
% extra credit ch2 2250, S2015
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 {\bf Name~}\underline{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~}\hfill
 {Class Time~}\underline{~~~~~~~~}\newline


\begin{center}
\bf Math 2250 Extra Credit Problems \\
\bf Chapter 2 \\
{\small\bf S2015}
\end{center}

{\bf Submitted work}. Please submit one stapled package with this sheet
on top. Kindly check-mark the problems submitted and
label the paper \fbox{\bf Extra Credit}. Label each solved problem with
its corresponding problem number, e.g., \fbox{Xc10.3-20}.

\Prob{Xc-Quadratics}{(Quadratic equations)}
Solve the following quadratic equations by hand and then check the
answer using technology. Techniques should include FOIL factoring, complete the square,
the quadratic formula, root and factor theorems, leading coefficient reconstruction.


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

\Prob{Xc-Plots}{(Plots using technology)}
Plot the following functions with technology. Print the plots in size
$2\times 2$ inches, approximately.

(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,
then sketch them along with 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 (carrying capacity).


\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 linear drag model (1) $v'=-32-0.15v$ before the chute opens and
quadratic drag 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 quadratic drag model $v'=-32+0.075v^2$ for a parachutist who
pulls the ripcord at 10000 feet at speed 10 feet per second. Find 
the terminal speed of the parachutist and the trip time to the ground
from 10000 feet.


\Prob{Xc-ExactSol1}{(Exact solution)}
Solve symbolically by chapter 1 methods the initial value problem
$y'=2xy^2$, $y(0)=1$. Do an answer check by hand.
Answer: $y=1/(1-x^2)$. This problem has no numerical work!

\Prob{Xc-NumSol1}{(Numerical solution by three methods)}
Solve $y'=2xy^2$, $y(0)=1$ numerically
for the value of $y(0.5)$ using (1) Euler's method,
(2) Heun's method and (3) the RK4 method. Use step size $h=0.1$. Include
computer code and a print of the data.  Report the answers in a table
for $x$-values $0$, $0.1$, $0.2$, $0.3$, $0,4$, $0.5$.


\Prob{Xc-ExactSol2}{(Exact solution)}
Solve symbolically by chapter 1 methods the initial value problem
$y'=e^{-y}$, $y(0)=0$. Do an answer check in maple or by hand.
Answer: $y=\ln(1+x)$. This problem has no numerical work!

\Prob{Xc-NumSol2}{(Numerical solution by three methods)}
Solve $y'=e^{-y}$, $y(0)=0$ numerically
for the value of $y(1.0)$ using (1) Euler's method,
(2) Heun's method and (3) the RK4 method. Use step size $h=0.001$. Include
a computer code appendix in the report, but do not print the data.
Report the answers in a table for $x$-values $0$, $0.2$, $0,4$, $0.6$,
$0.8$, $1.0$. Include the percentage error
$E=100|\ln(2)-y(1.0)|/|\ln(2)|$ in your report, one error report for
each of the three methods.




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

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