\def\wsn{11}
\input worksheet.tex

\itemitem{1.\hskip12pt a)}  In each of the following graphs label all
points of discontinuity and non-differentiability:

\medskip
\vbox{
\hbox{
\vbox{\hsize=.5truein i)\vskip1truein}
 \epsfxsize=1.1truein\epsfbox{/home/oehrtman/m210/graph11.a.eps}
\hskip.5in\vbox{\hsize=.5truein ii)\vskip1truein}
 \epsfxsize=1.1truein\epsfbox{/home/oehrtman/m210/graph11.b.eps}
\hskip.5in\vbox{\hsize=.5truein iii)\vskip1truein}
 \epsfxsize=1.1truein\epsfbox{/home/oehrtman/m210/graph11.c.eps}}
\vskip.2truein
\hbox{
\vbox{\hsize=.5truein iv)\vskip1truein}
 \epsfxsize=1.1truein\epsfbox{/home/oehrtman/m210/graph11.d.eps}
\hskip.5in\vbox{\hsize=.5truein v)\vskip1truein}
 \epsfxsize=1.1truein\epsfbox{/home/oehrtman/m210/graph11.e.eps}
\hskip.5in\vbox{\hsize=.5truein vi)\vskip1truein}
 \epsfxsize=1.1truein\epsfbox{/home/oehrtman/m210/graph11.f.eps}}}

\medskip
\itemitem{b)}  For each discontinuity, explain which parts of the
definition of continuity are not satisfied.

\medskip
\itemitem{c)}  For each point of non-differentiability, draw a blow-up
of the graph at that point.  Then draw a sequence (or two sequences if
necessary) of secant lines which show that
$\displaystyle\lim_{h\to0}{f(x+h)-f(x)\over h}$
does not exist.

\smallskip
\itemitem{}  If it is not possible to draw the appropriate secant
lines, explain why.

\bigskip
\item{2.}  Find the derivatives of the following functions:
$$
{\eqalign{
\pt a&f(x)=\root3\of x\cr\cr
\pt d&f(x)={x^2\over x^2+1}\cr\cr
\pt g&f(x)={x+2\over x^2-4}}}\qquad
{\eqalign{
\pt b&f(x)=x^2\cr\cr
\pt e&f(x)=\cases{{1-1/x\over1-1/x^2}&if $x\neq1$\cr
           {1\over2}&if $x=1$\cr}\cr\cr
\pt h&f(x)=\cases{0&if $x$ is rational\cr 
           x^2&if $x$ is irrational\cr}\cr}}\qquad
{\eqalign{
\pt c&f(x)=|4-x^2|\cr\cr
\pt f&f(x)=\cases{{1\over x}&if $x>0$\cr-x&if $x\leq0$\cr}\cr\cr
\pt i&f(x)={1\over x+3}+{1\over|x+3|}\cr}}\qquad
$$

\medskip
\itemitem{3.\hskip12pt a)}  Sketch a graph of the six trig functions.
Below each one, sketch a graph of its derivative using only graphical
information. 

\medskip
\itemitem{b)}  Rewrite each of the six trig functions in terms of sine
and cosine.  Use these formulas and the quotient and product rules to
compute the derivatives of all of the trig functions.  Simplify and
write the results next to the graphs of the appropriate derivatives.

\medskip
\itemitem{c)}  Algebraically compute places where the derivatives are
zero or are undefined.  Label these points on both graphs.

\bigskip
\item{4.}  Differentiate and simplify:
$$
{\eqalign{
\pt a&y=x\sqrt{1-x^2}\cr
\pt d&y={x^2-1\over x^2+1} \cr
\pt g&y={1\over(1-2x)^2}\cr}}\hskip.3in
{\eqalign{
\pt b&y=x^2(1-x^2)^2 \cr
\pt e&y={1-\sqrt x\over1+\sqrt x} \cr
\pt h&y=x^{2/3}(x+1)^{1/3} \cr}}\hskip.3in
{\eqalign{
\pt c&y=(1+\sqrt x)(1-\sqrt x) \cr
\pt f&y={x\over x^2+1} \cr
\pt i&y=x(x+1)^{1/3}  \cr}}
$$

\medskip
\item{5.}  Rewrite each of the quotients in Problem 4 as a product.
Then compute the derivatives using the product rule.
\eject

\bye