The term "e^at" represents an exponential function where "e" is the mathematical constant approximately equal to 2.71828, and "a" is a constant multiplier. This type of exponential function is commonly encountered in various fields of mathematics, science, and engineering.
The general form of the exponential function is given by:
(
)
=
f(t)=e
at
Here's a breakdown of the components:
(
)
f(t): This represents the value of the function at time "t."
e: This is the mathematical constant known as Euler's number, approximately equal to 2.71828. It's a fundamental constant that appears frequently in various mathematical contexts, including exponential growth and decay.
a: This is a constant multiplier that determines the rate of growth or decay of the function. A positive value of "a" leads to exponential growth, while a negative value of "a" leads to exponential decay.
t: This represents the independent variable, often representing time in various applications. It's the variable with respect to which the function's behavior is being analyzed.
Exponential functions of the form
e
at
have several important properties:
Exponential Growth: When
>
0
a>0, the function
e
at
exhibits exponential growth. As time increases, the function's value increases rapidly.
Exponential Decay: When
<
0
a<0, the function
e
at
exhibits exponential decay. As time increases, the function's value decreases rapidly.
Initial Value: The value of the function at
=
0
t=0 is
⋅
0
=
0
=
1
e
a⋅0
=e
0
=1. This is true regardless of the value of "a."
Rate of Change: The rate of change of the function is proportional to its current value. Mathematically,
=
⋅
dt
df
=a⋅e
at
, which means the rate of change is directly proportional to the value of the function.
Exponential functions have wide applications in various fields, such as physics, biology, economics, and engineering. They describe phenomena where a quantity grows or decays at a rate that is proportional to its current value, which is a common behavior in many natural and man-made systems.