Exam
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1)
A piece of molding 168 cm long is to be cut to form a rectangular picture frame. What dimensions
will enclose the largest area?
1)
A)
42 cm ×42 cm
B)
12.96 cm ×42 cm
C)
12.96 cm ×12.96 cm
D)
33.6 cm ×33.6 cm
Find dy/dx by implicit differentiation.
2)
x3+ 3x2y + y3= 8
2)
A)
–x2+ 3xy
x2+ y2
B)
–x2+ 2xy
x2+ y2
C)
x2+ 2xy
x2+ y2
D)
x2+ 3xy
x2+ y2
Solve the problem.
3)
A rectangular field is to be enclosed on four sides with a fence. Fencing costs $4 per foot for two
opposite sides, and $7 per foot for the other two sides. Find the dimensions of the field of area 830
ft2 that would be the cheapest to enclose.
3)
A)
38.1 ft @ $4 by 21.8 ft @ $7
B)
50.4 ft @ $4 by 16.5 ft @ $7
C)
21.8 ft @ $4 by 38.1 ft @ $7
D)
16.5 ft @ $4 by 50.4 ft @ $7
Find the equation of the tangent line at the given value of x on the curve.
4)
xy + x = 2, x = 1
4)
A)
y = – 1
2x +3
2
B)
y = 2x – 1
C)
y =1
2x +1
2
D)
y = – 2x + 3
1
Solve the problem.
5)
A grocery store estimates that the revenue (in dollars) from the sale of x cases of condensed soup is
given by R(x) = – 5600 +9.8x – 0.0017x2. Use the differential to approximate the change in revenue
from the sale of one more case of soup when 1100 cases are sold.
5)
A)
–$7.93
B)
$6.06
C)
$7.93
D)
$3123.00
Find dy/dx by implicit differentiation.
6)
x3+ y3= 5
6)
A)
x2
y2
B)
y2
x2
C)
–x2
y2
D)
–y2
x2
Find the equation of the tangent line at the given value of x on the curve.
7)
yx + 1 = 4, x = 3
7)
A)
2y =1
2x +1
2
B)
2y = – 1
2x +7
2
C)
y =1
4x +5
4
D)
y = – 1
4x +11
4
Solve the problem.
8)
Find two numbers x and y such that their sum is 420 and x2y is maximized.
8)
A)
x =105, y =315
B)
x =140, y =280
C)
x =315, y =105
D)
x =280, y =140
Use the differential to approximate the quantity to four decimal places.
9)
86
9)
A)
9.2778
B)
9.5556
C)
14.0000
D)
8.7222
2
Solve the problem.
10)
A piece of molding 163 cm long is to be cut to form a rectangular picture frame. What dimensions
will enclose the largest area?
10)
A)
32.6 cm ×32.6 cm
B)
12.77 cm ×12.77 cm
C)
12.77 cm ×40.75 cm
D)
40.75 cm ×40.75 cm
11)
The demand for boneless chicken breast, in dollars per pound, is given by q = – 0.6p +6, where p
represents the price per pound and q represents the average number of pounds purchased per
week per customer. Determine the price at which the demand for boneless chicken breast is unit
elastic.
11)
A)
$10.00 per pound
B)
$5.00 per pound
C)
$5.93 per pound
D)
The demand is not unit elastic at any price.
12)
A man 6 ft tall walks at a rate of 5 ft/s away from a lamppost that is 16 ft high. At what rate is the
length of his shadow changing when he is 30 ft away from the lamppost?
12)
A)
3 ft/s
B)
15
22 ft/s
C)
15
11 ft/s
D)
25 ft/s
13)
Boyle’s law states that if the temperature of a gas remains constant, then PV = c, where P is the
pressure, V is the volume, and c is a constant. Given a quantity of gas at constant temperature, if V
is decreasing at a rate of 10 in.3/s, at what rate is P increasing when P =60 lb/in.2 and V =70 in.3?
13)
A)
36
49 lb/in.2–s
B)
60
7 lb/in.2–s
C)
35
3 lb/in.2–s
D)
420 lb/in.2–s
Assume x and y are functions of t. Evaluate dy/dt.
14)
x + y
x – y = x2+ y2; dx/dt = 12, x = 1, y = 0
14)
A)
12
B)
1
12
C)
–12
D)
–1
12
Find the equation of the tangent line at the given point on the curve.
