Objectives: In this tutorial, we look at the definition of the exponential function and some of its properties. We investigate what it means to raise
a number to an irrational power. The graphs of various exponential functions
are compared; in addition, a comparison with the graphs of polynomial functions is made. The number e is introduced. There is an exploration which looks at
the approximation of the natural exponential function by polynomials.
After working through these materials, the student should be able
 to discuss the definition of the exponential function;
 in particular, to discuss what it means to raise a number to a power which is an irrational number;
 to describe the properties and graphs of exponential functions;
 to describe the relationship of the longterm behavior of exponential functions and polynomials;
 to describe the approximation of the natural exponential function by a certain family of polynomials;
 to recognize graphically and algebraically geometric transformations of the
exponential functions.
Modules:
Definition. Let a be a positive real number. The exponential function with base a
is the function:
f(x) = a^{x}.


 Discussion
[Using Flash]
 A Javascript module illustrating the numerical approximation
of an irrational power of a number.
 Comments:
 The domain of f(x) = a^{x} consists of all real numbers.
 The range of f(x) = a^{x} is the collection of all positive real numbers.
 If a < 1 then f is a decreasing function and if a > 1 then f is
an increasing function.
 Exponential functions satisfy the following properties:
 a^{r} a^{s} = a^{r+s}
 a^{r}/a^{s} = a^{rs}
 (a^{r})^{s} = a^{rs}
 a^{r} b^{r} = (ab)^{r}
 Examples:
 Use this LiveMath notebook to view an animation
showing the graphs of a parametrized family of exponential functions.
 Use this LiveMath notebook to determine which of exponential functions or polynomial functions grows more rapidly.
 Using a graphing calculator to determine which of exponential functions or polynomial functions grows more rapidly.
Definition. Let natural exponential function
is the function:
f(x) = e^{x}
where e = 2.71828182846... is a very interesting irrational number.


 Comments:
 We shall see later why e is a very interesting number.
 Since e > 1, the natural exponent function is an increasing function.
 Examples:
 LiveMath notebooks to explore graphically and symbolically the effect of transforming exponential functions.
 A Java applet to explore graphically and symbolically the effect of transforming exponential functions.
 A LiveMath Notebook to approximate the exponential function y = e^{x} by certain polynomials.