What is exponential function?
As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough.
What happens to the left tail of a graph?
Before graphing, identify the behavior and key points on the graph. Since is between zero and one, the left tail of the graph will increase without bound as decreases , and the right tail will approach the x -axis as increases. Since the graph of will be stretched by a factor of.
Using points to sketch an exponential graph
The best way to graph exponential functions is to find a few points on the graph and to sketch the graph based on these points.
Try it!
Identify points on the graph of the exponential function above and completing the table below.
Try it!
Consider the exponential function above. The -intercept of its graph, or the initial value of the function, is
Definition of exponential functions
In its most basic form, an exponential function can be thought of as a function where the variable appears in the exponent. The simplest exponential function is a function of the form , where b is a positive number.
Graphs of exponential functions
Let’s look at the following examples of how to graph exponential functions.
Limitation of b to positive numbers
The value of b is limited to numbers greater than 1 due to the following reasons:
What is the power series of exponential function?
The power series definition of the exponential function makes sense for square matrices (for which the function is called the matrix exponential) and more generally in any unital Banach algebra B. In this setting, e0 = 1, and ex is invertible with inverse e−x for any x in B. If xy = yx, then ex + y = exey, but this identity can fail for noncommuting x and y .
How to find derivative of exponential function?
The derivative of the exponential function is equal to the value of the function. From any point P on the curve (blue), let a tangent line (red), and a vertical line (green) with height h be drawn, forming a right triangle with a base b on the x -axis. Since the slope of the red tangent line (the derivative) at P is equal to the ratio of the triangle's height to the triangle's base (rise over run), and the derivative is equal to the value of the function, h must be equal to the ratio of h to b. Therefore, the base b must always be 1.