15)
x2+ 3y2= 13; (1, 2)
15)
A)
y = – 1
3x +7
3
B)
y = – 1
6x +13
6
C)
y =1
3x +5
3
D)
y =1
6x +11
6
Solve the problem.
16)
An architect needs to design a rectangular room with an area of 80 ft2. What dimensions should he
use in order to minimize the perimeter?
16)
A)
20 ft ×20 ft
B)
8.94 ft ×20 ft
C)
8.94 ft ×8.94 ft
D)
16 ft ×80 ft
Find the location of the indicated absolute extremum for the function.
17)
Maximum
17)
A)
No maximum
B)
x = – 1
C)
x = 2
D)
x = 0
Graph the function on the indicated domain, and use the capabilities of your calculator to find the location and value of
the indicated absolute extremum.
18)
f(x) =x3– 4x + 1
x4+x2+ 5 ; [–4, 1]
Maximum
18)
A)
–0.2 at x = 0.6
B)
–0.3 at x =0.9
C)
–0.2 at x = – 3.9
D)
0.6 at x = – 0.8
5
Find the location of the indicated absolute extremum for the function.
19)
Maximum
19)
A)
x = – 4
B)
x = 1
C)
x = 4
D)
No maximum
Solve the problem.
20)
Find the elasticity of demand E for the demand function q =7100 – 17p.
20)
A)
E =17p – 7100
17p
B)
E =7100 – 17p
17p
C)
E =17p
17p – 7100
D)
E =17p
7100 – 17p
Find dy for the given values of x and x.
21)
y = x3– 4x2+ 2x + 1; x = 8, x = – 0.3
21)
A)
39
B)
37
C)
–37
D)
–39
22)
y = 5x2– 2x + 3; x = 2, x = – 1
6
22)
A)
–6
B)
–3
C)
3
D)
6
6
Solve the problem.
23)
A private shipping company will accept a box for domestic shipment only if the sum of its length
and girth (distance around) does not exceed 114 in. What dimensions will give a box with a square
end the largest possible volume?
23)
A)
19 in. ×19 in. ×95 in.
B)
38 in. ×38 in. ×38 in.
C)
19 in. ×38 in. ×38 in.
D)
19 in. ×19 in. ×38 in.
24)
The concentration of a certain drug in the bloodstream x hours after being administered is
approximately C(x) =7x
15 + x2. Use the differential to approximate the change in concentration as x
changes from 1 to 1.25.
24)
A)
0.52
B)
0.26
C)
0.32
D)
0.10
25)
A certain company produces potting soil and sells it in 50 lb bags. Suppose that 100,000 bags are to
be produced each year. It costs $8 per year to store a bag of potting soil, and it costs $4000 to set up
the facility to produce a batch of bags. Find the number of bags per batch that should be produced.
25)
A)
14,121
B)
10,000
C)
100,000
D)
9574
26)
The energy cost of a speed burst as a function of the body weight of a dolphin is given by
E = 43.5w–0.61, where w is the weight of the dolphin (in kg) and E is the energy expenditure (in
kcal/kg/km). Suppose that the weight of a 500–kg dolphin is increasing at a rate of 5 kg/day. Find
the rate at which the energy expenditure is changing with respect to time.
26)
A)
–11.7541 kcal/kg/km/day
B)
–0.006 kcal/kg/km/day
C)
–2.9951 kcal/kg/km/day
D)
–0.0012 kcal/kg/km/day
Find the equation of the tangent line at the given point on the curve.
27)
x2+ y2= 25; (–4, 3)
27)
A)
y = – 4
3x –25
3
B)
y =4
3x –25
3
C)
y = – 4
3x +25
3
D)
y =4
3x +25
3
Assume x and y are functions of t. Evaluate dy/dt.
28)
x4/3 + y4/3 = 2; dx/dt = 6, x = 1, y = 1
28)
A)
6
B)
–1
6
C)
–6
D)
1
6
8
Find the location of the indicated absolute extremum for the function.
29)
Minimum
29)
A)
x = 5
B)
x = – 3
C)
x = 3
D)
x = – 5
Find dy/dx by implicit differentiation.
