Name: __________________
1 The figure shows the graphs of f , f
Which of the graphs is the graph of f
a.
Class:
' , and f '' .
'' ?
Graph 3
b.
Graph 2
c.
Graph 1
2 The figure shows the graphs of f , f
' , f '' , f '''
''' ?
Which of these graphs is the graph of f
a.
Graph 3
b.
Graph 4
c.
Graph 1
d.
Graph 2
PAGE 1
.
Date: _____________
Name: __________________
Class:
Date: _____________
3 The figure shows the graphs of three functions. One is the position function of a car, one is the velocity of the car, and one is its acceleration. Which of the graphs is the graph of position?
a.
Graph 2
b.
Graph 1
c.
Graph 3
4 Find the first and the second derivatives of the function
g ( u ) = u csc u
'
a.
g (u ) = b.
g ( u ) = csc ( u )
c.
g ( u ) = csc ( u )
d.
e.
''
csc ( u ) cot ( u ) + u csc ( u )
2
''
'
g (u ) = ''
u csc ( u ) 2
2
'
f (x ) = '
f (x ) = c.
f
d.
f
e.
f
PAGE 2
2 cot ( u )
u csc ( u ) cot ( u ) + csc ( u )
g ( u ) = csc ( u ) u csc ( u ) + u cot ( u ) 2
b.
2
u csc ( u ) + u cot ( u ) '
''
5 If f ( x ) = 8cosx + sin x, find f ( x ) and f ( x ).
a.
2
u cot ( u ) + 2 cot ( u )
8 sin ( 2 x ) + sin ( x )
8 sin ( x ) + sin ( 2 x )
''
(x ) = 8 cos ( x ) + 2 cos ( 2 x )
''
(x ) = 2 cos ( 2 x ) + 8 cos ( x )
''
(x ) = 8 cos ( 2 x ) + 2 cos ( x )
2 cot ( u )
Name: __________________
6
'''
Find y , if y =
a.
y
b.
y
c.
d.
y
y
Class:
6x + 7
.
3
2
'''
(x ) = 81 ( 6 x + 7 )
5
2
'''
( x ) = 81 ( 6 x + 7 )
'''
(x ) = 3 (6x + 7)
8
'''
7 If f ( x ) = (1
7x)
1/2
3
2
3 (6x + 7)
8
(x ) = '
5
2
''
'''
, find f ( 0 ), f ( 0 ),f ( 0 ), and f ( 0 ).
'
7, f
2
''
( 0 ) = 147 , f
4
a.
f ( 0 ) = 1, f ( 0 ) = '
7, f
f ( 0 ) = 1, f ( 0 ) =
2
''
b.
( 0 ) = 98 , f
4
'
f ( 0 ) = 1, f ( 0 ) = 7 , f
2
''
c.
( 0 ) = 147 , f
4
d.
f ( 0 ) = 1, f ( 0 ) = '
(n )
8 Find a formula for f
Date: _____________
7, f
2
''
'''
'''
5145
8
(0) = ( 0 ) = 2058
8
'''
( 0 ) = 98 , f
4
(0) =
'''
5145
8
(0) = 2058
8
(x)
f(x)=(3+x)
a.
f
b.
f
c.
f
d.
f
(n )
(x ) = ( 1)
(n + 1 )
(n )
(x ) = ( 1) n !(3 + x )
(n )
(x ) = ( 1)
(n )
(x ) = ( 1)
n ! (3 + x )
n
n
n
n ! (3 + x )
n
(n + 1 )
1
(n + 1 )
(n + 1 )
n
n !(3 + x )
n
9 The equation of motion is given for a particle, where s is in meters and t is in seconds. Find the acceleration after 2 seconds.
s = sin 2 t
a.
4 m/s
b.
16 c.
d.
0 m/s
e.
PAGE 3
2
2
m/s
2
16 2
m/s
2
4 m/s
2
2
Name: __________________
Class:
Date: _____________
10 A mass attached to a vertical spring has position function given by
y ( t ) = A cos t , where A is the amplitude of its oscillations and Find the velocity and acceleration as functions of time.
a.
v (t ) = A
b.
a (t ) = c.
a (t ) = A
cos ( d.
a (t ) = A
2
e.
v (t ) = 11 Find a third a.
