Choose the cubic polynomial for some experiments. Consider the red point and line. Move slowly from -1.17 to 3.9. What changes do you notice? Is there a relation? Write down your conclusion. Move slowly from -1.17 to 3.9 again. Make a table for the intervals in which is negative, zero, or positive.

What is the orientation of the tangent line (increasing, decreasing, horizontal) when is positive or negative?

What is the position of the tangent line if ?

How does change its sign around the local maximum? Around the local minimum? Moving the slider from left to right?

You found a relation between the value and the position of the tangent line.

Is this relationship also correct for the fourth-degree polynomial, the mixed exponential function , and the logistic function ?