30)
x4/3 + y4/3 = 1
30)
A)
x
y
1/3
B)
–y
x
1/3
C)
–x
y
1/3
D)
y
x
1/3
9
Solve the problem.
31)
In a certain state, the rate (per 500,000 inhabitants) at which automobiles were stolen each year
during the years 1990 – 2000 are given in the figure. Consider the closed interval [1990, 2000].
A(1990, 169) D (1993, 282) G (1996, 188) L(1999, 236)
B(1991, 204) E(1994, 211) H(1997, 257) M (2000, 270)
C(1992, 255) F(1995, 144) K (1998, 247)
Give the absolute maximum and minimum on the interval and the years when they occur.
31)
A)
Absolute maximum of 282 in 1993
Absolute minimum of 169 in 1990
B)
Absolute maximum of 282 in 1993
Absolute minimum of 144 in 1995
C)
Absolute maximum of 257 in 1997
Absolute minimum of 169 in 1990
D)
Absolute maximum of 270 in 2000
Absolute minimum of 144 in 1995
Find the equation of the tangent line at the given point on the curve.
32)
3x2+ 4xy + y2+ x – 2y = – 7; (–1, 3)
32)
A)
y = – x – 1
B)
x = – 1
C)
y = 3
D)
y = – x + 3
10
Find dy/dx by implicit differentiation.
33)
y5ex+ x =y3x
33)
A)
dy
dx =y3–y5ex– 1
5y4ex–3xy2– 1
B)
dy
dx =y3– 1
5y4ex–3xy2+ 1
C)
dy
dx =y3–y5ex– 1
5y4ex–3xy2
D)
dy
dx =y3– 1
5y4ex–3xy2
Solve the problem.
34)
Given the demand function q =365 – 4p, calculate the elasticity of demand when p =78.
34)
A)
0.46
B)
5.89
C)
2.17
D)
0.17
Find dy for the given values of x and x.
35)
y =x2
x2+ 21
; x = 10, x = 0.1
35)
A)
146
1331
B)
148
1331
C)
142
1331
D)
144
1331
Solve the problem.
36)
Given the demand function q =874 – 8p, determine the price where demand has unit elasticity.
36)
A)
p =27.32
B)
p =54.63
C)
p =59.12
D)
p =29.56
Find dy for the given values of x and x.
37)
y =1
4x3+ 3 x2+ 3; x = 1, x = 0.2
37)
A)
0.15
B)
0.05
C)
0.02
D)
0.5
Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain.
38)
f(x) =1
3x3– 2x2+ 3x – 4; [–2, 5]
Minimum
38)
A)
–8
3 at x = 1
B)
–62
3 at x = – 2
C)
–4 at x = 0
D)
–10
3 at x = 2
Find the equation of the tangent line at the given point on the curve.
39)
yx + 1 = 4; (3, 2)
39)
A)
y =1
2x +1
2
B)
y =1
4x +5
4
C)
2y = – 1
2x +7
2
D)
y = – 1
4x +11
4
Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain.
40)
f(x) = x3– 3x2; [0, 4]
Minimum
40)
A)
No absolute minimum
B)
0 at x = 0
C)
–4 at x = 2
D)
16 at x = 4
Find dy/dx by implicit differentiation.
41)
x ln y + y =x3y3
41)
A)
dy
dx =3x2y3– ln y
x –3x3y3+ y
B)
dy
dx =3x2y4– ln y
x –3x3y3+ 1
C)
dy
dx =3x2y4– y ln y
x –3x3y3+ y
D)
dy
dx =3x2y4– y ln y – 1
x –3x3y3+ y
Find dy for the given values of x and x.
42)
y = 2x5– 3x2+ x – 1; x = – 1, x =1
3
42)
A)
19
3
B)
22
3
C)
25
3
D)
17
3
Find the absolute extrema if they exist as well as where they occur.
43)
f(x) = – 3x4+16x3–18x2+9
43)
A)
No absolute extrema
B)
Absolute maximum of 4 at x = 1; no absolute minima
C)
Absolute maximum of 17 at x = 2; no absolute minima
D)
Absolute maximum of 36 at x =3; no absolute minima
Find the equation of the tangent line at the given point on the curve.
44)
xy2= 12; (3, –2)
44)
A)
y = – 1
3x – 3
B)
y =1
3x + 3
C)
y = – 1
3x + 3
D)
y =1
3x – 3
Find the equation of the tangent line at the given value of x on the curve.
