Weekend Reading: Adam Tooze: Goodbye to the American Century

Buildng Tools: Growth

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-MuḳābalaThe Compendious Book on Calculation by Completion and Balancing
  • Isaac Newton (1642-1727): Philosophiæ Naturalis Principia MathematicaMathematical 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…

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

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