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Find all the critical numbers of the function.
4
1.
8
12 n2.
2
n3.
4
12 n4.
none of these
5.
)4sin(4)( xxxg
Find all the critical numbers of the function.
1 2 3 4 5
20% 20% 20%20%20%
4
1.
8
12 n2.
2
n3.
4
12 n4.
none of these
5.
( ) 4 sin(4 )g x x x ( ) 4 4cos(4 ) 0
cos(4 ) 1
4 (2 1)
(2 1)
4
g x x
x
x n
nx
Find the absolute maximum value of .
1 2 3 4 5
20% 20% 20%20%20%
1. 7
2. 15
3. 0
4. 5
5. -6
7sin10
xy
(7)( ) cos 0
10 10
cos 010
(2 1)
10 2(2 1) 10
5(2 1)2
(5(2 1)) 7sin(5(2 1) )10
(5(2 1)) 7(1) 7 For any integer value of n
or 7(-1) these give mins
xy
x
x n
nx n
y n n
y n
1. x
2. 0
3. x
4. x
5. 1
Find the limit.1 2 3 4 5
20% 20% 20%20%20%
ln 2 ln 3ln 3 ln 2
t
tt
t
23lim
0
0 0
3 2 ln 3(3 ) ln 2(2 )lim = lim = ln 3 ln 2
1
t t t t
t tt
Find ƒ.
1. x
2. x
3. x
4. x
5. none of these
( ) 2 3sin , (0) 5f t t t f
2( ) 3cos 2f t t t 2( ) 3cosf t t t
( ) 2 3sin 5f t t t
2( ) 3cos 5f t t t
Find ƒ.
1 2 3 4 5
20% 20% 20%20%20%
1. x
2. x
3. x
4. x
5. none of these
( ) 2 3sin , (0) 5f t t t f
2( ) 3cos 2f t t t 2( ) 3cosf t t t
( ) 2 3sin 5f t t t
2( ) 3cos 5f t t t 2
2
2
( ) 2 3sin , (0) 5
( ) 2 3sin 3cos
5 0 3cos0
2
( ) 3cos 2
f t t t f
f t t tdt t t c
c
c
f t t t
Estimate the extreme values of the function.
1. 146, 182
2. 1,226, -223
3. 383.18, 200.95
4. 150, -223
5. 1,226, 182
27293
1)( 23 xxxxy
Estimate the extreme values of the function.
1 2 3 4 5
20% 20% 20%20%20%
1. 146, 182
2. 1,226, -223
3. 383.18, 200.95
4. 150, -223
5. 1,226, 182
27293
1)( 23 xxxxy
3 2
2
1( ) 9 72 2
3
( ) 18 72 0
6, 12
(6) 182, (12) 146
y x x x x
y x x x
x x
y y
1. x
2. x
3. x
4. x
5. x
Find the inflection points for the function given.
xxxf sin28)( , 30 x
,8,
,2,
,8,
,28,
,28,
216,2
216,2
16,2
216,2
16,2
1. x
2. x
3. x
4. x
5. x
Find the inflection points for the function given.1 2 3 4 5
20% 20% 20%20%20%
xxxf sin28)( , 30 x
,8,
,2,
,8,
,28,
,28,
216,2
216,2
16,2
216,2
16,2
( ) 8 2 sin
( ) 8 cos
( ) sin 0
0 2 3
0 0 0 0
f x x x
f x x
f x x
x n
x
f
f IP IP IP IP
Sketch the curve. 1 2 3
33% 33%33%
xxy 33 3 1.
2.
3.
3
2
3 3
9 3 0
18 0
0 for all x
0 for 0
0
0
0
y x x
y x
y x
y
y x
x
y
y
y IncConDw IncConUp
How many points of inflection are on the graph of the function?
1. 1
2. 2
3. 4
4. 3
5. 5
1712518)( 23 xxxxf
How many points of inflection are on the graph of the function? 1 2 3 4 5
20% 20% 20%20%20%
1. 1
2. 2
3. 4
4. 3
5. 5
1712518)( 23 xxxxf3 2
2
( ) 18 5 12 17
( ) 54 10 12
( ) 108 10 0
5
545
54( ) 0
f x x x x
f x x x
f x x
x
x
f x
ConDw IP ConUp
Evaluate the limit. 1 2 3 4 5
20% 20% 20%20%20%
1. ½
2. 2
3. 0
4. ∞
5. -2
x 1
1lim
1 ln
x
x x
x 1 x 1
x 1 x 1
x 1
2
1 ln 1 0lim lim
1 ln ( 1)(ln ) 0
ln 1 1 lnlim lim
1 1ln ln
11 1
lim1 ( 1) 1 1 2
x x x x
x x x x
x xx x
x xx x
xx x
x x
Find two positive numbers whose product is 144 and whose sum is a minimum.
