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## Lesson 21: Rates of Change and the Number e

Exponential functions are used in many areas in science and in business. Some examples of exponential functions are included below:

Here we are introduced to a number that is commonly found in various forms in nature. The number e is an irrational number that lies between 2.71 and 2.72. The number e is special because the derivative of e raised to the power x is also e raised to the power of x.

- The population of rats in a rat farm triples every n days. Then, a function to model the population of the rats is P(t)=I[(3)^(t/n)], where I stands for the initial population, and P represents the current population after t days.
- In the function M(x)=I[(1/2)^(x)], M represents the amount of remaining radioactive material, in grams, after x half-life periods, if I is the initial amount of radioactive material, in grams.
- In the function V(t)=10,000[(1.12)^(t)] $10,000 is earning compound interest in an account at a rate of 12% per year. V represents the value of the investment after t year.

Here we are introduced to a number that is commonly found in various forms in nature. The number e is an irrational number that lies between 2.71 and 2.72. The number e is special because the derivative of e raised to the power x is also e raised to the power of x.

The value of e is irrational, meaning that e cannot be written as a fraction. The exact value of e is given by the limit:

That's all there is to this lesson. In the next lesson, we will look at the inverse function of the number e.

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