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
Re-teach concepts as necessary