
% extra credit ch2 2280, S2016
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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline


\begin{center}
\bf Math 2280 Extra Credit Problems \\
\bf Chapter 2 \\
{\small\bf S2016}
\end{center}


{\bf Submitted work}. Please submit one stapled package per chapter.
Kindly label problems \fbox{\bf Extra Credit}. Label each problem with
its corresponding problem number, e.g., \fbox{Xc2.1-8}. Please attach this printed sheet to
simplify your work.

\Prob{XcL1.1}{(maple lab 1)}
Solve the following quadratic equations using maple, as in Example 1 of
maple lab 1, then use maple to reconstruct the quadratic equation from
the roots.
You may submit this problem only for score increases on maple lab 1.

(a) $x^2+x+1=0$ \\
(b) $8x^2+ 2x + 15=0$ \\
(c)  $5x^2-250x+3125$

\Prob{XcL1.2}{(maple lab 1)}
Plot the following functions in maple. Print the plots in size $2\times
2$ inches, approximately.
You may submit this problem only for score increases on maple lab 1.

(a) $f(x)=x^2-7x+5$ on $-1 \le x \le 3$\\
(b) $f(x)=|x-1|+|2x+3|$ on $-5 \le x \le -1$ \\
(c) $f(x)=e^x\sinh(x-1)+e^{-x}\sinh(2x-3)$ on $-2 \le x \le 1$ \\
(d) $f(x)=\ln|\sin(2x)+\cos(3x)+3|$ on $-2\pi \le x \le 3\pi$

\Prob{Xc2.1-8}{(Verhulst equation)}
Solve $x'(t)=4x(t)(7-x(t))$, $x(0)=11$ by separation of variables and partial fractions.
Sketch the solutions, including equilibria.
Check your answer from the textbook answer for 2.1-7.



\Prob{Xc2.1-16}{(population dynamics)}
Assume a population $P(t)$ of alligators with $P(0)=P_0$ satisfies the
Verhulst equation $P'=(aP-b)P$ with birth rate $aP_0^2=11$ and death
rate $bP_0=12$ [see 2.1-18 in the textbook]. Find the equilibrium solutions,
sketch them and a suitable sampling of other possible solutions. Determine
the time $t$ at which $P(t)=M/10$, where $M=b/a$ is the threshold population.



\Prob{Xc2.2-10}{(Separation of variables, Verhulst DE)}
Solve $x'(t)=x(t)(3-x(t))$ by separation of variables and partial fractions.
Sketch the solutions, including equilibria.
Check your answer from the textbook answer for 2.2-4.

\Prob{Xc2.3-10}{(Parachute)}
Assume model (1) $v'=-32-0.15v$ before the chute opens and model (2)
$v'=-32-1.5v$ after. Suppose the fall from 10000 feet uses (1) for 15 seconds.
Find the trip time from 10000 feet to the ground (0 feet).

\Prob{Xc2.3-20}{(Terminal speed)}
Assume model $v'=-32+0.075v^2$ for a parachutist who pulls the ripcord
at 10000 feet at speed 10 feet per second. Find his terminal speed and
the trip time to the ground from 10000 feet.




{\bf End of extra credit problems chapter 2.}

\end{document}




