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Continuous Growth and the Number e

Explore the concept of compounding multiple times per year and its limitations, leading to the introduction of continuous compounding using the base e. Learn how to calculate the continuous percent growth rate and graph functions using this concept. Convert between different forms of compounding equations.

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Continuous Growth and the Number e

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  1. Continuous Growth and the Number e Lesson 3.4

  2. Compounding Multiple Times Per Year • Given the following formula for compounding • P = initial investment • r = yearly rate • n = number of compounding periods • t = number of years

  3. Compounding Multiple Times Per Year • What if we invested$1000 for 5 yearsat 4% interest • Try the formula for different numbers of compounding periods • Monthly • Weekly (n = 52) • Daily (n = 365) • Hourly (n = 365 * 24) What phenomenon do you notice?

  4. Compounding Multiple Times Per Year • You should see that we seem to reach a limit as to how much multiple compounding periods increase the final amount • So we come up with continuous compounding

  5. Using e As the Base • We have used y = A * Bt • Consider letting B = ek • Then by substitution y = A * (ek)t • Recall B = (1 + r) (the growth factor) • It turns out that

  6. Continuous Growth • The constant k is called the continuous percent growth rate • For Q = a bt • k can be found by solving ek = b • Then Q = a ek*t • For positive a • if k > 0 then Q is an increasing function • if k < 0 then Q is a decreasing function

  7. Continuous Growth • For Q = a ek*t Assume a > 0 • k > 0 • k < 0

  8. Continuous Growth • For the functionwhat is thecontinuous growth rate? • The growth rate is the coefficient of t • Growth rate = 0.4 or 40% • Graph the function (predict what it looks like)

  9. Converting Between Forms • Change to the form Q = A*Bt • We know B = ek • Change to the form Q = A*ek*t • We will eventually discover that k = ln B

  10. Assignment • Lesson 3.4 • Page 133 • Exercises 1 – 25 odd

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