Derive Definition of Exponential Function (Euler's Number) From Compound Interest

Exponential Function Euler's Number Value

This example demonstrates how the concept of compound interest can be used to derive the exponential function by calculating a numeric value for Euler’s Number. Note, while this is a valid definition of the function, this site considers as shorthand for and instead defines the exponential function using a power series.

Steps

  1. Start with the formula for compound interest, shown below. This formula models an exponential growth curve that is dependent the principle amount , rate of interest , the frequency of investment and the time elapsed .

    For example, given an interest rate of , a principle amount of , total time of and a yearly investment strategy of per year this formula calculates the accumulated amount shown below.

    Given the same interest rate, principle amount and time elapsed, but with a quarterly investment strategy of per year, the formula yields a larger value.

    Given the same interest rate, principle amount and time elapsed, but with monthly investment strategy of per year the formula again yields a larger value.

  2. The question that mathematicians pose is “what happens when the interest compounds continuously?” Or, in other words, what happens when approaches infinity. This question can be represented by taking the limit of the function.

    Taking the limit in this form poses a couple challenges, so first we will manipulate the expression into an easier form to take the limit of. Observe that is a scalar value and can be moved outside the limit.

    Then, because we want to calculate an approximate value for the limit, apply the transformation so the limit is only in terms of one variable. This gives us the form below.

    Finally, apply the product property of exponentiation which allows the variables to be split from .

  3. Evaluate the approximate value of the limit.

    This value is equal to the numeric value of Euler’s number and is repsented with the symbol . Substituting this value back into the formula for the limit gives us the population growth formula in terms of the exponential function.

  4. To transform this function into the exponential function, observe that the variable scales the curve vertically and the growth rate constant scales the curve horizontally. Setting , and replacing the variable for leaves us with the exponential function.

    Note: On this site, this version of the exponential function where the symbol is given an approximate value is for demonstration. This site considers as shorthand for the exponential function where . The exponential function is instead defined using a power series as shown on this page.

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