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

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) = x^{2} + 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