143 Evaluate exponential functions with base e
As we saw earlier, the amount earned on an account increases as the compounding frequency increases. The table below shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to ask whether this pattern will continue.
Examine the value of $1 invested at 100% interest for 1 year, compounded at various frequencies.
| Frequency | [latex]A\left(t\right)={\left(1+\frac{1}{n}\right)}^{n}\\[/latex] | Value |
|---|---|---|
| Annually | [latex]{\left(1+\frac{1}{1}\right)}^{1}\\[/latex] | $2 |
| Semiannually | [latex]{\left(1+\frac{1}{2}\right)}^{2}\\[/latex] | $2.25 |
| Quarterly | [latex]{\left(1+\frac{1}{4}\right)}^{4}\\[/latex] | $2.441406 |
| Monthly | [latex]{\left(1+\frac{1}{12}\right)}^{12}\\[/latex] | $2.613035 |
| Daily | [latex]{\left(1+\frac{1}{365}\right)}^{365}\\[/latex] | $2.714567 |
| Hourly | [latex]{\left(1+\frac{1}{\text{8766}}\right)}^{\text{8766}}\\[/latex] | $2.718127 |
| Once per minute | [latex]{\left(1+\frac{1}{\text{525960}}\right)}^{\text{525960}}\\[/latex] | $2.718279 |
| Once per second | [latex]{\left(1+\frac{1}{31557600}\right)}^{31557600}\\[/latex] | $2.718282 |
These values appear to be approaching a limit as n increases without bound. In fact, as n gets larger and larger, the expression [latex]{\left(1+\frac{1}{n}\right)}^{n}\\[/latex] approaches a number used so frequently in mathematics that it has its own name: the letter [latex]e[/latex]. This value is an irrational number, which means that its decimal expansion goes on forever without repeating. Its approximation to six decimal places is shown below.
A General Note: The Number e
The letter e represents the irrational number
The letter e is used as a base for many real-world exponential models. To work with base e, we use the approximation, [latex]e\approx 2.718282\\[/latex]. The constant was named by the Swiss mathematician Leonhard Euler (1707–1783) who first investigated and discovered many of its properties.
Example 7: Using a Calculator to Find Powers of e
Calculate [latex]{e}^{3.14}\\[/latex]. Round to five decimal places.
Solution
On a calculator, press the button labeled [latex]\left[{e}^{x}\right]\\[/latex]. The window shows [e^(]. Type 3.14 and then close parenthesis, (]). Press [ENTER]. Rounding to 5 decimal places, [latex]{e}^{3.14}\approx 23.10387\\[/latex]. Caution: Many scientific calculators have an “Exp” button, which is used to enter numbers in scientific notation. It is not used to find powers of e.
Try It 9
Use a calculator to find [latex]{e}^{-0.5}\\[/latex]. Round to five decimal places.
Investigating Continuous Growth
So far we have worked with rational bases for exponential functions. For most real-world phenomena, however, e is used as the base for exponential functions. Exponential models that use e as the base are called continuous growth or decay models. We see these models in finance, computer science, and most of the sciences, such as physics, toxicology, and fluid dynamics.
A General Note: The Continuous Growth/Decay Formula
For all real numbers t, and all positive numbers a and r, continuous growth or decay is represented by the formula
where
- a is the initial value,
- r is the continuous growth rate per unit time,
- and t is the elapsed time.
If r > 0, then the formula represents continuous growth. If r < 0, then the formula represents continuous decay.
For business applications, the continuous growth formula is called the continuous compounding formula and takes the form
where
- P is the principal or the initial invested,
- r is the growth or interest rate per unit time,
- and t is the period or term of the investment.
How To: Given the initial value, rate of growth or decay, and time t, solve a continuous growth or decay function.
- Use the information in the problem to determine a, the initial value of the function.
- Use the information in the problem to determine the growth rate r.
- If the problem refers to continuous growth, then r > 0.
- If the problem refers to continuous decay, then r < 0.
- Use the information in the problem to determine the time t.
- Substitute the given information into the continuous growth formula and solve for A(t).
Example 8: Calculating Continuous Growth
A person invested $1,000 in an account earning a nominal 10% per year compounded continuously. How much was in the account at the end of one year?
Solution
Since the account is growing in value, this is a continuous compounding problem with growth rate r = 0.10. The initial investment was $1,000, so P = 1000. We use the continuous compounding formula to find the value after t = 1 year:
The account is worth $1,105.17 after one year.
Try It 10
A person invests $100,000 at a nominal 12% interest per year compounded continuously. What will be the value of the investment in 30 years?
Example 9: Calculating Continuous Decay
Radon-222 decays at a continuous rate of 17.3% per day. How much will 100 mg of Radon-222 decay to in 3 days?
Solution
Since the substance is decaying, the rate, 17.3%, is negative. So, r = –0.173. The initial amount of radon-222 was 100 mg, so a = 100. We use the continuous decay formula to find the value after t = 3 days:
So 59.5115 mg of radon-222 will remain.
