# Exponential Functions Applications

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Examples, videos, worksheets, and activities to help PreCalculus students learn how to apply exponential functions.

Compound Interest (Finite Number of Calculations)
One real world application of exponential equations is in compound interest. The formula for compound interest with a finite number of calculations is an exponential equation. We can solve for a parameter of this equation, and can use logarithms to access parameters in the exponent. Students may be asked to solve compound interest problems with interest compounded biannually, monthly, or daily.

Deriving the Annual Compound Interest Formula

Compound Interest - More than Once Per Year
This video discusses the formula, and do a few simple examples.
Example:
Suppose you deposit \$100 at APR of 12% compounded quarterly. How much do you have after 1 year? 2 years?

Compound Interest - More than Once Per Year - Part 2
This video gives another example using the compound interest formula.
Example:
Suppose you would like to have \$20,000 in bank 18 years from now. If you get an APR of 6% componded monthly, how much would you have to invest today?

Continuous Compound Interest
Problems that involve continuous compound interest use a different equation from problems that have finitely compounded interest, but the continuous compound interest equation is also an exponential equation. We use many of the same methods for calculating continuous compound interest as we do finitely compounded interest. To calculate compound interest, we can use logarithms and methods for solving exponential equations.
Example:
How long does it take for an investment to double in value if it is invested at 10% per annum compounded monthly? Compounded continuously?

Exponential Growth and Decay
Exponential growth refers to an amount of substance increasing exponentially. Exponential growth is a type of exponential function where instead of having a variable in the base of the function, it is in the exponent. Exponential decay and exponential growth are used in carbon dating and other real-life applications.
How to find the doubling time of a population when the growth rate is given?
Example:
The population of mice in the Duchy of Grand Fenwick grows at a rate of 6% per year. How long will it take for the population to double? Quadruple?

Word Problem Solving - Exponential Growth and Decay
Examples:

1. Suppose a radioactive substance decays at a rate of 3.5% per hour. What percent of the substance is left after 6 hours?
2. Nadia owns a chain of fast food restaurants that operated 200 stores in 1999. If the rate of increase is annually, how many stores does the restaurant operate in 2007?

Continuously Compound Interest
Example:
If your great great great grandfather owed a \$0.30 fine 100 years ago on an overdue library book you just found in you attic, and the fine grows exponentially at 5% annual interest rate, compounded continuously, how much would it take to pay the fine today?

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