Discover a surprising property involving the relative growth rate of an exponential function.
- Understand the distinction between growth rate (slope) and relative growth rate (slope/y -value) of a function
- Understand that the relative growth rate f'(x) / f(x) is constant for exponential functions of the form f(x) = bx
- Iidentify the natural logarithm of the base of an exponential function as this constant relative growth rate
- Describe the behavior of the graph of y = f(x) = bx for different values of b
- Understand what is special about the number e as a base for exponential functions in terms of growth rate
- Relative growth rate
- Exponential function
- Natural logarithm
- Natural exponential function and its base e
About the Lesson
The purpose of this lesson is to guide students to discover a surprising property involving the relative growth rate of an exponential function. Students will consider the ratio f'(x) / f(x) for f(x) = bx and see that for a given value b this ratio is a constant for all values of x. We define this ratio as the natural logarithm of b, denoted ln(b).