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: Using a Calculator to Find Powers of e
Calculate [latex]{e}^{3.14}[/latex]. Round to five decimal places.
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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: 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?
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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
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?
Try It
Using the data in Example 9, how much radon-222 will remain after one year?