Education Technology

Activities

• Subject Area

• Math: Calculus: Derivatives

9-12

60 Minutes

• Software

TI InterActive!™

• Other Materials
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.

• 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
• Re-teach concepts as necessary