## 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 $A\left(t\right)={\left(1+\frac{1}{n}\right)}^{n}$ Value Annually ${\left(1+\frac{1}{1}\right)}^{1}$$2
Semiannually ${\left(1+\frac{1}{2}\right)}^{2}$ $2.25 Quarterly ${\left(1+\frac{1}{4}\right)}^{4}$$2.441406
Monthly ${\left(1+\frac{1}{12}\right)}^{12}$ $2.613035 Daily ${\left(1+\frac{1}{365}\right)}^{365}$$2.714567
Hourly ${\left(1+\frac{1}{\text{8766}}\right)}^{\text{8766}}$ $2.718127 Once per minute ${\left(1+\frac{1}{\text{525960}}\right)}^{\text{525960}}$$2.718279

### Solution

Since the account is growing in value, this is a continuous compounding problem with growth rate = 0.10. The initial investment was $1,000, so = 1000. We use the continuous compounding formula to find the value after = 1 year: $\begin{cases}A\left(t\right)\hfill & =P{e}^{rt}\hfill & \text{Use the continuous compounding formula}.\hfill \\ \hfill & =1000{\left(e\right)}^{0.1} & \text{Substitute known values for }P, r,\text{ and }t.\hfill \\ \hfill & \approx 1105.17\hfill & \text{Use a calculator to approximate}.\hfill \end{cases}$ 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?

Solution

### 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, = –0.173. The initial amount of radon-222 was 100 mg, so = 100. We use the continuous decay formula to find the value after = 3 days:

$\begin{cases}A\left(t\right)\hfill & =a{e}^{rt}\hfill & \text{Use the continuous growth formula}.\hfill \\ \hfill & =100{e}^{-0.173\left(3\right)} & \text{Substitute known values for }a, r,\text{ and }t.\hfill \\ \hfill & \approx 59.5115\hfill & \text{Use a calculator to approximate}.\hfill \end{cases}$

So 59.5115 mg of radon-222 will remain.

### Try It 11

Using the data in Example 9, how much radon-222 will remain after one year?

Solution