# Activities

• • • ##### Subject Area

• Math: Calculus: Differential Equations

• ##### Author 9-12

45 Minutes

• ##### Device
• TI-Nspire™ CX/CX II
• TI-Nspire™ CX CAS/CX II CAS
• TI-Nspire™ Navigator™
• ##### Software

TI-Nspire™ CX
TI-Nspire™ CX CAS

5.0

• ##### Report an Issue

Exponential Functions and the Natural Logarithm

Updated on 06/13/2019

#### Activity Overview

Discover a surprising property involving the relative growth rate of an exponential function.

#### 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) = 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

#### Vocabulary

• Relative growth rate
• Exponential function
• Natural logarithm
• Natural exponential function and its base e