\magnification=1200
\baselineskip=12pt
\nopagenumbers

\noindent
MATH 180\hfill FALL 1995

%\noindent
%10:00 SECTION\hfill FONG

\bigskip\noindent
\centerline{ASSIGNMENT \#12}

\bigskip\noindent
2.5, \#1.
(a)  If $f''$ is positive on an interval, then $f'$ is
{\it increasing\/} on that interval, and $f$ is {\it concave-up\/}
on that interval.
\noindent
(b)  If $f''$ is negative on an interval, then $f'$ is
{\it decreasing\/} on that interval, and $f$ is {\it concave-down\/}
on that interval.


\bigskip\noindent
2.5, \#2.
The right half of a U-shaped curve
has first and second derivatives which are
everywhere positive;  the left half of the same curve has a first
derivative which is everywhere negative and a
second derivative which is everywhere positive.
The right half of a curve in the shape of an upside-down U
has first and second derivatives which are everywhere negative;
the left half of the curve has a first derivative which is
everywhere positive and a
second derivative which is everywhere negative.


\bigskip\noindent
2.5, \#3.  For (a) we can join two curves from \#2, beginning with
right half of a U followed by the left half
of an upside-down U.  (b) The graph of the first derivative
is positive everywhere with the shape of an upside-down U.
(c) The graph of the second derivative is a decreasing function
whose first half is positive and whose second half is negative.
So the graph of the second derivative could resemble a line of negative
slope.

\bigskip\noindent
2.5, \#8.  We approximate $f'(0.6)$ by the
average of the average rates of change of $f$ on the intervals
$[0.4,0.6]$ and $[0.6,0.8]$ since this uses the data
on both sides of $0.6$.
This gives $f'(0.6)\approx 1.25$.
We approximate $f'(0.5)$ by
the average rate of change of $f$ on $[0.4,0.6]$ since
$[0.4,0.6]$ is the smallest interval in the data containing $0.5$.
This gives $f'(0.5)\approx 2$.
Since we have values for $f'(0.6)$ and $f'(0.5)$, we may
approximate $f''(0.6)$ by $(f'(0.6)-f'(0.5))/(0.6-0.5)$.
This gives $f''(0.6)\approx -7.5$.
The maximum and minimum values of $f$ occur near
$x=0.8$ and somewhere between $0.2$ and $0.4$.

\bigskip\noindent
Comments:  A number of students found \#3 difficult, particularly
the part asking for a sketch of $f''(x)$.  The relation of $f''(x)$
to $f'(x)$ is the same as the relation of $f'(x)$ to $f(x)$.
So if the first part of the problem gave no difficulties,
neither should the second part.

\end