45)
y
4(1 – x) + x y+ 2x = 5, x = 1
45)
A)
y =33
2x – 9
B)
y = – 33
2x +51
2
C)
y = – 33
8x +105
8
D)
y = – 11
6x +65
6
Solve the problem.
46)
A product sells by word of mouth. The company that produces the product has noticed that
revenue from sales is given by R(t) =4 x, where x is the number of units produced and sold. If the
revenue keeps changing at a rate of $600 per month, how fast is the rate of sales changing when
1800 units have been made and sold? (Round to the nearest dollar per month.)
46)
A)
$12,728/month
B)
$7/month
C)
$6364/month
D)
$203,647/month
A
Find dy/dx by implicit differentiation.
47)
xy2= 4
47)
A)
–2y
x
B)
–y
2x
C)
x
2y
D)
2x
y
B
Assume x and y are functions of t. Evaluate dy/dt.
48)
x2 ln y = – 1+ xey; dx/dt =7, x =3, y = 1
48)
A)
7e
3– e
B)
1
C)
0
D)
7e
3(3 – e)
D
14
B
Find dy for the given values of x and x.
49)
y =1
x; x = 10, x = – 0.003
49)
A)
0.00003
B)
0.03
C)
0.0003
D)
0.003
Find the equation of the tangent line at the given point on the curve.
50)
x2+ y2+ 2y = 0; (0, –2)
50)
A)
y = – x
B)
x = 0
C)
y = – x – 2
D)
y = – 2
Solve the problem.
51)
P(x) = – x3+ 24x2– 144x + 50, x
2 is an approximation to the total profit (in thousands of dollars)
from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must
be sold to maximize profit.
51)
A)
12 hundred thousand
B)
2 hundred thousand
C)
10 hundred thousand
D)
4 hundred thousand
52)
The velocity of a particle (in ft
s) is given by v = t2– 5t + 4, where t is the time (in seconds) for which
it has traveled. Find the time at which the velocity is at a minimum.
52)
A)
2.5 sec
B)
4 sec
C)
5 sec
D)
2 sec
Assume x and y are functions of t. Evaluate dy/dt.
53)
x3+ y3= 9; dx/dt = – 3, x = 1, y = 2
53)
A)
4
3
B)
–4
3
C)
3
4
D)
–3
4
15
Solve the problem.
54)
The edge of a square is measured as 5.12 inches, with a possible error of ±0.05 inch. Estimate the
maximum error in the area of the square.
54)
A)
0.256 in.2
B)
0.512 in.2
C)
0.0025 in.2
D)
0.1 in.2
55)
A baseball team is trying to determine what price to charge for tickets. At a price of $10 per ticket,
it averages 45,000 people per game. For every increase of $1, it loses 5,000 people. Every person at
the game spends an average of $5 on concessions. What price per ticket should be charged in order
to maximize revenue?
55)
A)
$4.00
B)
$7.00
C)
$3.00
D)
$13.00
56)
A container, in the shape of an inverted right circular cone, has a radius of 8 inches at the top and a
height of 10 inches. At the instant when the water in the container is 9 inches deep, the surface level
is falling at the rate of –2 in./s. Find the rate at which water is being drained.
56)
A)
–311.04 in.3/s
B)
–286.5 in.3/s
C)
–325.71 in.3/s
D)
–456 in.3/s
Assume x and y are functions of t. Evaluate dy/dt.
57)
xy2= 4; dx/dt = – 5, x = 4, y = 1
57)
A)
–8
5
B)
–5
8
C)
8
5
D)
5
8
Find dy/dx by implicit differentiation.
58)
xy + x + y = x2y2
58)
A)
2xy2– y
2x2y + x
B)
2xy2+ y
2x2y – x
C)
2xy2+ y + 1
–2x2y – x – 1
D)
2xy2– y – 1
–2x2y + x + 1
16
Solve the problem.
59)
A company knows that unit cost C and unit revenue R from the production and sale of x units are
related by C =R2
112,000 + 5807. Find the rate of change of revenue per unit when the cost per unit is
changing by $12 and the revenue is $4000.
