Education Technology

What Is the Number "e"?

Published on 06/09/2008

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.

    Follow the Activity procedures:

  • 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