2
A
A
cos ( t )
t )
t )
sin ( t )
'
''
'''
degree polynomial Q such that Q ( 1 ) = 3, Q ( 1 ) = 5, Q ( 1 ) = 6, and Q ( 1 ) = 12.
3
2
3
2
Q ( x ) = 3x + 2x
3
Q ( x ) = 2x
3
d.
t )
cos ( 5x + 1
5x
Q ( x ) = 2x + 3x
b.
c.
sin ( is a constant.
Q ( x ) = 2x
2
5x
1
3x + 1
2
3x + 5x
1
''
'
''
12 The function g is a twice differentiable function. Find f in terms of g, g , and g .
f (x ) = g ( x )
a.
f
''
''
'
x g ( x ) + g ( x )
(x ) =
4 x
b.
f
''
''
(x ) =
4x
c.
f
''
'
g ( x ) x g ( x )
2
x
''
(x ) =
'
x g ( x ) 4x
g ( x )
x
13 A plane flying horizontally at an altitude of 2 mi and a speed of 460 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 5
mi away from the station.
a.
379 mi/h
b.
422 mi/h
c.
14
432 mi/h
d.
426 mi/h
e.
408 mi/h
2
If a snowball melts so that its surface area decreases at a rate of 1 cm
a.
1
96
cm/min
b.
1
48
cm/min
c.
1
98
cm/min
d.
1
95
cm/min
PAGE 4
/min , find the rate at which the diameter decreases when the diameter is 48 cm. Leave in your answer.
Name: __________________
Class:
Date: _____________
15 Two cars start moving from the same point. One travels south at 20 mi/h and the other travels west at 58 mi/h. At what rate is the distance between the cars increasing 2 hours later? Round the
result to the nearest hundredth.
a.
61.45 mi/h
b.
61.34 mi/h
c.
59.35 mi/h
d.
61.35 mi/h
e.
61.38 mi/h
f.
62.36 mi/h
16 A man starts walking north at 7 ft/s from a point P. Five minutes later, a woman starts walking south at 5 ft/s from a point 500 ft due east of P. At what rate are the people moving apart 30 min after
the woman starts walking? Round the result to the nearest hundredth.
17
a.
13.01 ft/s
b.
12.03 ft/s
c.
11.99 ft/s
d.
10 ft/s
e.
12.1 ft/s
f.
12 ft/s
2
The altitude of a triangle is increasing at a rate of 5 cm/min while the area of the triangle is increasing at a rate of 1 cm
/min . At what rate is the base of the triangle changing when the altitude is
2
5 cm and the area is 135 cm .
a.
60.6 cm/min
b.
52.6 cm/min
c.
48.6 cm/min
d.
55.6 cm/min
e.
43.6 cm/min
f.
53.6 cm/min
18 A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 5 m higher than the bow of the boat. If the rope is pulled in at a rate of 1 m/s
how fast is the boat approaching the dock when it is 7 m from the dock? Round the result to the nearest hundredth if necessary.
a.
1.21 m/s
b.
1.24 m/s
c.
1.34 m/s
d.
1.23 m/s
e.
1.18 m/s
f.
2.23 m/s
19 At noon, ship A is 120 km west of ship B. Ship A is sailing south at 38 km/h and ship B is sailing north at 27 km/h. How fast is the distance between the ships changing at 3:00 p.m.? Round the result
to the nearest thousandth if necessary.
a.
55.358 km/h
b.
55.468 km/h
c.
56.358 km/h
d.
55.308 km/h
e.
55.356 km/h
f.
55.359 km/h
PAGE 5
Name: __________________
Class:
20 A water trough is 10 m long and a cross 3
with water at the rate of 0.5 m
a.
Date: _____________
section has the shape of an isosceles trapezoid that is 35 cm wide at the bottom, 90 cm wide at the top, and has height 50 cm. If the trough is being filled
/min , how fast is the water level rising when the water is 45 cm deep? Round the result to the nearest hundredth.
