Name:
Instructor:
1. Find the interval(s) where
is increasing and the interval(s) where it is decreasing.
2. Let
1
2
)
(
+
-
=
x
x
x
f
.
(a) Find the interval(s) where
)
(
x
f
is concave upward.
(b) Find the interval(s) where
)
(
x
f
is concave downward.
(c) Find the x-coordinate(s) of any point(s) of inflection.
3. Sketch the graph of
2
3
x
y
x
-
=
+
.
4. A farmer wants to create a pen inside a barn like the one shown below. The sides of the barn will form two sides of the pen, while fencing material will be used for the other two sides. The farmer has a total of 80 feet of fencing material to enclose the pen. What is the maximum area that can be enclosed?
2
5
)
(
2
3
+
-
+
=
x
x
x
x
f
5. To make a rectangular trough, you bend the sides of a 32-inch wide sheet of metal to obtain the cross section pictured below. Find the dimensions of the cross section with the maximum area (this will result in the trough with the largest possible volume).
6. Simplify
(a)
(
)
(
)
84
63
xx
-
.
(b)
5
7
3
27
a
a
-
-
7. Use the laws of logarithms to expand and simplify the expression:
1
3
log
+
x
x
.
8. A quantity Q(t) is described by the exponential growth function
0.0035
()410
t
Qte
=
,
where t is measured in minutes.
(a) What quantity is initially present?
(b) What quantity is present in 50 minutes?
9. Find the derivative of the function
(
)
3
()1
fxxe
=+
.
10.
Use logarithmic differentiation to find the derivative of the function
x
x
x
e
y
ln
2
=
.
10. Find the indefinite integral
(
)
4
2543
xxdx
+-
ò
.
11. Find the indefinite integral
5
6
64
4
x
dx
xx
-
-
ò
.
12. Evaluate the definite integral
(
)
ò
+
+
1
0
2
1
2
dx
x
x
x
.
13. The demand function for a certain product is given by
10
2
.
0
01
.
0
2
+
-
-
=
x
x
p
, where p represents the unit price in dollars and x is the quantity demanded measured in units of a thousand. Determine the consumer surplus if the price is set at $2.
14. Find the amount of an annuity if $800/month is paid into it for a period of 20 years earning interest at the rate of 7%/year compounded continuously.
15. Evaluate the indefinite integral
(
)
6
2
ttdt
-
+
ò
.
16. Use a table of integrals to evaluate
(
)
2
ln4ln
dx
xxx
+
ò
.
17. Use the Trapezoidal Rule to approximate
1
0
6
1
dx
x
+
ò
with n = 6.
18. Use Simpson’s Rule to approximate
2
0
e
x
edx
-
ò
with n = 8.
19. A scholarship fund being set up in the name of a famous biologist. If the
scholarship is to award $1000/year and the fund earns interest at a rate of 7.5% compounded continuously, find the required amount for the endowment.
� EMBED Word.Picture.8 ���
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Micro-Scholarship, What it is, How can it help me.pdf
NameInstructor1. Find the interval(s) where i.docx
1. Name:
Instructor:
1. Find the interval(s) where
is increasing and the interval(s) where it is decreasing.
2. Let
1
2
)
(
+
-
=
x
x
x
f
.
(a) Find the interval(s) where
)
(
x
f
2. is concave upward.
(b) Find the interval(s) where
)
(
x
f
is concave downward.
(c) Find the x-coordinate(s) of any point(s) of inflection.
3. Sketch the graph of
2
3
x
y
x
-
=
+
.
4. A farmer wants to create a pen inside a barn like the one
shown below. The sides of the barn will form two sides of the
pen, while fencing material will be used for the other two sides.
The farmer has a total of 80 feet of fencing material to enclose
the pen. What is the maximum area that can be enclosed?
3. 2
5
)
(
2
3
+
-
+
=
x
x
x
x
f
5. To make a rectangular trough, you bend the sides of a 32-
inch wide sheet of metal to obtain the cross section pictured
below. Find the dimensions of the cross section with the
maximum area (this will result in the trough with the largest
possible volume).
6. Simplify
(a)
(
)
(
)
84
63
xx
-
4. .
(b)
5
7
3
27
a
a
-
-
7. Use the laws of logarithms to expand and simplify the
expression:
1
3
log
+
x
x
.
8. A quantity Q(t) is described by the exponential growth
function
0.0035
()410
t
Qte
5. =
,
where t is measured in minutes.
(a) What quantity is initially present?
(b) What quantity is present in 50 minutes?
9. Find the derivative of the function
(
)
3
()1
fxxe
=+
.
10.
Use logarithmic differentiation to find the derivative of the
function
x
x
x
e
y
ln
2
6. =
.
10. Find the indefinite integral
(
)
4
2543
xxdx
+-
ò
.
11. Find the indefinite integral
5
6
64
4
x
dx
xx
-
-
ò
.
12. Evaluate the definite integral
8. , where p represents the unit price in dollars and x is the
quantity demanded measured in units of a thousand. Determine
the consumer surplus if the price is set at $2.
14. Find the amount of an annuity if $800/month is paid into it
for a period of 20 years earning interest at the rate of 7%/year
compounded continuously.
15. Evaluate the indefinite integral
(
)
6
2
ttdt
-
+
ò
.
16. Use a table of integrals to evaluate
(
)
2
ln4ln
dx
xxx
+
ò
.
9. 17. Use the Trapezoidal Rule to approximate
1
0
6
1
dx
x
+
ò
with n = 6.
18. Use Simpson’s Rule to approximate
2
0
e
x
edx
-
ò
with n = 8.
19. A scholarship fund being set up in the name of a famous
biologist. If the
scholarship is to award $1000/year and the fund earns interest at
a rate of 7.5% compounded continuously, find the required
amount for the endowment.
� EMBED Word.Picture.8 ���
