Type any function into the input bar on the right of the applet ( f(x)=sin(x) is a useful example that has already been provided, but you may enter what you'd like). [list] [*]Point A is a point on your function, and can be changed using the slider below the input box [*]Point B is a point on the derivative of your function, and will trace the movements of A. [*]Using the show f'(x) check box you can toggle the graph of the derivative on and off. [*]The value [b]s[/b] shows you the slope of the tangent line of the function at any given point A [*]The play button will animate the slider, and show you how the points move together. Feel free to hit pause at any time. [/list]
Play with the applet above, and use your observations to answer the following questions: 1)The value [b]s[/b] shows you the slope of the tangent line of the function at any given point A. How does this slope ([b]s[/b]) relate to the location of point B? 2)Look at the position of point B when [b]s[/b] is positive versus when [b]s[/b] is negative. Do you see a pattern? What is it? (Hint: look at B's position with respect to the x axis) 3)Whenever B crosses the x axis something in particular happens to our original function. In mathematical terms, what could we call point A at any moment that point B is actually on the x axis? (Hint: Point A would be either a relative ___________ or a relative ___________ for our function, depending on if B crosses the x axis from positive to negative or from negative to positive.)