CS 170: Computing for the Sciences and MathematicsAdministriviaUnconstrained GrowthExample of Unconstrained GrowthAnalytic SolutionExponential DecayWhere’s The Computation?Finite difference equationApproximating Unconstrained GrowthSystems Dynamics ToolStock/Box Variable/ReservoirFlowConverter/Variable/FormulaConnector/Arrow/ArcWith system dynamics toolHOMEWORK!Unconstrained Growth and DecayCS 170:Computing for the Sciences and MathematicsAdministriviaLast timeError and Basic differential calculusAssigned HW 2TodayHW2 due!Unconstrained GrowthWE HAVE CLASS ON MONDAYUnconstrained GrowthPopulation growth without constraintsExamples?Example of Unconstrained GrowthRate of change of population is directly proportional to number of individuals in the population (P)dP/dt = rPwhere r is the growth rate.Analytic Solution“Closed Form” solutionCan determine with a computer algebra systemLike MapleP = P0ertExponential DecayRate of change of mass of radioactive substance proportional to mass of substanceConstant of proportionality(rate) is negativeRadioactive Carbon-14: -0.000120968(about .0120968% per year)dQ/dt = -0.000120968 QQ = Q0 e-0.000120968tWhy Carbon-14?Where’s The Computation?An analytic solution is always preferable!But…finding it can be very hardInstead of solving the relationship, we’ll approximate it.Finite difference equationnew = old + changepopulation(t) = population(t - ∆t) + ∆populationIf I repeat this calculation a lot (moving the time up a bit each pass), I can see the trend of population over timeApproximating Unconstrained Growthinitialize simulationLe ng th, population, growth Rate, ∆tnumIterations  simulationLength / ∆tfor i going from 1 to numIterations do the following:growth  growthRate * populationpopulation  population + growth * ∆tt  i * ∆tdisplay t, growth, and populationUNITS ARE IMPORTANTDoes this give me the exact answer?Systems Dynamics ToolHelps to modelPerforms simulationWhat happens at one time step influences what happens at nextStock/Box Variable/ReservoirAnything that accumulates, buffer, resourceExamplesPopulationRadioactivityPhosphateBody fatLaborFlowRepresents activitiesExamplesBirthing, dying with populationIntaking & expending calories with body fatMoving from one population to anotherDiffusion ReactionsConverter/Variable/FormulaContains equations that generate output for each time periodConverts inputs into outputsTakes in information & transforms for use by another variableExamplesGrowth rate with population & growthCalories in a foodRates of reaction/diffusionConnector/Arrow/ArcLinkTransmits information & inputsRegulates flowsShows dependenceWith system dynamics toolEnter equationsRun simulationsProduce graphsProduce tablesHOMEWORK!READ Module 3.2 in the textbookYES CLASS on