
know most of the key ideas here, but we need to ``synthesize''
everything and give statements of some of the general patterns
that can be deduced from what we have learned.

A function $f(x)$ is said to have a {\it local 
maximum\/} at $x = a$ if $f(a) \ge f(x)$ for all $x$ near $a$ 
(for instance all $x$ on some interval containing $a$).  Similarly,
$f(x)$ is said to have a {\it local minimum\/} at $x = a$ if 
$f(a) \le f(x)$ for all $x$ near $a$.  (Note:  since these
words are taken from the Latin language, the plurals are
formed according to Latin rules:  the plural of ``maximum''
is ``maxima'' and the plural of ``minimum'' is ``minima''.) 
\bigskip
\noindent
{\it Discussion Questions}
\bigskip
\item{A)}  Estimate the $x$-coordinates of the 
local maxima and local minima of the following function:
\vglue 2.0truein
\item{B)}  What is true about the derivative of $f$ at
each of the points you found in part A?  (Note: there are
{\it two different possibilities}.  Be careful!)
\bigskip
\item{C)}  Suppose you had a formula or a graph of the 
derivative function $f'(x)$.  How could you use only that
to determine where the local maxima and local minima
of the function $f(x)$ are located?  Give a collection of 
``rules'' to follow, in your own words, to determine where 
the maxima and where minima are, using {\it only information
from $f'(x)$}.  Explain with the following
example, showing $y = f'(x)$:
\vglue 2.0truein
\bigskip
\item{D)} Now, suppose that in addition to the formula
or graph for $f'(x)$ we also had a formula or graph of 
the second derivative $f''(x)$.  How could you use this 
information to locate the maxima and minima of $f(x)$, 
{\it assuming that $f'(a)$ exists whenever 
$x=a$ is a local maximum or minimum of the function $f$
in question}?
Again, give a collection of 
``rules'' to follow, in your own words, to determine where 
the maxima and where minima are, but now using both
$f'(x)$ and $f''(x)$.  Explain using the example from question C.
(You'll need first to use that graph to sketch a qualitative
graph of $y = f''(x)$.)
\bigskip
\item{E)}  Find all local maxima and minima of $f(x) = x^2 e^{-2x}$
and $f(x) = x^4 - 6x^2 + 1$.
following your ``rules'' from both questions C and D.  Is
either of your methods clearly simpler than the other?
\bigskip
\noindent
{\it Assignment}
\bigskip
Group write-ups due in class on Monday, November 6.
\bye
