In this applet, a quartic (4th degree) polynomial is graphed and we'll explore the
relationships between the characteristics of the graph (increasing, decreasing,
concave up, concave down) and the characteristics of its derivatives.
See how the positioning sliders work. One shifts the graph of `f` left and
right, and the other shifts it up and down. Now check the "Show `f'`" checkbox
and see the effect of shifting on the derivative. Is this what you expected?
Can you explain the behavior?
Play with the shaping sliders, to see the different ways you can shape `f`.
(Incidentally, after appropriate shifting, `f` is a polynomial of the form
`f(x) = ax^4 + bx^3 + cx^2 + dx`.)
From the power rule, we know that if `f` is a 4th degree polynomial, then
`f'` will be a 3rd degree polynomial and `f''` will be a 2nd degree polynomial.
Can you see how this is reflected in the applet?
Check the boxes to show where `f` is increasing and decreasing. Now see how
the derivative changes as `f` changes. Specifically, how are the increasing
sections of the `f`-graph related to the `f'`-graph? And the
Check the boxes to show where `f` is concave up and concave down. How are the
concave up sections of `f` related to the `f'`-graph?
How are the concave up sections of `f` related to the `f''`-graph?