\documentclass[11pt]{article}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\usepackage{latexsym,amsmath,amssymb,float,graphicx,subfig}
\setlength{\textwidth}{6.5in}
\setlength{\oddsidemargin}{0in}
\setlength{\textheight}{8.6in}
\setlength{\topmargin}{-0.6in}
\begin{document}
\centerline{MATH 135 -- Calculus 1}
\centerline{Curve Sketching}
\centerline{November 20, 2019}
\vskip 10pt
\noindent
{\it Background}
\vskip 10pt
When mathematicians and scientists want to understand graphs of 
functions and other curves these days, they almost always turn to graphing 
calculators and mathematical software to generate the plots. So learning to 
plot functions by hand can seem somewhat pointless. However, in my experience, 
some experience with curve sketching by hand is a great way to solidify your
understanding of the First and Second Derivative Tests, concavity, asymptotes, 
etc. So while you might never need to do this again in ``real life,'' 
it's still a worthwhile educational exercise.

Here's a checklist of things to do and look for, given $y = f(x)$ to plot:
\begin{enumerate}
\item Determine the domain of your function and note whether the graph has any vertical asymptotes. Draw those in if there are any.
\item Compute $f'(x)$ and find the critical points $x = c$, and the critical values $f(c)$.
\item On each interval between adjacent critical points, determine the 
sign of $f'(x)$. Recall that intervals where $f'(x) > 0$ are intervals where $f$ 
is increasing and intervals where $f'(x) < 0$ are intervals where $f$ is decreasing.
\item Compute $f''(x)$ and determine any $x$ where $f''(x) = 0$ or 
where $f''(x)$ does not exist.
\item Determine the intervals where $f$ is concave up and concave 
down using the sign of $f''$ (recall that inflection points are places 
where the concavity changes. If your function has inflection
points, find them and the function values there.
\item Determine horizontal asymptotes (if any), by taking $\lim_{x\to\pm\infty}f(x)$.
(L'Hopital's Rule will be useful for some of these. One or other of these directions might not make sense for some functions if the 
domain does not extend to $\pm\infty$.)
\item The critical points and the inflection points can be thought of as 
\emph{transition points} as described
in Section 4.6 of the text.
\item You should now be prepared to sketch the graph using the 
transition points, the information about where the function is increasing, 
decreasing, concave up, and concave down. See Figures
1 and 2 on page 231 for the four basic local shapes of graphs 
and an example of how they can
be put together.
\end{enumerate}
\vskip 10pt
\noindent
{\it Questions}
\vskip 10pt
Find the transition points (critical points and points of inflection), intervals of increase and decrease,
intervals of upward and downward concavity, asymptotes, and sketch the graph:
\begin{enumerate}
\item  $y = x^5 - 15 x^3$ (Hint: This one has three critical points, three points of 
inflection, and no asymptotes.)
\item $y = x - 2 \ln(x)$ (Hint: $\lim_{x\to 0+} \ln(x) = -\infty$.)
\item $y = \displaystyle \frac{1}{x^2 - 2x}$
(Hint: This one has both horizontal and vertical asymptotes. Be sure to simplify
$f'(x)$ before computing the second derivative.)
\item $y = x^2 e^{-2x}$ (Hint: This has a horizontal asymptote in one direction but
not the other.) 
\end{enumerate}
\end{document}
