Calculus: Exponential Functions and the Natural Logarithm

Objectives

• 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) = b^x
• 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) = b^x for different values of b
• Understand what is special about the number e as a base for exponential functions in terms of growth rate

Vocabulary

• 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) = b^x 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).