MA 3972-MA-Book April 11, 2018 17:21Graphs of Functions and Derivatives 173- (See Figure 8.8-9.)
[–π, π] by [–2, 2]
Figure 8.8-9Entery 1 =cos(x^2 )
Using the [Inflection] function of your
calculator, you obtain three points of
inflection on [0,π]. The points of
inflection occur atx= 1 .35521, 2.1945,
and 2.81373. Sincey 1 =cos(x^2 ) is an even
function, there is a total of 6 points of
inflection on [−π,π]. An alternate
solution is to enter
y 2 =
d^2
dx^2(
y 1 (x),x,2). The graph ofy 2
indicates that there are 6 zeros on [−π,π].
19. Entery 1 = 3 ∗e∧(−x∧ 2 /2). Note that
the graph has a symmetry about they-axis.
Using the functions of the calculator, you
will find:
(a) a relative maximum point at (0, 3),
which is also the absolute maximum
point;
(b) points of inflection at (−1, 1.819) and
(1, 1.819);
(c) y=0 (thex-axis) a horizontal
asymptote;
(d) y 1 increasing on (−∞, 0] and
decreasing on [0,∞); and
(e) y 1 concave upward on (−∞,−1) and
(1,∞) and concave downward on (−1,
1). (See Figure 8.8-10.)[–4, 4] by [–1, 4]
Figure 8.8-10- (See Figure 8.8-11.) Entery 1 =cos(x)∗
(sin(x))∧2. A fundamental domain ofy 1
is [0, 2π]. Using the functions of the
calculator, you will find:
[–1, 9.4] by [–1, 1]
Figure 8.8-11(a) relative maximum points at (0.955,
0.385), (π, 0), and (5.328, 0.385),
and relative minimum points at
(2.186,− 0 .385) and (4.097,− 0 .385);
(b) points of inflection at (0.491, 0.196),(
π
2,0
)
, (2.651,− 0 .196),
(3(.632,− 0 .196),
3 π
2,0
)
, and (5.792, 0.196);
(c) no asymptote;
(d) function is increasing on intervals
(0, 0.955), (2.186,π), and
(4.097, 5.328), and decreasing on
intervals (0.955, 2.186), (π, 4.097),
and (5.328, 2π);