What does f '(x) say about f(x)?

The applet "What does f '(x) say about f(x)?" illustrates the relationship between a function and its derivative.

Use the Applet:

Enter the coefficients of a 3^rd order polynomial, select the "Draw Curve" button, and move the "x" slider.  As the "x" slider is moved a plot of the tangent line to f(x) at the current value of x is drawn as well a plot of f '(x).  

The value of x, f(x), and f ' (x) are calculated and presented.

Select the "Relationship" button for a description of the relationship between the function and its derivative.

Points of Interest:

• f '(x) > 0, on an interval, f(x) is increasing
• f '(x) > 0, the slope of the tangent to f(x) is positive
• f '(x) < 0, on an interval, f(x) is decreasing
• f '(x) < 0, the slope of the tangent to f(x) is negative
• f '(x) = 0,  the tangent to f(x) is a horizontal line or has a slope of zero

 Use Applet (author Garrett Heath)