This Demonstration shows the instantaneous rate of change of
for different
values for polynomial functions of degree 2, 3, and 4, an exponential function, and a logistic function.
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
?
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