Su, Yung – Exam 1 – Due: Oct 2 2007, 11:00 pm – Inst: Shinko Harper
2
since sin(
π
4
) =
1
√
2
.
keywords: integral, FTC
003
(part 1 of 1) 10 points
Determine
F
0
(
x
) when
F
(
x
) =
Z
√
x
2
6 cos
t
t
dt.
1.
F
0
(
x
) =
6 cos
x
√
x
2.
F
0
(
x
) =

6 sin(
√
x
)
√
x
3.
F
0
(
x
) =

6 sin
x
x
4.
F
0
(
x
) =
3 sin(
√
x
)
x
5.
F
0
(
x
) =

6 sin
x
√
x
6.
F
0
(
x
) =
3 cos
x
x
7.
F
0
(
x
) =
3 cos(
√
x
)
x
correct
8.
F
0
(
x
) =

3 cos(
√
x
)
√
x
Explanation:
By the Fundamental Theorem of Calculus
and the Chain Rule,
d
dx
‡
Z
g
(
x
)
a
f
(
t
)
dt
·
=
f
(
g
(
x
))
g
0
(
x
)
.
When
F
(
x
) =
Z
√
x
2
6 cos
t
t
dt,
therefore,
F
0
(
x
) =
6 cos(
√
x
)
√
x
‡
d
dx
√
x
·
.
Consequently,
F
0
(
x
) =
3 cos(
√
x
)
x
,
since
d
dx
√
x
=
1
2
√
x
.
keywords: Stewart5e, FTC, Chain Rule
004
(part 1 of 1) 10 points
If
w
0
(
t
) is the rate of growth of Mira’s weight
(in pounds per year), what does the de±nite
integral
I
=
Z
8
4
w
0
(
t
)
dt
represent?
1.
decrease in Mira’s weight from age 4 to 8
2.
average of Mira’s weight from age 4 to 8
3.
Mira’s weight at age 8
4.
increase in Mira’s weight from age 4 to 8
correct
5.
Mira’s weight at age 4
Explanation:
By the Fundamental theorem of Calculus,
Z
b
a
w
0
(
x
)
dx
=
w
(
b
)

w
(
a
)
,
in other words, the value of the integral is the
net change,
w
(
b
)

w
(
a
), in
w
over the interval
[
a, b
].
Consequently,
I
is the
increase in Mira’s weight from age 4 to 8
.
keywords: integral, rate growth, FTC, weight,
net change,
005
(part 1 of 1) 10 points