5.72 cm/min
b.
4.85 cm/min
c.
5.92 cm/min
d.
6.02 cm/min
e.
16.02 cm/min
f.
5.97 cm/min
21
3
Gravel is being dumped from a conveyor belt at a rate of 25 ft
/min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How
fast is the height of the pile increasing when the pile is 7 ft high? Round the result to the nearest hundredth.
a.
0.59 ft/min
b.
0.72 ft/min
c.
3 ft/min
d.
0.82 ft/min
e.
1.69 ft/min
f.
0.65 ft/min
22
Two sides of a triangle have lengths 14 m and 19 m. The angle between them is increasing at a rate of 3
of fixed length is 60
a.
0.707 m/min
b.
0.758 m/min
c.
1.771 m/min
d.
1.037 m/min
e.
1.729 m/min
f.
0.745 m/min
PAGE 6
? Round the result to the nearest thousandth if necessary.
/ min. How fast is the length of the third side increasing when the angle between the sides
Name: __________________
Class:
Date: _____________
23 Boyle's Law states that when a sample of gas is compressed at a constant temperature, the pressure and volume satisfy the equation PV = C, where C is a constant. Suppose that at a certain
3
instant the volume is 550 cm , the pressure is 165 kPa, and the pressure is increasing at a rate of 20 kPa/min. At what rate is the volume decreasing at this instant? Round the result to the
nearest thousandth if necessary.
24
3
a.
66.65 cm /min
b.
66.307 cm /min
c.
66.777 cm /min
d.
66.694 cm /min
e.
67.887 cm /min
f.
66.667 cm /min
3
3
3
3
3
When air expands adiabatically (without gaining or losing heat), its pressure and volume are related by the equation PV
1.4
=C
, where C is a constant. Suppose that at a certain instant the
3
volume is 345 cm
necessary.
a.
25
and the pressure is 90 kPa and is decreasing at a rate of 12 kPa/min. At what rate is the volume increasing at this instant? Round the result to the nearest thousandth if
3
32.857 cm
3
b.
32.825 cm
c.
33.17 cm
d.
32.962 cm
e.
33.878 cm
f.
32.687 cm
3
/min
/min
/min
3
3
3
/min
/min
/min
If two resistors with resistances R
1
and R
2
are connected in parallel, as in the figure, then the total resistance R measured in ohms ( ), is given by
1 = 1 + 1
R
R
R
1
increasing at rates of 0.1 a.
0.164 /s
b.
0.224 /s
c.
0.111 /s
d.
0.173 /s
e.
1.201 /s
f.
0.15 PAGE 7
/s
/ s and 0.4 / s respectively, how fast is R changing when R
1
= 85 and R
2
. If R
2
= 105 ? Round the result to the nearest thousandth if necessary.
1
and R
2
are
Name: __________________
Class:
Date: _____________
26 Two carts, A and B, are connected by a rope 35 ft long that passes over a pulley (see the figure below). The point Q is on the floor 11 ft directly beneath and between the carts. Cart A is being pulled
away from Q at a speed of 3 ft/s. How fast is cart B moving toward Q at the instant when cart A is 2 ft from Q? Round the result to the nearest hundredth.
a.
1.62 ft/s
b.
0.61 ft/s
c.
0.41 ft/s
d.
0.62 ft/s
e.
0.54 ft/s
f.
1.11 ft/s
27 A television camera is positioned 4800 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the
mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let's assume the rocket rises vertically and its speed is 680 ft/s when it has
risen 4000 ft. If the television camera is always kept aimed at the rocket, how fast is the camera's angle of elevation changing at this moment? Round the result to the nearest thousandth if
necessary.
a.
0.073 rad/s
b.
0.254 rad/s
c.
0.146 rad/s
d.
0.084 rad/s
e.
0.107 rad/s
f.
1.114 rad/s
28
A plane flying with a constant speed of 240 km/h passes over a ground radar station at an altitude of 2 km and climbs at an angle of 30
station increasing a minute later? Round the result to the nearest integer if necessary.
a.
228 km/h
b.
238 km/h
c.
