College Algebra & Trigonometry, 2018a

3.1. EXPONENTIAL AND LOGISTIC APPLICATIONS 141
Exponential Relationships
The next type of relationship is the focus of this chapter - the exponential relationship.
In this situation, the rate of change of a quantity is proportional to the size
of that quantity. This relationship can be explored in more depth in an integral
calculus course, but we will discuss the basics here.
In a linear or proportional relationship, the slope, or rate of change, is constant.
For example, in the equation y =3x+1, the slope is always three, no matter what
the values of x and y are. In an exponential relationship, the rate of change (also
called ”y prime” or y ′ ) is proportional to the value of y. In this case, we say that
y ′ = k ∗ y.
This is what is known as a differential equation. This is an equation in which the
variable and its rate of change are related. Through the processes of differential
and integral calculus, we can solve the equation above y ′ = k ∗ y as:
y = Ae kt
In the equation above, A is the value of y at time t = 0, k is a constant that
determines how fast the quantity y increases or decreases and t plays the role
of the independent variable (as x often does) and represents the time that has
passed. If k is positive, then the quantity y is growing because its rate of change
is positive. If k is negative, then the quantity y is decreasing because the rate of
change is negative.
The quantity represented by e in the above equation is a mathematical constant
(like π) that is often used to represent exponential relationships. The best way to
understand the value of e and what it represents is directly related to fundamental
questions from differential and integral calculus.
Differential Calculus is concerned primarily with the question of slopes. We discussed
earlier that a linear relationship has a constant slope. Polynomial and exponential
relationships have slopes that depend on the value of x and/or y. This
is what makes them curves rather than lines. If we consider the slopes of some
different exponential relationships, we can see one aspect of where the value for
e comes from.