1. 4, 36
2. 2, 72
3. 12, 12
4. 3, 48
5. 6, 24
Find two positive numbers whose product is 144 and whose sum is a minimum.
1 2 3 4 5
20% 20% 20%20%20%
1. 4, 36
2. 2, 72
3. 12, 12
4. 3, 48
5. 6, 24
2
144
Minimize
144( )
144( ) 1 0
12
0 12
0
24
xy
x y
f x xx
f xx
x
x
f
f Dec Inc
A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $12 per square meter. Material for the sides costs $5 per square meter. Find the cost of materials for the cheapest such container.
1. $153.92
2. $158.10
3. $152.90
4. $151.60
5. $153.90
6. $152.40
A rectangular storage container with an open top is to have a volume of 10 m3. The length of its base is twice the width. Material for the base costs $12 per square meter. Material for the sides costs $5 per square meter. Find the cost of materials for the cheapest such container.
1 2 3 4 5 6
17% 17% 17%17%17%17%
1. $153.92
2. $158.10
3. $152.90
4. $151.60
5. $153.90
6. $152.40
2xx
y2
2
2
2
22
2
2
2
3
2
3
3
(2 )( )( )
10 2
10
2( , ) 12(2 )( ) 5(2)(2 )( ) 5(2)( )( )
( , ) 24 20 10
( , ) 24 30
10( ) 24 30 ( )
2150
( ) 24
150( ) 48
15048 0
150
48
150 150( ) 24 153.8
48 15048
v x x y
x y
yx
c x y x x x y x y
c x y x xy xy
c x y x xy
c x x xx
c x xx
c x xx
xx
x
c x
9783
Find the most general antiderivative of the function.
1. x
2. x
3. x
4. x
5. x
91418)( 2 xxxf
CxxxxF 92830)( 45
CxxxxF 976)( 23
CxxxxF 91418)( 23
CxxF 1436)(
CxxxF 146)( 2
Find the most general antiderivative of the function.
1 2 3 4 5
20% 20% 20%20%20%
1. x
2. x
3. x
4. x
5. x
91418)( 2 xxxf
CxxxxF 92830)( 45
CxxxxF 976)( 23
CxxxxF 91418)( 23
CxxF 1436)(
CxxxF 146)( 2
2
3 2
18 14 9
18 14 93 2
x x dx
x xx c
Given that the graph of ƒ passes through the point (4, 69) and that the slope of its tangent line at is 10x - 4, find ƒ(1).
1. 1
2. 12
3. 11
4. 6
5. 0
)(, xfx
Given that the graph of ƒ passes through the point (4, 69) and that the slope of its tangent line at is 10x - 4, find ƒ(1).
1 2 3 4 5
20% 20% 20%20%20%
1. 1
2. 12
3. 11
4. 6
5. 0
)(, xfx
2
2
2
f (x)=10x 4
f(x)=5x 4
69 5(4) 4(4)
5
f(x)=5x 4 5
(1) 6
x c
c
c
x
f
Find the critical numbers of
7
12,2,0
1.
11
12,3,0
2.
7
12,3,0
3.
11
12,2,0
4.
12
7,2,0
5.
.3)( 34 xxxf
Find the critical numbers of
1 2 3 4 5
20% 20% 20%20%20%
7
12,2,0
1.
11
12,3,0
2.
7
12,3,0
3.
11
12,2,0
4.
12
7,2,0
5.
34( ) 3 .f x x x
3 23 4
2 3 4
2 4 3 4
2 4 3
( ) 4 3 3 3 ( )
( ) 3 (4 ( 3) 3 ))
( ) 3 (4 12 3 )
( ) 3 (7 12 ) 0
120, 3, x=
7
f x x x x x
f x x x x x
f x x x x x
f x x x x
x x
Find the absolute maximum value of on the interval [- 6, 6]. 1 2 3 4 5
20% 20% 20%20%20%
1. 5
2. 6
3. 7
4. 0
5. 1
236 xy 2
2 2
36
2
2 36 360, x= 6
6 0 6
0
0 6 0
y x
x xy
x xx
x
y ud ud
y
Estimate the absolute maximum value of the
function to two decimal places on
the interval [0, 2].
