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MAT1300D
Solution to Final Examination
Fall 2007
1
Solution to Final Examination
MAT1300D, Fall 2007
Part I.
MultipleChoice Questions
(30 marks)
1.
Find the equation of the tangent line to the graph of
y
= 3
x

1 when
x
= 4.
(A)
y
=
1
1
2
x
+
;
(B)
y
=
1
3
2
x
+
;
(C)
y
=
3
7
4
x
+
;
(D)
y
=
3
2
4
x
+
;
(E)
y
=
3
1
4
x

.
Solution
.
y'
=
3
2
x
.
When
x
= 4,
y
= 5, and
y'
=
3
4
.
Hence, the equation of the tangent
line is
y
=
3
4
(
x

4) + 5, or
y
=
3
4
x
+ 2.
2.
Calculate
2
3
12
lim
3
x
x
x
x
→


+
.
(A)
1
5
(B)
3
(C)
7
4
(D)
4
9
(E)

7.
Solution
.
x
2

x

12 = (
x
+ 3)(
x

4).
2
3
3
3
12
(
3)(
4)
lim
lim
lim(
4)
7.
3
3
x
x
x
x
x
x
x
x
x
x
→
→
→


+

=
=

= 
+
+
3.
On what interval is the function
g
(
x
) =

2
x
3
+ 12
x
2

36
x
+ 3 concave down?
(A)
(2,
∞
)
(B)
(2, 3)
(C)
(

1,
∞
)
(D)
(
∞
,

1)
(E)
(

2, 4).
Solution
.
g'
=

6
x
2
+ 24
x

36,
g"
=

12
x
+ 24.
g''
= 0 implies
x
= 2.
When
x
< 2,
g"
<
0, the graph of
g
(
x
) is concave down.
4.
Which of the following statements is true for the function
g
(
x
) = 2
x
3
+ 3
x
2

36
x
+ 2?
(A)
x
=

1 is a local minimum.
(B)
x
=

3 is a local maximum.
(C)
x
=

3 is a local minimum.
(D)
x
= 1 is a local minimum.
(E)
x
= 1 is a local maximum.
D
1.
E
2.
A
3.
B
4.
C
5.
A
6.
C
7.
E
8.
D
9.
C
10.
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View Full DocumentMAT1300D
Solution to Final Examination
Fall 2007
2
Solution
.
g'
= 6
x
2
+ 6
x

36.
Critical points are the roots of
x
2
+
x

6 = 0,
x
=

3, 2.
When
∞
<
x
<

3,
g'
> 0; when

3 <
x
< 2,
g'
< 0; when 2 <
x
<
∞
,
g'
> 0.
g
attains a
local maximum at
x
=

3.
5.
Calculate
4
0
(3
1)
x
dx
+
∫
(A)
10
(B)
15
(C)
20
(D)
25
(E)
30
Solution
.
4
4
4
1/ 2
3/ 2
0
0
0
2
(3
1)
3
3
4
4
16
4
20
3
x
dx
x
dx
dx
+
=
+
=
+
=
+
=
∫
∫
∫
.
6a.
Suppose that for a certain product, the demand function is given by
D
(
x
) = 11

x
2
and the supply function is given by
S
(
x
) = 2
x
+ 3.
Calculate the producer's surplus.
(A)
1;
(B)
4;
(C)
9;
(D)
13;
(E)
22.
Solution
.
The equilibrium point is 11

x
2
= 2
x
+ 3,
x
= 2,
p
= 7.
The producer's surplus
is
2
2
2
2
0
0
0
(7
2
3)
(4
2 )
4
4
x
x
dx
x dx
x
x
=


=

=

=
∫
∫
.
6c.
Suppose that for a certain product the demand function is given by
p
=
180
x

, 0
≤
x
≤
180.
For which values of
x
the demand is elastic?
(A)
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 Fall '09
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