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\title{Math 241, Lecture 23}
\maketitle


\section{ Curve sketching}  

We can assemble which comes through our analysis of 
a function and its first and second derivatives in order to 
get a good picture of the graph of a function.  
%The steps we take are as follows:

\begin{itemize}
\item Plot a few values of the function, including $f(0)$, which is where the
function crosses the $y$-axis.  
\item Determine $\lim_{x \to \infty} f(x)$ and $\lim_{x \to -\infty} f(x)$, 
if they exist, in order to see where the graph ``begins'' and ``ends'' and draw in 
these ends if they can be determined. 
\item Find and plot the places where the function crosses the $x$-axis
by solving the equation $f(x) = 0$.  
\item Calculate the first derivative and find where it is zero, positive positive and negative 
in order to find the critical points of $f$ and determine where $f$ is increasing
and decreasing.  
\item Calculate the second derivative and find where it is zero, positive and
negative in order to determine where $f$ is concave upward and downward.  
\item Check where the sign of the derivative changes or use the second derivative
test to determine which critical points are
local maxima or minima.  
\item Draw in ``cups'' at local minima, ``caps'' at local maxima, and one of four kinds
of curve, as sketched on the board, in the regions where the the signs of the first
and second derivative do not change.  
\item Fill in the parts of the graph in between the curves you have put in.  
\end{itemize}

\begin{example}
Sketch the graph of $f(x) = 2x^3 + 3x^2 - 12x - 7$,   and of $x e^{1-x}$.  
\end{example}

\section{ Related rates }  

In some problems all quantities in question are functions of some variable
(most often time)  but this dependence is not explicit; only a relationship is known.  
   The techniques we have used for implicit differentiation can be used to find the 
derivative of one quantity if we know the derivative of the other.

The classic related rates problem, one used to torture calculus
students for as long as the subject has been taught, is the problem
of the ladder.   

 Try not to let this problem and its cousins
make its way into your nightmares.  

\begin{example}
A ladder, which is ten feet long, is leaning against a wall.  Its
feet begin to slide out from under it, and its top falls at a constant
rate of one foot per second.   How fast is the foot of the ladder
moving when the top of the latter is at $8$ feet?    What about 
$4$ feet?  
\end{example}


The notion of related rates, like that of implicit differentiation, is 
based on the fact that taking the derivative of both sides of a valid
equation gives rise to a valid equation   (as long as the variable
with respect to the derivative is being taken is clear, and that the
chain rule is applied properly).  



\begin{example}
At ACME Anvils, output is $Q = 60 K^{\frac{1}{3}} L^{\frac{2}{3}}$ where $K$
is the capital inverstment (in thousands of dollars) and $L$ is the size of the
labor force, measured in worker-hours.    If output is kept constant, at
what rate must capital investment change at a time when $K = 8$,
$L = 1000$ and $L$ is increasing at the rate of $25$ worker-hours per
week?  
\end{example}



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