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This is Activity 14 from the EXPLORATIONS Book: TI InterActive! Math for High School

Discovering the Derivative of the Sine and Cosine Functions

Activity Overview

In this activity, students discover the derivative of sin(x) and cos(x) by analyzing a scatter plot of x-values and the function's numerical derivatives at these x-values.

Before the Activity

See the attached PDF file for detailed instructions for this activity

Print pages 123 - 126 from the attached PDF file for your class

During the Activity

Distribute the pages to the class.

Follow the Activity procedures:

Define f(x) = sin(x), x1 = 1, and y1(x) = f(x)

Store x1 in L1 and derivative of f(x1) in L2

Set up a scatter plot

Sketch the graph on grid provided

Change values of x from 1 to {x, x, -2π, 2π, π/12} , {0, 1, 2, 3, 4, 5, 6} and { -6, -5 , -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 }, and add new slopes to the scatter plot on the grid

Define a function y2(x) that will connect the data in the scatter plot

Observe that the function that connects the data in the scatter plot is cos(x)

Define y3(x) as the derivative of f(x), use nDeriv function to find the derivative of f(x)

Note that y2(x) = y3(x)

Observe that if f(x) = sin(x) then f'(x) = cos(x)

Redefine f(x) = cos(x) and observe the effect on the graph

Observe that the graph of f(x) and the scatter plot of the numerical derivatives have changed

Also note, since y2 = cos(x), the graph of f(x) is the same as the graph of y2

Again define a function y2(x) that will connect the data in the scatter plot

Observe that the function that connects the data in the scatter plot is -sin(x)

Observe that if f(x) = cos(x) then f'(x) = -sin(x)

After the Activity

Students complete the Student Activity sheet.

Review student results:

As a class, discuss questions that appeared to be more challenging