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CS 170: Computing for the Sciences and Mathematics

This module explores unconstrained growth and decay, focusing on population dynamics and decay processes like radioactive decay. We apply basic differential calculus to understand how the rate of change in populations is proportional to their size. The understanding of constants like growth and decay rates is made through analytical solutions and numerical approximations. We will also discuss systems dynamics tools that simulate populations' changes over time, utilizing finite difference equations. The session will conclude with essential homework related to these concepts for deeper understanding.

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CS 170: Computing for the Sciences and Mathematics

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  1. CS 170:Computing for the Sciences and Mathematics Unconstrained Growth and Decay

  2. Administrivia • Last time • Error and Basic differential calculus • Assigned HW 2 • Today • HW2 due! • Unconstrained Growth • WE HAVE CLASS ON MONDAY

  3. Unconstrained Growth • Population growth without constraints • Examples?

  4. Example of Unconstrained Growth • Rate of change of population is directly proportional to number of individuals in the population (P) dP/dt = rP where r is the growth rate.

  5. Analytic Solution • “Closed Form” solution • Can determine with a computer algebra system • Like Maple • P = P0ert

  6. Exponential 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?

  7. 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.

  8. Finite difference equation new = 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 time

  9. Approximating Unconstrained Growth initialize simulationLength, population, growthRate, ∆t numIterations  simulationLength / ∆t fori going from 1 to numIterations do the following: growth  growthRate * population population  population + growth * ∆t t  i * ∆t display t, growth, and population • UNITS ARE IMPORTANT • Does this give me the exact answer?

  10. Systems Dynamics Tool • Helps to model • Performs simulation • What happens at one time step influences what happens at next

  11. Stock/Box Variable/Reservoir • Anything that accumulates, buffer, resource • Examples • Population • Radioactivity • Phosphate • Body fat • Labor

  12. Flow • Represents activities • Examples • Birthing, dying with population • Intaking & expending calories with body fat • Moving from one population to another • Diffusion • Reactions

  13. Converter/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/diffusion

  14. Connector/Arrow/Arc • Link • Transmits information & inputs • Regulates flows • Shows dependence

  15. With system dynamics tool • Enter equations • Run simulations • Produce graphs • Produce tables

  16. HOMEWORK! • READ Module 3.2 in the textbook • YES CLASS on Monday

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