1. 0.87
2. 1.3
3. 1.95
4. 1.5
5. -0.87
22y x x x
Estimate the absolute maximum value of the
function to two decimal places on
the interval [0, 2].
1 2 3 4 5
20% 20% 20%20%20%
1. 0.87
2. 1.3
3. 1.95
4. 1.5
5. -0.87
22y x x x 2
2
2
2 2 2
2 2
2 2
2 2
2
(2 2 )(1) 2
2 2
2(2 ) (2 2 ) 4 2 2 2
2 2 2 2
6 4 3 2
2 2 23
0, x= , 22
30 220 0
0 1.299 0
y x x x
xy x x x
x x
x x x x x x x xy
x x x x
x x x xy
x x x x
x x
x
y ud
y
Find any absolute or local maximum and minimum values of if
1. -4 is an absolute maximum
2. 6 is an absolute minimum
3. 6 is an absolute maximum
4. 6 is a local minimum
5. -4 is a local maximum
6. -4 is an absolute minimum
xxf 28)( .6x
Find any absolute or local maximum and minimum values of if 1 2 3 4 5 6
17% 17% 17%17%17%17%
1. -4 is an absolute maximum
2. 6 is an absolute minimum
3. 6 is an absolute maximum
4. 6 is a local minimum
5. -4 is a local maximum
6. -4 is an absolute minimum
( ) 8 2f x x .6x( ) 8 2
( ) 2
6
4
f x x
f x
x
f
f
Absolute max
Find the critical numbers of the function.
1 2 3 4 5
20% 20% 20%20%20%
1. 5, 0
2. -2
3. 20
4. 0
5. -1
xxy 205 2
25 20
10 20 0
2
y x x
y x
x
How many points of inflection are on the graph of the function?
1. 3
2. 1
3. 4
4. 2
5. 5
971412)( 23 xxxxf
How many points of inflection are on the graph of the function? 1 2 3 4 5
20% 20% 20%20%20%
1. 3
2. 1
3. 4
4. 2
5. 5
971412)( 23 xxxxf3 2
2
( ) 12 14 7 9
( ) 36 28 7 0
.98, .2
( ) 72 28 0
.39
.98 .39 .2
0 0
0
f x x x x
f x x x
x x
f x x
x
x
f
f
f IP
Find the maximum or minimum point(s) of the function.
1. (0, 1)
2. (-8.6, 6)
3. (-8, 2)
4. (8, 0)
5. (16, 1)
222 61)( xxxF
Find the maximum or minimum point(s) of the function.
1 2 3 4 5
20% 20% 20%20%20%
1. (0, 1)
2. (-8.6, 6)
3. (-8, 2)
4. (8, 0)
5. (16, 1)
222 61)( xxxF
22 2
2
3 3
( ) 1 6
( ) 2 1 ( 2 ) 12
( ) 4 4 12 4 8 0
0
F x x x
F x x x x
F x x x x x x
x
Let and be polynomials.
Find if the degree of is 5 and
the degree of is 9.
1. -4
2. 9
3. 4
4. 0
5. 5
)(xP
)(xP )(xQ
)(xQ)(
)(lim
xQ
xPx
Let and be polynomials.
Find if the degree of is 5 and
the degree of is 9. 1 2 3 4 5
20% 20% 20%20%20%
1. -4
2. 9
3. 4
4. 0
5. 5
)(xP
)(xP )(xQ
)(xQ)(
)(lim
xQ
xPx
Sketch the curve.1 2 3
33% 33%33%
1.
2.
3.
3/22xxy 2/3
13
1/31 1
3 3
13
13
23
23
2
2 4 3 41 (2)( ) 1 0
3 3 3
3 4 0
4 64
3 274 1 4
3 3 964
0270
0Re
y x x
xy x
x x
x
x x
y xx
x
y ud
y ud
y DecConDw IncConUplMax
1. x
2. x
3. x
4. x
5. x
For what values of c does the curve have maximum and minimum points?
xcxxxF 105)( 23
15|| c
150|| c
500,1|| c
30|| c
750|| c
1. x
2. x
3. x
4. x
5. x
For what values of c does the curve have maximum and minimum points? 1 2 3 4 5
20% 20% 20%20%20%
xcxxxF 105)( 23
15|| c
150|| c
500,1|| c
30|| c
750|| c
3 2
2
2 2
2 2
2
2
( ) 5 10
( ) 15 2 10
4 (2 ) 4(15)(10)
4 4(15)(10) 4 600
4 600 0
150 0
150
F x x cx x
F x x cx
b ac c
c c
c
c
c
Let . Show that . This shows that the graph of ƒ approaches the graph of y = x2, and we say that the curve y = f (x) is asymptotic to the parabola y = x2. Use this fact to help sketch the graph of ƒ.
