11.3 The Number e Objective: Use the exponential function y = ex.

Euler’s Number (pronounced oiler) e is an irrational constant. e ≈ 2.718281828459045… Does the number e have any real physical meaning, or is it just a mathematical convenience? Yes, the number e does have physical meaning. It occurs naturally in any situation where a quantity increases at a rate proportional to its value, such as a bank account producing interest, or a population increasing as its members reproduce. Obviously, the quantity will increase more if the increase is based on the total current quantity (including previous increases), than if it is only based on the original quantity (with previous increases not counted). How much more? The number e answers this question.

Exponential Growth or Decay (in terms of e) 𝑁= 𝑁 𝑂 𝑒 𝑘𝑡

Continuously Compounded Interest 𝐴=𝑃 𝑒 𝑟𝑡

𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 𝐴=𝑃 𝑒 𝑟𝑡 𝐴=10000 1+ .0675 2 (2∙25) 𝐴=10,000 𝑒 .0675∙25 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 𝐴=𝑃 𝑒 𝑟𝑡 𝐴=10000 1+ .0675 2 (2∙25) 𝐴=10,000 𝑒 .0675∙25 𝐴=$54,059.49 𝐴=$52,574.62 The continuously compounded account earns $1484.87 more than the account that is compounded semiannually.

𝑁= 𝑁 𝑂 𝑒 𝑘𝑡 𝑎) 𝑁=87,806 𝑒 −.004𝑡 𝑏) 𝑁=87,806 𝑒 −.004∙12 ≈83,691 people

𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 𝐴=𝑃 𝑒 𝑟𝑡 𝐴=5000 1+ .085 4 4∙10 𝐴=5000 𝑒 .085∙10 𝐴=𝑃 1+ 𝑟 𝑛 𝑛𝑡 𝐴=𝑃 𝑒 𝑟𝑡 𝐴=5000 1+ .085 4 4∙10 𝐴=5000 𝑒 .085∙10 𝐴=$11,698.23 𝐴= $11594.52 The continuously compounded account earns $103.71 more than the account that is compounded semiannually.

Assignment 11.3 Practice Worksheet 11.3 pg 714 #1, 2, 4, 6, 7, 9