Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
1)
Which of the following graphs could represent a function with the following properties?
I. f(x) > 0, for x < 0
II. f(x) 0, for all x
III. f(0) = 0
1)
A)
B)
C)
D)
none of these
2)
A rectangular garden of area 50 square feet is to be surrounded on three sides by a fence costing $2
per running foot and on one side by a brick wall costing $6 per running foot. Let x be the length of
the brick wall side. Which of the following represents the total cost of the material?
2)
A)
3x + (50 – x)
B)
6x +50
x
C)
3x
50 +1
50x
D)
6x +300
x
E)
none of these
3)
The graph of a function f(x) is sketched below. Which of the following is f(x)?
3)
A)
f(x) =1
5 – x + 2
B)
f(x) =x – 5 – 2
C)
f(x) =x2– 5x – 2
D)
f(x) =1
x – 5
4)
Suppose f(x) is the function graphed below. Which of the following is f(x)?
4)
A)
f(x) = – x2+ 2x + 5
B)
f(x) =x3+ 6x2+ 9x
C)
f(x) =1
x+x2+ 3x
D)
f(x) =x3+ 5
E)
f(x) =x2+ 3x
5)
Which of the following most resembles the graph of f(x) =1
(2 – 3x)3 near x =2
3?
5)
A)
B)
C)
D)
6)
Let a, b, and c be fixed numbers with a > 0 and let f(x) =ax2+ bx + c. Which of the following
properties is true of the graph of f(x) ?
6)
A)
f(x) has one relative maximum
B)
f(x) has one inflection point
C)
f(x) is always concave up
D)
f(x) has either a relative maximum or inflection point
E)
none of these
Choose the graph of a function having the given properties.
4
7)
From x = 0 to x =25 the function increases and the slope decreases. When x >25 the function
decreases and the slope decreases (note: the slope is negative and becomes more negative).
7)
A)
25
B)
25
C)
25
5
D)
25
8)
Which of the following is the graph of y =1
x+ 2x + 5?
8)
A)
B)
C)
D)
Choose the graph of a function having the given properties.
9)
Relative minimum points at x =8 and x =40; relative maximum point at x =24; inflection points at
x =16 and x =32.
9)
A)
816 24 32 40
B)
816 24 32 40
C)
816 24 32 40
D)
816 24 32 40
10)
Which of the following could represent a function having the given properties?
(I) increasing slope for x < 4
(II) f'(x) > 0 for all x (x
4)
(III) asymptote at x = 4
10)
A)
B)
C)
D)
11)
A toll road averages 36,000 cars per day when charging $1 (100 cents) per car. A survey concludes
that changing the toll will result in 300 fewer cars for each cent of increase in price. Which of the
following represents the revenue that will result from an increase of x cents in the price of the toll?
11)
A)
(36,000 – 300x)(100 + x)
B)
36,000(100 + x) – 300 x
C)
36,000 + 300x + (100 + x)
D)
36,000 · 100 – x(300 – x)
E)
none of these
Solve the problem.
8
12)
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Using the following properties of a twice–differentiable function y = f(x), select a possible graph of
f.
x y Derivatives
x < 2 y> 0, y< 0
–211 y= 0, y< 0
–2 < x < 0 y< 0, y< 0
0–5 y < 0, y= 0
0 < x < 2 y< 0, y> 0
2–21 y= 0, y> 0
x > 2 y> 0, y> 0
12)
A)
B)
C)
D)
13)
Which of the following best represents the graph of y = 2x3– 3x2– 12x + 17 near the point (2, –3)?
13)
A)
B)
C)
D)
10
14)
Which of the following is the graph of F(x) = 2x –1
x + 2 + 5?
14)
A)
B)
C)
D)
15)
Where is the function f(x) =5
(2x – 4)3 increasing?
15)
A)
for x > 2
B)
at x = 30
C)
for all x
D)
none of these
16)
Which of the following is the graph of f(x) =x4– 9?
16)
A)
B)
C)
D)
17)
Which of the following graphs could represent a function having the given properties?
(I) asymptotes x = 2 and x = – 2
(II) f”(x) > 0 for all x
(III) f(x) > 0 for all x > 4
17)
A)
B)
C)
D)
18)
The graph of a function f(x) is sketched below. Which of the following is f(x)?
18)
A)
f(x) =1
x2
B)
f(x) =1
x + 2 – x
C)
f(x) =1
x–x2
D)
f(x) = – x2–x3
19)
What are the x–intercepts of the graph of y = 3x3+ 24?
19)
A)
(0, –2)
B)
(24, 0)
C)
(2, 0)
D)
(0, 24)
E)
none of these
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
20)
Is this the graph of f(x) =(x – 2)3+ 1?
Enter just the word “yes” or “no” (lower case).
20)
21)
Compute the minimal average cost if the total cost function is C(x) = 9x2+ 5x + 100.
Enter just an integer.
21)
65
22)
Is this the graph of y =3x + 4?
Enter just the word “yes” or “no” (lower case).
22)
yes
14
E
23)
A large rectangular garden is to be enclosed by a fence and divided into 5 regions by 4
parallel fences across the interior of the garden as shown below.
Find the values of l and w for which the total area of the garden is as large as possible,
assuming that 120 ft of fencing is available.
23)
24)
Find the maximum value of the function f(x) = – x3+ 6x2+ 10 for x 0.
Enter your answer exactly as: f(a) = b rel max
24)
25)
Determine the minimum value of x + y when xy = 100 (x > 0, y > 0).
