\input epsf
\input graphicx
\magnification 1300
\vsize 10truein
\nopagenumbers
\parskip 10pt
\parindent 0 pt


Find the following for $f(x) = e^{-x^2} + 1$ (if 
they 
exist; if they don't exist, state so).  Use this information to graph $f$.

Note $f'(x) = -2x e^{-x^2}$, $f''(x) = 2 e^{-x^2}[2x^2 - 1]$

Is $f$ even, odd, periodic?  What is the domain and range of $f$?
\vskip -2pt

[1.5] 1a.) critical numbers: $\underline{\hskip 1.5in}$
		

[1.5] 1b.) local maximum(s) occur at $x = \underline{\hskip 1.5in}$

[1.5] 1c.) local minimum(s) occur at $x = \underline{\hskip 1.5in}$

[1.5] 1d.)  The global maximum of $f$ on the interval [0, 5] is $\underline{\hskip .5in}$ and occurs at $x = \underline{\hskip 1in}$

[1.5] 1e.)  The global minimum of $f$ on the interval [0, 5] is $\underline{\hskip .5in}$ and occurs at $x = \underline{\hskip 1in}$

[1.5] 1f.) Inflection point(s) occur at $x = \underline{\hskip 1.5in}$


[1.5] 1g.)  $f$ increasing on the intervals $\underline{\hskip 1.5in}$

[1.5] 1h.)  $f$ decreasing on the intervals $\underline{\hskip 1.5in}$

[1.5] 1i.)  $f$ is concave up on the intervals $\underline{\hskip 1.5in}$

[1.5] 1j.)  $f$ is concave down on the intervals$\underline{\hskip 1.5in}$

[1.5] 1k.)  Equation(s) of vertical asymptote(s)$\underline{\hskip 1.5in}$

[4] 1l.)  Equation(s) of horizontal and/or 
slant asymptote(s)$\underline{\hskip 1.5in}$


\vskip -4pt
[4.5] 1m.)  Graph $f$
\vskip -8pt
\includegraphics[width=36ex]{grapht}

\end
$f(x) = x^3 - 3x^2 + 33$ 

$f'(x) = {3x^2 - 6x} = 3x(x - 2) = 0 $ or DNE,  critical points: $x = 0, 2$

Check increasing/decreasing between critical points, points of discontinuity, and singleton points not in domain (where function could change between increasing/decreasing).

\vskip 100pt

$f''(x) = {6x - 6} = 0 $ or DNE, so possible inflection point:  $x = 1$

Check concave up/down between possible inflection points, points of discontinuity, and singleton points not in domain (where function could change between concave up/down).


\end
