Math 1310: CSM
Spring 2017
Mini-Project 4: DIY (Do-It-Yourself ) Modeling, First Draft
Due Wednesday, April 19
Please note that this is Part I of a two-part Mini Project. For Part II, which will be the
last Mini Project of the semester (there will be one other Mini Project in between), you will
be asked to refine and improve upon Part I, based on suggestions and comments you will
receive when this assignment is handed back.
Part I and Part II will each be worth as much as any other Mini Project. Neither Part I nor
Part II can be dropped. So please do a good job!
You’ll have more time to do Part I than Part II, so shoot for perfection on Part I, and you’ll
have less work to do later on.
For this assignment, you need to:
1. Describe, in words, an original “real-world” situation that may be modeled by a system of
differential equations and a set of initial conditions – that is, by an initial value problem;
2. Write down the differential equations that model this situation;
3. Write a Sage program that solves your initial value problem numerically, using Euler’s
method;
4. Write up carefully, and hand in, your work, including: your verbal description of the
phenomena being modeled, your differential equations and initial conditions (these should
be part of your write-up, not just part of your Sage code), a copy of the Sage code, a copy
of the graphical output, AND a brief analysis (just a few sentences) of the results.
Some hints and notes:
(a) You needn’t (and perhaps shouldn’t) start from scratch!! You can use, as a template/starting point for your work, the SIR project (Mini Project #2) or the Fermentation project (Mini Project #3). For either of these projects, you should already have a
complete set of differential equations, and a complete Sage program, that you can modify
for this assignment. But your modifications must be significant! In particular:
(b) Be ORIGINAL! It’s OK to use the ideas mentioned above as launching points, but
your scenario, and the differential equations that model it, should differ in significant,
interesting ways from anything you’ve seen before in this class.
(c) Be CREATIVE!!! Interpret the adjective “real-world” broadly. You can be as silly, wacky,
etc. as you’d like. It’s not important that you describe anything particularly realistic.
What’s important is “internal consistency.” That is, no matter how fantastical your
scenario is, your differential equations should match the words you use to describe this
scenario. And your Sage program should WORK – in the end, you should get graphical
output that actually LOOKS like something.
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Math 1310: CSM
Spring 2017
Besides the second and third Mini Projects, you might want to look at the following for
inspiration/ideas: the Week 8 and Week 9 tutorials; Individual Homework #7, the “Some
modeling problems” review sheet for Exam 3.
For further inspiration, see the sample Mini Projects attached at the end of
this assignment. These are actual Mini Projects handed in by groups from previous
semesters. (Apologies if they’re a bit low-res.)
DISCLAIMER: these samples are excellent, but they are in “Part I” (first draft) form.
Think carefully about your own work, to make sure you don’t replicate any imperfections.
(And of course, you’ll get a chance to correct any of your own imperfections for Part II.)
(d) It’s OK, and maybe not a bad idea, to work backwards, sort of. That is: you might start,
for example, with your yeast-alcohol-sugar code, and make a series of minor changes to
the differential equations and/or parameter values in that code, making sure that, with
each change, you still get nice graphical output. Once you’ve made sufficiently many
changes that things really look different, try to imagine a situation that your resulting
differential equations describe. Again, this approach is fine, as long as your end result
really looks like something new.
(e) The system you create should involve at least three dependent variables (as in the SIR or
fermentation projects), and therefore at least three differential equations. You can have
more than three if you want, but you’ll find that things get very complex (and hard to
make work) very quickly if you have too many variables.
(f) It will probably take some trial and error to get things to work. (You can tell things
are working if your graph looks good in the end, meaning you can see how each of your
variables evolves with time. If one or more or your variables doesn’t evolve – it doesn’t
show up on the graph at all, or it stays flat, or it immediately drops down to zero and
stays there, etc. – then you’ll probably need to tweak something.)
(g) Make sure you label your axes, and specify units for all of your variables and parameters.
(h) In “real life,” you’d need to think carefully about the values of your parameters, and
they’d need to be chosen realistically; you can’t just plug in any numbers you want. For
this project, the only question you should be asking yourself about the parameters is:
what values will give me nice graphs? This will likely take some experimentation.