229 km/h
d.
226 km/h
e.
225 km/h
f.
227 km/h
. At what rate is the distance from the plane to the radar
29 A runner sprints around a circular track of radius 180 m at a constant speed of 4 m/s. The runner's friend is standing at a distance 210 m from the center of the track. How fast is the distance
between the friends changing when the distance between them is 210 m? Round the result to the nearest thousandth if necessary.
a.
3.567 m/s
b.
3.562 m/s
c.
3.614 m/s
d.
3.642 m/s
e.
3.724 m/s
f.
4.824 m/s
PAGE 8
Name: __________________
Class:
Date: _____________
30 Atmospheric pressure V decreases as altitude h increases. At a temperature of 10 C the pressure is 100 kilopascals (kPa) at sea level, 86 kPa at h = 1 km, and 74.4 kPa at h = 2 km. Use a linear
approximation to estimate the atmospheric pressure at an altitude of 3 km.
a.
58.9
b.
64.3
c.
65.2
d.
62.8
e.
64
f.
65.4
31 The table lists the amount of U.S. cash per capita in circulation as of June 30 in the given year. Use a linear approximation to estimate the amount of cash per capita in circulation in the year 2000.
t
1960
1970
1980
1990
C(t)
$164
$254
$564
$1068
a.
$1572
b.
$1661
c.
$1616
d.
$1565
e.
$1517
f.
$1593
32 Find the linearization L(x) of f at a.
f (x ) =
1
, a = 0.
6 + x
a.
1
L (x ) = x
+
6
b.
L (x ) =
6 L (x ) =
1
c.
12 6 x
+
6
d.
L (x ) =
1
6
33
12 6
x
12 6
x
12 6
Use the linear approximation of the function f
a.
6.03
b.
1.78
c.
2.26
d.
0.04
e.
6.59
f.
2.84
PAGE 9
(x ) =
5 x
at a = 0 to approximate the number
5.09
.
Name: __________________
34
Class:
Use the linear approximation of the function f
a.
1.19
b.
5.58
c.
5.03
d.
4.22
e.
6.12
f.
2.58
(x ) =
8
4 + x
Date: _____________
at a = 0 to approximate the number
8
3.97
.
35 Find the differential of the function.
y = x
a.
dy = ( 5x b.
dy = ( 5x
c.
dy = ( x
d.
dy = ( 5x
36 Compute y
4
4
5
+ 4x
4 ) dx
+ 4 ) dx
+ 4 ) dx
5
+ 4 ) dx
and dy for the given values of x and dx
= x. .
2
y = x , x = 3, x = 0.5
a.
y = 2.25, dy = 3
b.
y = 2.25, dy = 2
c.
y = 3.25, dy = 3
d.
y = 3.25, dy = 2
37 Find any absolute and local maximum and minimum values of f(x) = 7
a.
4 is an absolute minimum
b.
4 is a local minimum
c.
5x if x
4.
Begin by sketching its graph by hand.
13 is an absolute maximum
d.
e.
4 is an absolute maximum
f.
13 is an absolute minimum
13 is a local maximum
38 Find all the critical numbers of the function:
g ( x ) = 7x + sin(7x )
a.
7
b.
2 n
7
c.
(2n + 1)
14
d.
(2n + 1)
7
PAGE 10
Name: __________________
39
40
Find the local minimum value of y
a.
b.
0
c.
Class:
= 9x
2
if Date: _____________
6 < x < 6
24
6
Find the absolute maximum value of y
a.
3
b.
2
c.
9
d.
0
= 2sin( x )
6
41 Find the critical numbers of the function:
y = 10x
a.
10
b.
c.
0
d.
60
PAGE 11
3
2
+ 60x
Name: __________________
Class:
Date: _____________
42 Find the critical numbers of the function:
x
y =
x
a.
7,
b.
49,
c.
7, 0
d.
0,
2
+ 49
7
49
7
43 Find the critical numbers of the function:
4
5
2
F ( x ) = x (x a.
3)
3, 0
7
b.
3, 3 , 0
7
c.
3, 12
7
d.
