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