3.2 The Natural Exponential Function

In Section 3.1 we used the formula for compound interest A = P ( 1 + r / n )^{n t} , where A is the amount owed, P is the original amount borrowed, r is the annual interest rate and n is the number of compounding intervals per year.

The base of this exponential function is b = ( 1 + r / n )ⁿ. Let us consider the special case when
r = 100% = 1. That is, let b = ( 1 + 1 / n )ⁿ.

Exercise 3.2.1

Compute to six decimal places the value of b when n = 1, n = 10, n = 100, n = 1000,
n = 1,000,000

Solution

If we let n increase without bound, then b will approach a limiting value of approximately
2.718281828. This limiting number is sufficiently important in mathematics that it is given a special symbol: e. It is referred to as the natural base. Scientific calculators are capable of computing e^x for any input number x.

Using methods from calculus it can be shown that the limiting value of b = ( 1 + r / n )ⁿ when n increases without bound is equal to e^r. In the formula for compound interest, allowing n to increase without bound corresponds to compounding the interest continuously.

The formula for computing the amount owed when interest compounds continously at an annual rate of r is

A = P e^{r t}

Exercise 3.2.2

Find the amount owed on a loan of $1000 compounded continuously if the annual rate is 15%.

Solution

Exercise 3.2.3

Sketch the graph of f ( x ) = e^x for input values between - 1 and 1.

Solution

Return