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

2. Administrivia Last time Today WE HAVE CLASS ON MONDAY
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:
(about % per year)
dQ/dt = Q
Q = Q0 e t
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
for i 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