3, 6 , 0
7
44 Find the critical numbers of the function:
G(x) =
a.
3
x
2
2x
2, 0
b.
2, 4, 0
c.
1, 0
d.
2, 1, 0
45 Find the absolute minimum value of:
y = 2x
2
+ 4
x
on the interval [0 , 4].
a.
2
b.
6
c.
8
d.
4
46 Find the absolute maximum value of
y =
on the interval [ a.
10
b.
9
c.
0
d.
8
PAGE 12
9, 9].
81 x
2
Name: __________________
Class:
Date: _____________
47 Find the absolute maximum of the function:
f ( x ) = sin(2x ) + cos(2x )
on the interval [0 ,
]
6
a.
f( ) =
8
b.
f( ) = 1
8
c.
f( ) = 2
4
d.
f( ) =
2
2
2
48 Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.
f(x)=x
a.
b.
3
21x
2
+ 104x + 3, [ 0, 13 ]
129
c = 7 3
129
c = 7 +
3
c.
d.
c
c
129
= 7 +
1
, c
3
= 129 +
1
7
, c
3
2
= 7 2
= 129 129
3
7
3
49 Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.
f ( x ) = sin 5 x , [ a.
c
b.
c
c.
c
d.
c
1 , c = 2
10
= 1
1 , c = 2
10
=
1
2, 2 ]
5 5
1
10
3
10
1
= 1 , c = 2
10
3
10
1
= 1 , c = 3
2
10
10
50 Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem.
f (x ) = x
a.
c=
13
b.
c=
14
c.
c=
d.
c = 11
PAGE 13
12
x + 18 , [ 18, 0 ]
Name: __________________
Class:
Date: _____________
51 Use the graph of f to estimate all values of c that satisfy the conclusion of the Mean Value Theorem for the interval [ 0, 14 ].
a.
c = 0.6
b.
c = 11.4
c.
c = 2.4
d.
c = 1.4
e.
c = 7.8
f.
c = 12.8
g.
c=6
h.
c = 10.6
52 Find the exact values of the numbers c that satisfy the conclusion of The Mean Value Theorem for the function f ( x ) = x 3
a.
c = 4 3
b.
c = c.
c =
4x for the interval [
4, 4 ].
4 3
3
4 3
3
d.
c = 4 3
3
53 Verify that the function satisfies the hypotheses of The Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of The Mean Value Theorem.
f ( x ) = 4x
a.
c=5
b.
c=4
c.
c=0
d.
c=2
PAGE 14
2
+ 2x + 5, [
4, 4 ]
Name: __________________
Class:
Date: _____________
54 Verify that the function satisfies the hypotheses of The Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of The Mean Value Theorem.
f (x ) =
a.
c =
b.
c =
4
x,
[ 0, 1 ]
1
16
3
16
16
3
3
c =
c.
16
c =
d.
16
16
55 Verify that the function satisfies the hypotheses of The Mean Value Theorem on the given interval. Then find all numbers c that satisfy the conclusion of The Mean Value Theorem.
f (x ) =
x
,
[ 0, 1 ]
x + 3
56
a.
c =
b.
c = c.
c =
d.
c = 3
3 +
12
12
12 +
3
5
How many real roots does the equation x
a.
no real roots
b.
exactly two real roots
c.
exactly one real root
d.
exactly three real roots
57 How many real roots does the equation x 5
[
+ 5x + 1 = 0 have ?
8x + c = 0 have in the interval
1, 1 ] ?
a.
at most two real root
b.
no real roots
c.
at most three real root
d.
at most one real root
58 If f ( 3 ) = 10 and f
'(x ) a.
f ( 7 ) = 96
b.
f ( 7 ) = 33
c.
f ( 7 ) = 26
d.
f ( 7 ) = 14
4 for 3 x 7 , how small can f ( 7 ) be?
59 Does there exist a function f that satisfies the following condition sets ?
f ( 0 ) = 5, f ( 2 ) = 6 and
f ' ( x ) 3 for all x
a.
no
b.
yes
PAGE 15
Name: __________________
Class:
Date: _____________
60 Let f ( x ) = 6 / x and
g (x ) =
{
6
x
1 + 6
x
if x > 0
if x < 0
Is it true or false that f ' ( x ) = g ' ( x ) for all x in their domains ?
a.