59)
A)
$374.35
B)
$240.00
C)
$168.00
D)
$580.70
60)
Find the elasticity of demand E for the demand function q =49,000 – 10p2
60)
A)
E =2p2
49,000 –p2
B)
E =–p
49,000 – 10p2
C)
E =2p2
4900 –p2
D)
E =–2p2
4900 –p2
Find the absolute extrema if they exist as well as where they occur.
61)
f(x) =x –1
x2+5x +10
61)
A)
Absolute minimum of – 1 at x = – 3; absolute maximum of 1
15 at x =5
B)
No absolute extrema
C)
Absolute minimum of – 1 at x = – 3; no absolute maxima
D)
Absolute minimum of –5
6 at x = – 4; absolute maximum of 1
15 at x =5
Use the differential to approximate the quantity to four decimal places.
62)
ln 1.03
62)
A)
0.0296
B)
0.0300
C)
–0.0300
D)
–0.0305
Solve the problem.
63)
P(x) = – x3+ 12x2– 21x + 100, x
4 is an approximation to the total profit (in thousands of dollars)
from the sale of x hundred thousand tires. Find the number of hundred thousands of tires that must
be sold to maximize profit.
63)
A)
13 hundred thousand
B)
7 hundred thousand
C)
10 hundred thousand
D)
4 hundred thousand
Find the equation of the tangent line at the given point on the curve.
64)
2xy – 2x + y = – 14; (2, –2)
64)
A)
y =6
5x –22
5
B)
y = – 6
5x +22
5
C)
y = – 5
6x +22
6
D)
y =5
6x –22
6
Solve the problem.
65)
Water is discharged from a pipeline at a velocity v given by v =1270p(1/2), where p is the pressure
(in psi). If the water pressure is changing at a rate of 0.280 psi/second, find the acceleration (dv/dt)
of the water when p =53 psi.
65)
A)
24.42 ft/s2
B)
46.23 ft/s2
C)
1294.4 ft/s2
D)
87.22 ft/s2
66)
Supertankers off–load oil at a docking facility shore point 3 miles offshore. The nearest refinery is
10 miles east of the docking facility. A pipeline must be constructed connecting the docking facility
with the refinery. The pipeline costs $300,000 per mile if constructed underwater and $200,000 per
mile if over land.
3 mi
10 mi
Locate point B to minimize the cost of construction.
66)
A)
Point B is 2.68 miles from Point A.
B)
Point B is 9.15 miles from Point A.
C)
Point B is 5.06 miles from Point A.
D)
Point B is 5.67 miles from Point A.
Assume x and y are functions of t. Evaluate dy/dt.
67)
yx + 1 = 12; dx/dt = 8, x = 15, y = 3
67)
A)
4
3
B)
3
4
C)
–3
4
D)
–4
3
Solve the problem.
68)
The demand for ground chuck (hamburger) in a certain region of the United States is given by
q =3.45p–0.14. Is the demand for ground chuck elastic or inelastic?
68)
A)
Elastic
B)
The demand has unit elasticity.
C)
Inelastic
D)
None of these
Use the differential to approximate the quantity to four decimal places.
69)
e0.48
69)
A)
1.6161
B)
1.4800
C)
.5200
D)
0.6188
Find the indicated absolute extremum as well as all values of x where it occurs on the specified domain.
70)
f(x) =x2e–0.25x; [3,10]
Maximum
70)
A)
4.2513 at x = 3
B)
8.2085 at x = 10
C)
0 at x = 0
D)
8.6615 at x = 8
71)
f(x) = (x + 1)2(x – 2); [–2, 1]
Maximum
71)
A)
0 at x = – 1
B)
–2 at x = 0
C)
No absolute maximum
D)
–4 at x = – 2
Find dy/dx by implicit differentiation.
72)
x1/3 – y1/3 = 1
72)
A)
–x
y
2/3
B)
–y
x
2/3
C)
y
x
2/3
D)
x
y
2/3
Find the absolute extrema if they exist as well as where they occur.
73)
f(x) =3– x –25/x, x > 0
73)
A)
Absolute minimum of –7 at x =5; no absolute maximum
B)
Absolute maximum of –23 at x = 1; no absolute minimum
C)
Absolute maximum of –7 at x =5; no absolute minimum
D)
Absolute maximum of 13 at x = – 5; absolute minimum of 3 at x = 0