1.
2.
3.
3 2( )
xf x
x
2lim ( ) 0
xf x x
Let . Show that . This shows that the graph of ƒ approaches the graph of y = x2, and we say that the curve y = f (x) is asymptotic to the parabola y = x2. Use this fact to help sketch the graph of ƒ.
1 2 3
33% 33%33%1.
2.
3.
3 2( )
xf x
x
2lim ( ) 0
xf x x
Find the point on the line y = 4x +8 that is closest to the origin.
17
10,
17
32
1.
17
9,
17
342.
17
8,
17
323.
17
8,2
4.
17
8,
17
315.
Find the point on the line y = 4x +8 that is closest to the origin. 1 2 3 4 5
20% 20% 20%20%20%
17
10,
17
32
1.
17
9,
17
342.
17
8,
17
323.
17
8,2
4.
17
8,
17
315.
2 2
2 2
2 2 2 2
( , 4 8) & (0,0)
( ) ( 0) (4 8)
( ) (4 8)
2 2(4 8)(4) 17 32( ) 0
2 (4 8) 2 (4 8)
32
1732 8
( )17 17
x x
d x x x
d x x x
x x xd x
x x x x
x
y
1. x
2. x
3. x
4. x
5. x
Find f . 22412)( xxxf
DCxxxxf 43 46)(
DCxxxxf 43 22)(
DCxxxxf 43 84)(
DCxxxxf 432)(
DCxxxxf 43 4)(
1. x
2. x
3. x
4. x
5. x
Find f .1 2 3 4 5
20% 20% 20%20%20%
22412)( xxxf
DCxxxxf 43 46)(
DCxxxxf 43 22)(
DCxxxxf 43 84)(
DCxxxxf 432)(
DCxxxxf 43 4)(
2
2 3 2 3
3 4
( ) 12 24
12 24( ) 6 8
2 36 8
( )3 4
f x x x
f x x x C x x C
f x x x Cx D
Evaluate the limit.1 2 3 4 5
20% 20% 20%20%20%
2x 0
1 coslim
x
x x
1. -5
2. 0
3. π
4. 1
5. ∞
2x 0
x 0
1 cos 0lim
0sin 0
lim 02 1 1
x
x xx
x
Sketch the curve.1 2 3
33% 33%33%
1.
2
3 6
xy
x
2
2 2
2 2
3 6
2 (3 6) 3 3 120
(3 6) (3 6)
0, 4, 2
xy
x
x x x x xy
x x
x x x
Find the inflection points for the function.
1. x
2. x
3. x
4. x
5. x
30,sin256)( xxxxf
10,2,65,
610,2,5,
610,2,65,
10,2,5,
610,2,6,
Find the inflection points for the function.
1 2 3 4 5
20% 20% 20%20%20%
1. x
2. x
3. x
4. x
5. x
30,sin256)( xxxxf
10,2,65,
610,2,5,
610,2,65,
10,2,5,
610,2,6,
( ) 6 5 2sin
( ) 5 2cos
( ) 2sin 0
, 2
f x x x
f x x
f x x
x n
x x
A particle is moving with the given data. Find the position of the particle.
1. x
2. x
3. x
4. x
5. x
( ) sin cos , (0) 0v t t t s
( ) cos sins t t t
( ) 1 sins t t t
2( ) 1 coss t t
( ) 1 cos sins t t t
( ) 1 cos sins t t t
A particle is moving with the given data. Find the position of the particle.
1 2 3 4 5
20% 20% 20%20%20%
1. x
2. x
3. x
4. x
5. x
( ) sin cos , (0) 0v t t t s
( ) cos sins t t t
( ) 1 sins t t t
2( ) 1 coss t t
( ) 1 cos sins t t t
( ) 1 cos sins t t t
( ) sin cos , (0) 0
( ) sin cos cos sin
0 cos 0 sin 0
1
( ) cos sin 1
v t t t s
s t t tdt t t c
c
c
s t t t
Use the linear approximation of the function
at a = 0 to approximate the number .
1. 3.02
2. 0.15
3. 7.44
4. 7.4
5. 2.25
xxf 9)(09.9