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\centerline{Mathematics 133 -- Intensive Calculus for Science 1}
\centerline{Discussion 4 -- Let's Do It Again!}
\centerline{October 14, 2005}
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{\it Background}
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Over the past two days, we have introduced the derivative function 
$f'(x)$ for a function $f(x)$ and seen how to get quantitative 
and qualitative information about $f'(x)$ from a formula, graph, or
table of values of $f(x)$.  Today, we are going to do the same
process to $f'(x)$ and compute what is usually called the {\it second
derivative} of $f$, also written $(f')'(x) = f''(x)$.
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{\it Discussion Questions}
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The following graph shows $y = f(x)$.  
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{\it ``Supply your favorite graph''}
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{\it Note:  You are now going to sketch two graphs 
produced from this one.  It will probably help to ``stack'' the three graphs
so that the same $x$-values line up vertically on all three graphs.}
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\item{A)}  Give a ``qualitative'' sketch of the graph $y = f'(x)$
for the same interval of $x$-values.  Note that there is no $y$-axis
scale here.  So you will not be able to get exact values for $f'(x)$.
But determine the intervals where $f'(x) > 0$, where $f'(x) < 0$, and
the points where $f'(x) = 0$.  
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\item{B)}  Next, apply the same process to your graph from part A -- 
make a qualitative sketch of the derivative of $f'(x)$ -- this is
$f''(x)$.  
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\item{C)}  Answer the following questions:
\itemitem{1)}  Over which intervals is $f''(x) > 0$? 
Over which intervals is $f''(x) < 0$?
\itemitem{2)}  What is $f'(x)$ doing when $f''(x) > 0$?  when $f''(x) < 0$?
\itemitem{3)}  What is true about the original graph $y = f(x)$
when $f''(x) > 0$?  when $f''(x) < 0$?

\bye

