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 long-term 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
Definition. Let a be a positive real number. The exponential function with base a
is the function:
f(x) = ax.|
of an irrational power of a number.
- The domain of f(x) = ax consists of all real numbers.
- The range of f(x) = ax 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:
- ar as = ar+s
- ar/as = ar-s
- (ar)s = ars
- ar br = (ab)r
- 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) = ex|
where e = 2.71828182846... is a very interesting irrational number.
- We shall see later why e is a very interesting number.
- Since e > 1, the natural exponent function is an increasing function.
- 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 = ex by certain polynomials.