Education Technology

Investigating One Definition of Derivative

Published on 06/09/2008

Activity Overview

In this activity, students investigate limit h ---> 0 [f (x + h) - f (x)]/[(x + h) - (x)] definition of the derivative.

Before the Activity

  • See the attached PDF file for detailed instructions for this activity
  • Print pages 107 - 113 from the attached PDF file for your class
  • During the Activity

    Distribute the pages to the class.

    Follow the Activity procedures:

  • Define the function f as f(x) = x2 + 1
  • Find the slope m of the secant line that contains the points (1, f(1)), and (2, f(2))
  • Define h = 1 and ms as the slope of the secant line that contains the points (1, f(1)), and (1 + h, f(1 + h))
  • Note ms = [f(1 + h) - f(1)]/[(1 + h) - 1]
  • Observe that when h = 1, (2, f(2)), and (1 + h, f(1 + h)) are the same point
  • Enter the coordinates of the two points on the secant line and the slope, ms in the table
  • Record the slope, ms of the secant line and the coordinates of the two points that contains the points (1, f(1)), and (1 + h, f(1 + h)) when h = 0.5 in the table
  • Change h to 0.1, 0.01, -1, -0.5, -0.1, and -0.01 and record the information in the table
  • Observe that as x + h gets closer to x, h approaches 0 and the slope approaches 2
  • Note as h ----> 0, the slope appears to be approaching 2
  • Redefine h = 1
  • Define the function g as the equation of the secant line g(x) = ms*(x - 1) + f(1)
  • Enter {1, 1 + h} in L1 and {f(1), f(1 + h)} in L2
  • Graph f(x), g(x), and the scatter plot L1, L2
  • Sketch f(x), g(x), and the scatter plot on the provided grid
  • Change h to 0.5 and notice that (1, f(1 + h)) gets closer to (1, f(1)) and the secant line changes
  • Sketch the new g(x) and the scatter plot L1, L2 on the same grid
  • Enter the equation of the line tangent x = 1 by defining, y3(x): = mt*(x - 1) + f(1)
  • Repeat the above steps for h = 0.1, 0.01, -1, -0.5, and -0.01 and observe that as h ----> 0, the secant lines approach the tangent line
  • Observe that when h is very small, g(x), the secant line, and t(x), the tangent line, appear to be the same line
  • After the Activity

    Students complete the Student Activity sheet.

    Review student results:

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