Cheung, Anthony – Homework 3 – Due: Sep 19 2006, 3:00 am – Inst: David Benzvi
1
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printout
should
have
22
questions.
Multiplechoice questions may continue on
the next column or page – fnd all choices
beFore answering.
The due time is Central
time.
001
(part 1 oF 1) 10 points
The graph oF
f
is shown in the fgure
2
4
6
8
2
4

2
IF the Function
g
is defned by
g
(
x
) =
Z
x
1
f
(
t
)
dt,
For what value oF
x
does
g
(
x
) have a maxi
mum?
1.
x
= 5
correct
2.
x
=

1
3.
x
= 2
.
5
4.
x
=

7
5.
not enough inFormation given
Explanation:
By the ±undamental theorem oF calculus, iF
g
(
x
) =
Z
x
1
f
(
t
)
dt,
then
g
0
(
x
) =
f
(
x
). Thus the critical points
oF
g
occur at the zeros oF
f
,
i.e.
, at the
x

intercepts oF the graph oF
f
.
To determine
which oF these gives a local maximum oF
g
we
use the sign chart
g
0
+

1
5
7
For
g
0
. This shows that
max
g
(
x
) at
x
= 5 ,
since the sign oF
g
0
changes From positive to
negative at
x
= 5.
keywords:
±TC, integral, sign chart, maxi
mum
002
(part 1 oF 1) 10 points
±ind
g
0
(
x
) when
g
(
x
) =
Z
x
π
(8 + cos
t
)
dt .
1.
g
0
(
x
) = 8
x
+ sin
x
2.
g
0
(
x
) = 8

sin
x
3.
g
0
(
x
) =

sin
x
4.
g
0
(
x
) = 8 + cos
x
correct
5.
g
0
(
x
) = 8
x

cos
x
Explanation:
By the ±undamental theorem oF Calculus,
iF
g
(
x
) =
Z
x
a
f
(
t
)
dt ,
then
g
0
(
x
) =
d
dx
Z
x
a
g
(
t
)
dt
=
f
(
x
)
.
In the given example, thereFore,
g
0
(
x
) = 8 + cos
x
.
keywords:
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2
003
(part 1 of 1) 10 points
If
F
(
x
) =
Z
x
0
2
e
6 sin
2
θ
dθ ,
Fnd the value of
F
0
(
π/
4).
1.
F
0
(
π/
4) = 3
e
2
2.
F
0
(
π/
4) = 3
e
6
3.
F
0
(
π/
4) = 2
e
6
4.
F
0
(
π/
4) = 2
e
3
correct
5.
F
0
(
π/
4) = 3
e
3
Explanation:
By the ±undamental theorem of calculus,
F
0
(
x
) = 2
e
6 sin
2
x
.
At
x
=
π/
4, therefore,
F
0
(
π/
4) = 2
e
3
since sin(
π
4
) =
1
√
2
.
keywords: integral, ±TC
004
(part 1 of 1) 10 points
Evaluate the deFnite integral
I
=
Z
π/
2
0
(4 cos
x

sin
x
)
dx .
1.
I
= 1
2.
I
= 4
3.
I
= 2
4.
I
= 5
5.
I
= 3
correct
Explanation:
By the ±undamental Theorem of Calculus,
I
=
h
F
(
x
)
i
π/
2
0
=
F
(
π
2
)

F
(0)
for any antiderivative
F
of
f
(
x
) = 4 cos
x

sin
x .
Taking
F
(
x
) = 4 sin
x
+ cos
x
and using the fact that
cos 0 = sin
π
2
= 1
,
sin 0 = cos
π
2
= 0
,
we thus see that
I
= 3
.
keywords: integral, ±TC, trig function
005
(part 1 of 1) 10 points
Evaluate the deFnite integral
I
=
Z
1
0
(5 + 8
x

9
x
2
)
dx .
Correct answer: 6 .
Explanation:
By linearity,
Z
1
0
(5 + 8
x

9
x
2
)
dx
= 5
Z
1
0
dx
+ 8
Z
1
0
x dx

9
Z
1
0
x
2
dx .
But
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 Calculus, Trigonometry, Fundamental Theorem Of Calculus, dx

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