# Activities

• • • ##### Subject Area

• Math: Algebra II: Logarithms and Exponentials

• ##### Author 9-12

60 Minutes

• ##### Device
• TI-92 Plus / Voyage™ 200
• ##### Other Materials
This is Activity 12 from the EXPLORATIONS Book:
Discovering Math on the Voyage 200.
• ##### Report an Issue

What Is the Number "e"?

#### Activity Overview

In this activity, students explore the fundamental theorem of calculus. They apply this theorem to understand the number "e" - the base of the natural logarithm. They also use integration and natural logarithmic function to solve a typical problem in exponential growth.

#### Before the Activity

• See the attached PDF file for detailed instructions for this activity
• Print pages 89 - 99 from the attached PDF file for your class
• #### During the Activity

Distribute the pages to the class.

• Use the calculator to graph and calculate the area under the curve
• Find the definite integral of a function
• Find a value of b such that integral b to1 of (1/x) dx is equal to 1
• Observe that changing the value of b changes the area under the curve
• Graph the function myst(x) = (1 / t )dt
• Find where myst(u) intersects the lines y = 0,1,2,3 to solve integral u to 1 (1/t ) dt = 1, integral from u to2 (1 / t) dt = 2 and so on
• Observe that the point of intersection is (1, 0)
• Find 3 other points of intersection
• Observe that the values of y increase linearly and values of x increase rapidly
• Find the common ratios of the x values and observe this value defines the number e
• Enter the differential equation to solve a problem in exponential growth
• Integrate each side of the equation separately with a constant k of integration on the right
• Find the value of constant k
• Enter the value of k in the exponential equation and find the result
• #### After the Activity

Review student results:

• As a class, discuss questions that appeared to be more challenging
• Re-teach concepts as necessary