False
b.
True
61 Find the inflection points of the following function:
2x+2 0 < x < 3
f(x)=
a.
( , 2 ) , ( 2 , 4 + 2 )
b.
( , 2 ) , ( 2 , 4 + 2 )
c.
( , 2 ) , ( 2 , 4 )
d.
( , 2 + 2 ) , ( 2 , 4 )
e.
( , 2 + 2 ) , ( 2 , 4 + 2 )
62 Find the critical numbers of f ( x ) = x4 ( x
a.
0 , 5 , 16
7
b.
0 , 4 , 16
7
c.
0 , 5 , 16
11
d.
0 , 4 , 16
11
4)
3
63 The graph of the derivative f '(x) of a continuous function f is shown. On what intervals is f decreasing?
a.
( 1 , 3) ( 4 , 5)
b.
( 0 , 1) ( 3 , 4) c.
( PAGE 16
, 4) (5 , )
(4 , )
2 sin x
Name: __________________
Class:
Date: _____________
64 The graph of the derivative f ' (x) of a continuous function f is shown. On what intervals is f concave downward?
a.
( 10 , 20 ) b.
( 5 , 15 ) c.
( 0 , 10 )
( 20 , )
( 20 , 21)
65 The graph of the derivative f'(x) of a continuous function f is shown. Find the x a.
4,12,17
b.
16
c.
8
d.
0
coordinate(s) of the point(s) of inflection.
66 Find the intervals of increase or decrease of the following function:
y(x) = x
a.
f decreasing on (
b.
f increasing on (
c.
f decreasing on (
d.
f increasing on (
PAGE 17
, 4) and (8,
, 4) and (8,
), and increasing on (4, 8)
), and decreasing on (4, 8)
, 12) and (24,
, 12) and (24,
), and increasing on (12, 24)
), and decreasing on (12, 24)
3
18 x
2
+ 96 x
Name: __________________
Class:
67
Find a, b, c, and d in the cubic function
a.
a = 2, b =
b.
a = 5, b = 24, c = 15, d = 2
c.
a = 2, b = 15, c = 24, d = 5
d.
a = 5, b = 5, c = 25, d =
15, c = 24, d =
f(x) = a x
3
bx
2
+ cx Date: _____________
d
if the function has a local maximum value of 6 at 1 and a local minimum value of
5
15
68 Explain the meaning of the expression.
lim
x a.
As x becomes large, f (x) approaches 9
b.
As x becomes large negative, f (x) approaches 9
f (x ) = 9
69 For the function f whose graph is given, state the limit.
lim f (x )
x
a.
4
b.
c.
3
d.
PAGE 18
4
21 at 4.
Name: __________________
Class:
Date: _____________
70 For the function f whose graph is given, state the limit.
lim
x a.
4
b.
c.
d.
3
f (x )
3
71 For the function f whose graph is given, state the limit.
lim
x a.
b.
c.
3
d.
4
PAGE 19
f (x )
3
+
Name: __________________
Class:
Date: _____________
72 For the function f whose graph is given, state the limit.
lim f (x )
x
a.
b.
5
c.
3
d.
73 Find the value of the limit
lim
x
7
x
x
7
a.
7
b.
c.
0
74 Evaluate the limit
x + 6
lim
x
a.
b.
7
c.
1
d.
6
e.
0
f.
6
7
PAGE 20
x
2
2x + 7
Name: __________________
Class:
Date: _____________
75 Find the limit
lim
x
1
a.
5
b.
5
6
c.
1
6
d.
e.
1
76 Find the limit.
(x7
lim
x
a.
0
b.
c.
3
d.
77 Find the limit.
lim
x
x
9
x
8
a.
b.
1
c.
d.
0
PAGE 21
4
+ 5
3x
6
)
x
2
+ 6x
6x + 1
Name: __________________
Class:
Date: _____________
78 What is the graph of a function that satisfies the following conditions:
x
lim f(x) = 0; lim f(x) = x 0
lim f(x) = ; lim f(x) = ; f (6) = 0;
x
7
x
+
7
a.
b.
