Amount Function

As a health-conscious person, you’re interested in tracking your weight. You decide to step on the scale every day and record the result. After three days, your data might look like this:

$$w(1) = 150, \quad w(2) = 151, \quad w(3) = 149$$

where the numbers in parentheses represent the day number, and the values represent your weight in pounds.

This creates a weight function, denoted $w(d)$. This function takes a day number $d$ as input and returns your weight on that particular day as output. In other words, $w(d)$ tells you your total weight on day $d$.

Once you understand the weight function, you can apply the same logic to other measurements. For instance, you could create a height function $h(d)$ that tracks your height over time (though this changes much more slowly than weight!).

This same principle appears in finance, where we care about the amount function $A(t)$. This function tracks the total value of an investment at time $t$, typically measured in years. For example:

$$A(0) = 1000, \quad A(1) = 1050, \quad A(2) = 1102.50$$

might represent an investment that starts at $ \$1,000$ and grows over time. Just as $w(d)$ tells you your weight on a given day, $A(t)$ tells you how much money you have at a given time.