(a) Enable the graphs of
f and
f ' and drag the
a0 slider to change the constant term. What happens to the graph of
f ?
When a0 increases, the graph of f is
, and when a0 decreases, the graph of f is
.
Observe that varying the value of
a0 has no visible effect on the graph of
f ' . Can you explain why this is to be expected for any polynomial function?
(b) Enable all three graphs and change the coefficient of
x by dragging the
a1 slider. What effect does this have on the graphs of
f,
f ' and
f '' ?
When a1 increases, the graph of f '' is
, the graph of f ' is
, and the graph of f is
.
Can you give reasons for what you observe in the graph of
f ' ?
Why is the graph of
f '' not affected at all?
(c) Use the
a3 slider to vary the coefficient of
x3. What happens to the shape of the graph of
f as the value of
a3 increases?
At what value does the graph appear to lose the peaks and valleys you originally see?
a3 ≈
(d) If you vary the value of
a3, what is the effect on the graph of
f' ?
When does the graph appear to lose the two valleys you originally see?
a3 ≈
As
a3 increases from
−3, which graph loses its peaks and valleys first,
f or
f ' ?
What is the effect on the graph of
f '' ?
(e) Use the
a5 slider to vary the coefficient of
x5. What happens to the shape of the graph of
f as the value of
a5 increases? At what value does the graph appear to lose the peaks and valleys you originally see? What happens when
a5 < 0?
(f) If you vary the value of
a5, what is the effect on the graphs of
f ' and
f '' ?