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{\bf Quiz3 Problem 1}.
A graphic called a {\bf phase diagram}
displays the behavior of all solutions of $u'=F(u)$.
A {\bf phase line diagram} is an abbreviation for a direction field on the
vertical axis ($u$-axis). It consists of equilibrium points and signs of
$F(u)$ between equilibria.
A phase diagram can be created solely from a phase line diagram, using
just three drawing rules:
\begin{itemize}
\item[{\bf 1}.] Solutions don't cross.
\item[{\bf 2}.] Equilibrium solutions are horizontal lines $u=c$. All other solutions are increasing or decreasing.
\item[{\bf 3}.] A solution curve can be moved rigidly left or right to create another solution curve.
\end{itemize}
Use these tools on the equation $u'=(u-1)(u-2)^2(u+2)$ to make a phase line
diagram, and then make a phase diagram with at least 8 threaded solutions.
Label the equilibria as stable, unstable, funnel, spout, node.

{\bf References}. Edwards-Penney section 2.2.
Course document on
\href{http://www.math.utah.edu/~gustafso/s2014/2250/2250phaseline.pdf}{Stability},
%{\bf References}. Edwards-Penney section 2.3.
%Serway and Vuille, {\em College Physics 9/E}, Brooks-Cole (2011), ISBN-10: 0840062060.

{\bf Quiz3 Problem 2}.
An autonomous differential equation $\frac{dy}{dx}=F(x)$ with initial
condition $y(0)=y_0$ has a formal solution
$$y(x) = y_0 + \int_0^x F(u)du.$$
The integral may not be solvable by calculus methods. In this case, the integral
is evaluated numerically to compute $y(x)$ or to plot a graphic. There are
three basic numerical methods that apply,  the rectangular rule (RECT), the trapezoidal rule (TRAP)and Simpson's rule (SIMP).

Apply the three methods for $F(x)=\cos(x^2)$ and $y_0=0$ using step size $h=0.2$ from $x=0$ to $x=1$.
Then fill in the blanks in the following table. Use technology if it saves time. Lastly, compare the four data sets in a plot, using technology.
$$\small\rm
\begin{array}{|l|l|l|l|l|l|l|}
\hline
x-\mbox{values} & 0.0 & 0.2 & 0.4 & 0.6 & 0.8 & 1.0\\
y-\mbox{\normalsize\rm to 10 digits} &
0.0& 0.1999680024& 0.3989772129& 0.5922705167& 0.7678475376& 0.9045242379
\\
y-\mbox{\normalsize\rm RECT values} &
0.0& 0.2& 0.3998400213& 0.5972854780& \fbox{\phantom{0.7844648427}}& 0.9448839943
\\
y-\mbox{\normalsize\rm TRAP values} &
0.0& 0.1999200107& 0.3985627497& \fbox{\phantom{0.5908751604}}& 0.7646744186& 0.8989142250
 \\
y-\mbox{\normalsize\rm SIMP values} &
0.0& 0.1999666703& 0.3989746144& 0.5922670741& 0.7678445414& \fbox{\phantom{0.9045242680}}
   \\
   \hline
\end{array}
$$
{\bf References}. Edwards-Penney Sections 2.4, 2.5, 2.6, because methods Euler, Modified Euler and RK4 reduce to RECT, TRAP, SIMP methods when $f(x,y)$ is independent of $y$, i.e., an equation $y'=F(x)$. Course document on numerical solution of $y'=F(x)$
\href{http://www.math.utah.edu/~gustafso/s2014/2250/solve-quadrature-numerically.pdf}
{RECT, TRAP, SIMP methods}.
Wolfram Alpha at \href{http://www.wolframalpha.com/}{\tt http://www.wolframalpha.com/} can do the RECT rule and graphics with input string
\begin{quote}
{\tt integrate  cos(x\^{}2) using left endpoint method with interval width 0.2 from x=0 to x=1}
\end{quote}

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