# Activities

• • • ##### Subject Area

• Math: Calculus: Derivatives

• ##### Author 9-12

60 Minutes

• ##### Software

TI InterActive!™

• ##### Other Materials
This is Activity 13 from the EXPLORATIONS Book:
TI InterActive! Math for High School

## Investigating One Definition of Derivative

#### 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.

• 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