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Explore Continuous Growth with Natural Base e

Understand continuous compounding, investments, and population growth using the natural base e (2.71828). Learn the formula A = Pert and practical applications in Tacoma and Portland populations.

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Explore Continuous Growth with Natural Base e

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  1. Suppose Marcello invests $500 at 1.2% annually. How long will it take for that amount to double?

  2. Better than investing annually?

  3. e (2.71828…):the natural baseIt represents the base rate of growth shared by all continuously growing processes.

  4. e (2.71828…):the natural baseQ: Why the natural base?A: Because it shows up in population growth, radioactive decay, and in systems that exhibit continuous growth or decay.

  5. Suppose we applied a (theoretical) continuous growth rate to an investment. Continuously Compounding Interest Formula:A = Pert

  6. A = PertNote that the expression has been substituted with er now.

  7. Take Tacoma’s population (as of 2012) of 202,010 which was growing at 1.8%. If the growth rate remains the same, what will its population be in 2013? Continuously Compounding Interest Formula:A = Pert= 202,010e0.018*3= ~213218

  8. If Portland, with its 2012 population of 603,106, is continuously growing at 1.7% then what will its population be in 2017? It’s e-asy!

  9. How e-nteresting Portland, with its 2012 population of 603,106, is continuously growing at 1.7%. When what will its population surpass 700,000?700,000 = 603,106e0.017t[solving for t!]1.16065… = e0.017t[divide by 603,106]loge1.16065 = 0.017t[put into log form,and with a base of e]0.148987… = 0.017t[evaluate dat log]8.76 396… = t[divide by 0.017]

  10. Portland, with its 2012 population of 603,106, is continuously growing at 1.7%. When what will its population surpass 800,000? e-nsane in the Membrane

  11. oh btw logex = ln(x)The natural basedeserves a natural logarithm.So from now on, instead of using loge(3),a shortcut to use is now ln(3).

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