Key Equations
definition of the exponential function | [latex]f\left(x\right)={b}^{x}\text{, where }b>0, b\ne 1[/latex] |
definition of exponential growth | [latex]f\left(x\right)=a{b}^{x},\text{ where }a>0,b>0,b\ne 1[/latex] |
compound interest formula | [latex]\begin{cases}A\left(t\right)=P{\left(1+\frac{r}{n}\right)}^{nt} ,\text{ where}\hfill \\ A\left(t\right)\text{ is the account value at time }t\hfill \\ t\text{ is the number of years}\hfill \\ P\text{ is the initial investment, often called the principal}\hfill \\ r\text{ is the annual percentage rate (APR), or nominal rate}\hfill \\ n\text{ is the number of compounding periods in one year}\hfill \end{cases}[/latex] |
continuous growth formula |
[latex]A\left(t\right)=a{e}^{rt},\text{ where}[/latex]t is the number of unit time periods of growth a is the starting amount (in the continuous compounding formula a is replaced with P, the principal) e is the mathematical constant, [latex] e\approx 2.718282[/latex] |
Key Concepts
- An exponential function is defined as a function with a positive constant other than 1 raised to a variable exponent.
- A function is evaluated by solving at a specific value.
- An exponential model can be found when the growth rate and initial value are known.
- An exponential model can be found when the two data points from the model are known.
- An exponential model can be found using two data points from the graph of the model.
- An exponential model can be found using two data points from the graph and a calculator.
- The value of an account at any time t can be calculated using the compound interest formula when the principal, annual interest rate, and compounding periods are known.
- The initial investment of an account can be found using the compound interest formula when the value of the account, annual interest rate, compounding periods, and life span of the account are known.
- The number e is a mathematical constant often used as the base of real world exponential growth and decay models. Its decimal approximation is [latex]e\approx 2.718282[/latex].
- Scientific and graphing calculators have the key [latex]\left[{e}^{x}\right][/latex] or [latex]\left[\mathrm{exp}\left(x\right)\right][/latex] for calculating powers of e.
- Continuous growth or decay models are exponential models that use e as the base. Continuous growth and decay models can be found when the initial value and growth or decay rate are known.
Glossary
- annual percentage rate (APR)
- the yearly interest rate earned by an investment account, also called nominal rate
- compound interest
- interest earned on the total balance, not just the principal
- exponential growth
- a model that grows by a rate proportional to the amount present
- nominal rate
- the yearly interest rate earned by an investment account, also called annual percentage rate