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white (taw933) – Review 2 – benzvi – (55600)
1
This printout should have 18 questions.
Multiplechoice questions may continue on
the next column or page – fnd all choices
beFore answering.
001
10.0 points
Which one oF the Following properties does
f
(
x
)=
x

2
x
2
+12
have?
1.
local min at
x
=

6
2.
local max at
x
=6
correct
3.
local max at
x
=2
4.
local max at
x
=

6
5.
local min at
x
6.
local min at
x
Explanation:
By the Quotient rule,
f
±
(
x
x
2

2
x
(
x

2)
(
x
2
+12)
2
=
12 + 4
x

x
2
(
x
2
2
.
The critical points oF
f
occur when
f
±
(
x
)=0,
i.e.
,atthesolutionsoF
f
±
(
x
(2 +
x
)(6

x
)
(
x
2
2
=0
.
Thus the critical points oF
f
are
x
=

2and
x
=6.
To classiFy these critical points we use the
±irst Derivative Test. But the sign oF
f
±
de
pends only on the numerator, so it is enough,
thereFore, to look only at a sign chart For
(2 +
x
)(6

x
):

26


+
±rom this it Follows that
f
is decreasing on
(
∞
,

2), increasing on (

2
,
6), and de
creasing on (6
,
∞
). Consequently,
f
has a
local maximum at
x
.
keywords: local maximum, local minimum,
critical point, quotient rule, ±irst Derivative
Test, rational Function
002
10.0 points
Determine iF the limit
lim
x
→∞
x
+3
x
2

4
x
+5
exists, and iF it does, fnd its value.
1.
limit =
3
5
2.
limit = 5
3.
limit = 0
correct
4.
limit = 3
5.
limit doesn’t exist
6.
limit =

1
4
Explanation:
Dividing in the numerator and denominator
by
x
2
,theh
ighestpower,weseethat
x
x
2

4
x
=
1
x
+
3
x
2
1

4
x
+
5
x
2
.
On the other hand,
lim
x
1
x
=l
i
m
x
1
x
2
.
By Properties oF limits, thereFore, the limit
exists and
limit = 0
.
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View Full Document white (taw933) – Review 2 – benzvi – (55600)
2
003
10.0 points
Which of the following functions has
4
3
2
3
as its graph.
1.
f
(
x
)=
3
x
+4
x

2
2.
f
(
x
3
x

4
x

2
3.
f
(
x
4

3
x
x
+2
4.
f
(
x
3
x
x
5.
f
(
x
3
x

4
x
correct
Explanation:
The graph of any function
f
(
x
ax

b
x
+
c
has the properties
(i) a vertical asymptote
x
=

c
,
(ii) a horizontal asymptote
y
=
a
,
(iii) an
x
intercept at
x
=
b
a
.
From the graph above, therefore,
c
=2
,a
=3
,
b
a
=
4
3
.
Hence the graph is that of
f
(
x
3
x

4
x
.
004
10.0 points
Use calculus to decide which of the follow
ing is the graph of
f
(
x
)=3
x
2
/
3

2
x.
1.
x
y
2.
x
y
correct
3.
x
y
4.
x
y
white (taw933) – Review 2 – benzvi – (55600)
3
5.
x
y
Explanation:
After diFerentiation of
f
(
x
)=3
x
2
/
3

2
x
we see that
f
±
(
x
)=
2
x
1
/
3

2=
2(1

x
1
/
3
)
x
1
/
3
;
in particular,
(i)
f
has critical points at
x
=0
,
1.
DiFerentiating again we next see that
f
±
(
x

2
3
x
4
/
3
,
from which it follows that
(ii)
f
±
<
0
,
so
f
concave down, on (
∞
,
0)
,
and again
(iii)
f
±
<
0
,
so
f
concave down, on (0
,
∞
);
in particular,
f
has a
local maximum
at
x
=1
.
Of the ±ve graphs only
x
y
has these properties.
005
10.0 points
If a tank holds 2000 gallons of water, and
the water can drain from the tank in 40 min
utes, then Torricelli’s Law gives the volume
V
of water remaining in the tank after
t
minutes
as
V
=2
0
0
0
±
1

t
40
²
2
.
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This note was uploaded on 12/05/2011 for the course M 408k taught by Professor Schultz during the Spring '08 term at University of Texas at Austin.
 Spring '08
 schultz
 Differential Calculus

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