(i) To get full credit, your Sage code should be neat and commented nicely. See the solutions
to Mini Projects #’s 2 and 3.
(j) Get started early. Give your group’s brains time to play around with this. And allow
time for things to not work on the first go, because they probably won’t.
(k) Have fun with it!!!
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ASampleDIYModelingMini-Project
Weanalyzedaracebetweenthreeextremelymentalathletes.Ourfirstcompetitoris
DiscouragedDan,asnailwhoentersastheundisputedunderdog.Everybodylovesa
Cinderellastory,butforthissnailwhopreferstogoby“DD”,anupsetisunlikely.
Nextwehaveaveterancrowdfavorite,TimmyTwo-Toes,thesloth.Heisoneofthe
animalkingdom’smostrespectedlow-speedathletes,andisracingforhisworld
record10thwin.RoundingoutthepackisGreasyGarry,aturtlewhomakesupfor
hislackofspeedwithcunning,andtrickery.AsTimmyoncesaid,“IfGarryain’t
cheatin’,thenyou’vegotthewrongGarry,becausethisGaryisseriouslyalways
cheating.“
Asmentioned,theseareincrediblymentalathletes,whoseperformancesdepend
heavilyontheirstatesofmind.
Here’swhatweknow:
--WecancountonGreasyGarrytimingthestartingshotalltoowell,andgettinga
(relative)runningstart,ata(relatively)highspeedof15meters/hour.
--Dancomplainshiswayintoaslidingstart,andwithaboostfromTannerTen
Point,thebuck,ourunderdogsnailstartstheraceatacool10meters/hour.
--Weseesomanyathletesusethe“everyonedoesit”excuse,butexpectTimmy
Two-Toestostartstandingstill,inadisplayoftrueveteranclass.
--Absentofotherfactors,Dan’sspeedincreaseslogisticallywithagrowthrateof
.001,withamaximumspeedthatisproportionaltoGarry’sspeed.Themental
aspectofracingcomesintoplayhere.DiscouragedDandoesn’tgethisnameforjust
anyreason.Thefasterhisopponentsget,theworseheperforms.Hismaximum
performanceisessentiallylimitedbyGarry’ssuccess.DiscouragedDanreallylooks
uptoTimmy,andisnegativelyaffectedbythesloth’sstellarperformance.
Specifically,helosesspeedatarateproportionalto.001ofTimmy’sspeed,in
proportiontohisownspeed.
--Timmydisplaystheextremedrivenecessaryforsuccessinthecompetitivelowspeedracingworld.Hecouldn’tcarelessabouttheslowsnailtohisright.But,Garry
gotaheadstart,andTimmyTwoToesismotivatedandprepared.Thedetermined
slothincreaseshisspeedatarateproportionalto2%ofGarry’sspeed.
--Garrynormallycouldn’tkeepthespeedwithwhichhestarted.Absentoutside
factors,hewouldlosehisspeedatarateof.25meters/hourperfoot.However,heis
acompetitiveanimalaswell.AsTimmygetsstarted,andespeciallyasheaccelerates
quicklybehind,Garryspeedsupaccordingly.Inanattempttoholdhislead,Garry
increaseshisspeedatarateproportionalto4%ofTimmy’sspeed.
DifferentialEquations
SageCode
GraphandAnalysis
Garrygothisheadstart,butthenbegantoslow,aspredicted.But,onceTimmy
startedtoaccelerateweseeGarrypickhisspeedup.Timmyhasthefastest
acceleration,astheslopeofhisspeedisthegreatest,butwithGarry’sheadstart,
Timmynevercatchesup.HisspeedneverevenreachesahigherpointthanGarry’s.
Dan,ourdiscouragedsnail,livesuptohisname.Hetoohadaheadstart,butitdidn’t
dohimmuchgood.Assoonashesawhiscompetitorstakeoff,hedeceleratedquite
rapidly.TimmyTwo-Toeswilllivetoraceanotherday,andhopefullygethisworld
record.Butfornow,GreasyGarryisthelowspeedchampionoftheanimalkingdom.
Another sample
DIY modeling
mini
-
project