**Growth: Exponential, Convergent, Logistic:** How much of this will my students this semester know? How much of this will I have to remind them? And how much of this will I have to teach them for the first time?

**The Uses of Math**

- Muḥammad ibn Mūsā al-Khwārizmī (c. 780-850):
*Al-Kitāb al-Mukhtaṣar fī Hisāb al-Jabr wa’l-Muḳābala*—*The Compendious Book on Calculation by Completion and Balancing* - Isaac Newton (1642-1727):
*Philosophiæ Naturalis Principia Mathematica*—*Mathematical Principles of Natural Philosophy* - Arithmetic and accounting
- Algebra and calculus
- What-if machines—ways of doing a huge number of potential calculations all at once…

**Exponential Growth**

- dy/dt = g(y - a)
- does nothing for a long time—stays very near a—then explodes
- And keeps on exploding…
- Rules of thumb for an annual growth rate g:
- (y-a) doubles every 0.693/g years
- (y-a) grows a thousandfold every 6.91/g years

**Exponential Convergence**

- dy/dt = g(k - y)
- heads rapidly for k
- and then stays there
- (k-y) halves in… guess what? 0.693/g
- (k-y) shrinks to a thousandth of its initial value in… guess what? 6.91/g

**Combine the Two: Logistic Growth**

- Math
- dy/dt = g(y-a)(k-y)/k
- y = a + (k-a)[exp(gx)]/(k-a+exp(gx)-1)

- a is the initial population
- k is the carrying capacity
- g is the unimpeded growth rate (you’ll see this called “r”)
- Pierre-Francois Verhulst in 1838, building a mathematical model of Thomas Malthus’ Essay on the Principal of Population
- Rediscovered by McKendrick, by Pearl and Reed, and by Lotka

**Logistic Growth: Things to Remember**

- Asymptote: a (in the negative direction, for growth and logistic)
- Asymptote: k (in the positive direction, for convergence and logistic)
- Rule of 72: 72 divided by the growth rate gives you the doubling (for growth) or halving (for convergence) time
- Rule of 720: Multiply the doubling time by 10 to get the thousand-fold time
- Why 72? Why not 0.693?
- 72 is easier to do in your head
- 72 = 36x2=24x3=18x4=12x6=9x8
- If things aren’t continuous but come in steps…

https://www.icloud.com/keynote/0GIwRdnkZgEVFuB0Jmj2EQn4Q#2017-01-14_Tools-Growth_.TCEH_.AEH