MA140-01
4/8/09
Page 1
Major Quiz 5 Solutions
No Calculator Allowed
________________________ in da house...
1.) Find the points of inflection of the graph of the function, if there are any. Determine where
the graph is concave upward and where it is concave downward. Identify other essential
information such as extrema and intervals where the function is increasing and decreasing.
Draw a sketch of the graph. [5 points]
3 2
( ) 6 20
f x x x
= − +
2
'( ) 3 12 3 ( 4)
f x x x x x
= − = −
Critical numbers:
0, 4
x
=
"(0) 12, "(4) 12
f f
= − =
"( ) 6 12 6( 2)
f x x x
= − = −
"( ) 0 when 2, "( ) 0 when 2
f x x f x x
< < > >
2
x
=
is an inflection point
(0)
f
is a relative maximum and
(4)
f
is a relative minimum.
The function is concave down on
(
)
, 2
−∞
and concave up on
(
)
2,
∞
.
The function is increasing on
(
)
(
)
,0 4,
−∞ ∪ ∞
and decreasing on
(
)
0, 4
.
2.)
Find the general antiderivative for each of the following. [5 points]
(a)
3 2
( 3 4 3)
x x x dx
+ − +
(b)
( 2sin )
x dx
−
4 3 2
3 4 3
4 3 2
x x x
x C
+ − + +
( 2)( cos )
x C
− − +
4
3 2
2 3
4
x
x x x C
+ − + +
2 cos
x C
+
(c)
2
4 5 sec tan 6
x x x x x
dx
x
− +
(d)
dx
x
x
2
cos
sin
2
4 5 sec tan 6
x x x x x
dx dx dx
x x x
− +
sin 1
cos cos
x
dx
x x
⋅
1
2
4 5 sec tan 6
xdx x xdx x dx
−
− +
(
)
tan sec
x x dx
⋅
1
22
4 5sec 6 1
2
2
x x
x C
− + +
sec
x C
+
1
22
2 5sec 12
x x x C
− + +
(e)
−dx
x
xx
cos
cos4tan3
2
2
tan cos
3 4
cos cos
x x
dx dx
x x
−
1
3 tan 4 cos
cos
x dx xdx
x
⋅ −
(
)
3 tan sec 4 cos
x x dx xdx
⋅ −
3sec 4(sin ) 3sec 4 sin
x x C x x C
− + → − +