Derivative of Sine Function (Intuitive Discovery)

Instructions: In the following applet, the graph of the function y = sin(x) is shown. You'll also notice there are 9 red points. The y-coordinate of each red point is displayed. These y-coordinate of each red point is the slope of the line drawn to the graph of this function (and appears either directly above or below it). Your job is to move these 9 red points up and/or down so that ALL the black lines APPEAR TANGENT to the graph of the function y = sin(x). Do this task now. After completing this task, check the "Show Graph of Derivative Function" checkbox to see how you did. The graph of the derivative of this function is shown in red. Please see the follow-up questions below this applet and answer them.
Questions: 1) Does the graph of this derivative function look familiar? 2) Write a function rule, f'(x), for the derivative of this function. 3) Prove, using the definition of a derivative, that the derivative function you wrote in (2) above is correct.