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 exponential functions.

Modules:

Definition. Let a be a positive real number. The exponential function with base a is the function: f(x) = ax.

• 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:
    1. a^r a^s = a^r+s
    2. a^r/a^s = a^r-s
    3. (a^r)^s = a^rs
    4. 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.
  • TI-86 Graphing Calculator [Using Flash]
  • TI-85 Graphing Calculator

  Definition. Let natural exponential function is the function: f(x) = ex 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.