79 Find the horizontal asymptotes of the curve.
y = 5 2x
1 + x
a.
y=2
b.
y= 0
c.
y=5
d.
y=
2
e.
y=
5
PAGE 22
Name: __________________
Class:
Date: _____________
80 Sketch the graph of the function that satisfies all of the given conditions.
f '(2) = 0, f (2) = 2, f (0)=0, f '(x ) < 0 if 0 < x < 2, f ' (x ) > 0 if x > 2, f ''(x ) < 0 if 0 f ''(x ) > 0 if 1 < x < 6, lim f (x ) = 2, f ( x ) = f (x ) for all x .
x
a.
b.
c.
PAGE 23
x < 1 or if x > 6,
Name: __________________
Class:
81
Let P(x) and Q(x) be polynomials. Find
lim P(x)
Q(x)
x
a.
b.
8
c.
3
d.
0
e.
5
3
82
Find
lim f(x) if
x
3x x
2 < f(x) < 3x
6
b.
3
c.
10
PAGE 24
3x + 7
x
x > 10.
a.
2
2
Date: _____________
if the degree of P(x) is 5 and the degree of Q(x) is 8.
Name: __________________
Class:
Date: _____________
83 Sketch the curve.
y = x tan 3x , a.
b.
c.
PAGE 25
< x < 2
2
Name: __________________
Class:
Date: _____________
84 Sketch the curve.
y = 1 x 3
a.
b.
c.
PAGE 26
sin x ,
0 < x < 3
Name: __________________
Class:
Date: _____________
85 Sketch the curve.
y =
a.
b.
c.
PAGE 27
cos x
6 + sin x
Name: __________________
Class:
Date: _____________
86 Sketch the curve.
x
a.
b.
c.
PAGE 28
3
x
y =
2
1
Name: __________________
Class:
Date: _____________
87 Sketch the curve.
2
y =
a.
b.
c.
PAGE 29
x
4x + 7
Name: __________________
Class:
Date: _____________
88 Sketch the curve.
y = x a.
b.
c.
PAGE 30
3
x
Name: __________________
89
Let f ( x ) =
x
3
Class:
Date: _____________
+ 2 . Show that
x
lim
x 2
f (x ) x
2
= 0
2
This shows that the graph of f approaches the graph of y = x , and we say that the curve y = f ( x ) is asymptotic to the parabola y = x . Use this fact to help sketch the graph of f.
a.
b.
c.
PAGE 31
Name: __________________
Class:
Date: _____________
90 Discuss the asymptotic behavior of
f (x ) = x
Find the asymptote of f.
2
a.
y=x
b.
y=x
c.
y=1
d.
y=x
PAGE 32
3
3
+ 6.
x
ANSWER KEY
Name: __________________
Class:
Date: _____________
1.
a
11.
d
21.
f
31.
a
41.
b
51.
2.
3.
4.
5.
6.
7.
8.
9.
10.
d
a
c,d
b,c
b
c
b
d
b,e
12.
13.
14.
15.
16.
17.
18.
19.
20.
c
b
a
d
f
f
d
a
c
22.
23.
24.
25.
26.
27.
28.
29.
30.
a
f
a
c
b
d
f
c
d
32.
33.
34.
35.
36.
37.
38.
39.
40.
d
c
a
b
c
c
d
b
b
42.
43.
44.
45.
46.
47.
48.
49.
50.
a
d
d
b
b
a
c
c
c
52.
53.
54.
55.
56.
57.
58.
59.
60.
PAGE 1
b,d,e,g
d
c
b
b
c
d
c
a
b
61.
e
71.
a
81.
d
62.
63.
64.
65.
66.
67.
68.
69.
70.
b
a
c
c
b
c
b
d
b
72.
73.
74.
75.
76.
77.
78.
79.
80.
c
c
e
c
b
c
a
d
a
82.
83.
84.
85.
86.
87.
88.
89.
90.
b
c
c
c
a
b
c
b
a