Enter just an integer.
25)
26)
Determine the relative maximum point of f(x) = – 2x2+ 4x + 1.
Enter your answer exactly as just an ordered pair of integers: (a, b)
26)
27)
Determine the relative maximum point of f(x) =x3– 6x2+ 9x – 3.
Enter your answer as exactly just an ordered pair of integers: (a, b)
27)
28)
Is this the graph of a function having the following properties?
(I) f'(x) < 0 for all x
(II) f”(x) > 0 for all x < 0
(III) (0, 1) is a point on the graph
Enter your answer as just “yes” or “no” (lower case).
28)
29)
This is the graph of f(x) =1
3x3–1
2x2– 6x. What is the inflection point?
Enter your answer exactly as just an ordered pair of reduced fractions of form a
b, or
integers (a, b), or the word “none”.
29)
30)
Determine the relative minimum point of f(x) =x3– 3x2+ 1.
Enter your answer as exactly just an ordered pair of integers: (a, b)
30)
Answer:
(2, –3)
31)
A rectangular garden is to be fenced in and divided into three parallel sections. The
fencing for the boundary costs $20 per foot whereas the fencing for the dividing fences
costs $5 per foot. Consider the problem of finding the dimensions of the largest garden
possible if the gardener can spend $2000 for the fencing. Find the values of l and w for
which the total area of the garden is as large as possible.
31)
Answer:
32)
Is this the graph of f(x) =x3– 3x2+ 2?
Enter just the word “yes” or “no” (lower case).
32)
Answer:
yes
16
Answer:
Explanation:
33)
An open box with square ends is to be constructed with a volume of 125 cubic inches. The
bottom is to be made of a material that weighs twice as much per square inch as the
material used for the sides. What should the dimensions of the box be in order to minimize
its weight? (Note: the actual weights of the materials do not matter.)
Enter your answer exactly as a, b, c where these integers represent length, width, and
height (no units or × symbols)
33)
34)
Is this the graph of a function having the following properties?
(I) concave down for all x
(II) asymptotic to the line x = 0
Enter your answer as just the word “yes” or the word “no” (lower case)
34)
35)
Determine all maximum and minimum values of f(x) = – x3+ 3x2+ 9x – 1 on –2 x 2.
Enter your answer exactly as: a,b both integers where a is the minimum of f and b is the
maximum of f.
35)
36)
Is this the graph of the function f(x) = – x3+ 3x2+ 9x – 15?
Enter just the word “yes” or “no” (lower case).
36)
17
37)
A sports retailer expects to sell 120 sweat suits at a steady rate over the course of the
coming year. The cost of placing an order with the wholesaler is $40. The annual
inventory cost per sweat suit is $6 based on the average inventory level. Determine the
economic order quantity for the suits (i.e., determine the order size that minimizes
ordering and inventory expenses).
Enter just an integer (no words or units).
37)
38)
Is this the graph of g(x) = 2x3+ 3x2– 12x – 5?
Enter just “yes” or “no” (lower case).
38)
39)
Find the interval(s) where f is decreasing and concave up for f(x) =x3– 3x2– 24x + 3.
Enter your answer in standard interval notation as either : (a, b) or (a, b)(c,d) for a<c.
39)
40)
Is this the graph of a function having the following properties?
(I) asymptotic to the line y =1
2x – 1
(II) relative maximum at x = 0
Enter just the word “yes” or the word “no” (lower case).
40)
41)
If the second derivative of a function is f ”(x) =(x – 2)2(x2– 9) is (2, f(2)) an
inflection point?
Enter just the word “yes” or “no” (lower case).
41)
42)
Is this the graph of a function having the following properties ?
(I) f(0) = – 5
(II) f’(0) = 0
(III) f”(0) = 12
(IV) f( 7) = f(–7) = – 12
Enter your answer as just “yes” or “no” (lower case).
42)
43)
Find the interval(s) where f is increasing and concave down for f(x) =1
4x4–x3+x2.
Enter your answer in standard interval notation as either: (a, b) or (a, b)
(c, d) where a<c
with any irrational numbers in lowest terms of form: e ±f
g
43)
44)
A fast food restaurant is establishing its inventory policy for ordering frozen french fries.
In the coming year, they expect to sell 2500 lbs of french fries. It costs $4 to place an order
and the carrying costs for a year are $2 per pound based on the average amount in storage.
The restaurant wishes to determine how many pounds of fries to order to minimize its
ordering and inventory costs. Determine the optimal number of orders that should be
placed and the optimal size of those orders.
Enter your answer exactly as just: a, b (both integers) where a represents the orders and b
represents the lbs/order (no words or units).
44)
45)
Is this the graph of h(x) = x + 10 +9
x?
Enter just the word “yes” or “no” (lower case).
45)
46)
Determine all the values of x where relative maximum and minimum points of the
function f(x) =1
3x3–3
2x2– 10x occur. Distinguish the maxima from the minima using the
second derivative rule. Enter your answer exactly as: f(a) rel max, f(b) rel min in that
order.
46)
47)
Compute the maximum profit for the profit function P(x) =x3
3–9
2x2+ 8x.
Enter your answer as just a real number rounded off to two decimal places (no label).
47)
48)
Determine the maximum value of f(x) = – 2x3+ 6x + 3 on 0
x 2.
Enter just an integer.
48)
49)
Determine the minimum value of f(x) =(x2– 2x + 2)2+ 2(x2– 2x + 2).
Enter just an integer.
49)
20