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CS 170: Computing for the Sciences and Mathematics

CS 170: Computing for the Sciences and Mathematics. Unconstrained Growth and Decay. Administrivia. Last time Error and Basic differential calculus Assigned HW 2 Today HW2 due! Unconstrained Growth WE HAVE CLASS ON MONDAY. Unconstrained Growth. Population growth without constraints

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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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