[list][*]Active the box [b]Trace derivative[/b] to sketch the derivative.[/*][*]Slowly drag the point to observe how the slope of the tangent segment changes. (If you can't see the drag button, click on "controls")[/*][*]Observe what happens to [math]f'\left(x\right)[/math] when [math]f\left(x\right)[/math] is increasing and decreasing. [/*][*]Change the function [math]f\left(x\right)[/math]. Repeat the previous step. Examples:[/*][list][*][math]f\left(x\right)=x^2[/math][br][math]f\left(x\right)=x^3[/math][br][/*][/list][*]After trying out some functions, answer the questions at the bottom. [/*][/list]
Using the graph of [math]f\left(x\right)=x^2[/math], what can you say about [math]f'\left(3\right)[/math]?
Using the graph of [math]f\left(x\right)=x^2[/math], what can you say about [math]f'\left(-3\right)[/math]?
Using the graph of [math]f\left(x\right)=x^2[/math], what can you say about [math]f'\left(0\right)[/math]?
After trying different functions, what do you notice about [math]f'\left(x\right)[/math]? In other words, when is [math]f'\left(x\right)[/math] positive and when is [math]f'\left(x\right)[/math] negative?
[math]f'\left(x\right)[/math] is positive when [math]f\left(x\right)[/math] is increasing and [math]f'\left(x\right)[/math] is negative when [math]f\left(x\right)[/math] is decreasing.
Can you come up with a function where the derivative is negative at all x values?