Find all the critical numbers of the function.

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. 7

2. 15

3. 0

4. 5

5. -6

10

sin7

y

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.

ln 2 ln 3ln 3 ln 2

t

tt

t

23lim

0

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. xxy 33 3 1.

2.

3.

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

Sketch the curve.1

x

xy

1.

2.

3.

Sketch the curve.1 2 3

33% 33%33%

1

x

xy

3.

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

Sketch the curve. ,

1.

2.

3.

xxy 2tan22

x

Sketch the curve. ,1 2 3

33% 33%33%3 .

xxy 2tan22

x

Evaluate the limit.

1. ½

2. 2

3. 0

4. ∞

5. -2

x 1

1lim

1 ln

x

x x

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. 5

2. 6

3. 7

4. 0

5. 1

236 xy

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. 5, 0

2. -2

3. 20

4. 0

5. -1

xxy 205 2

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.

3/22xxy

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.

2x 0

1 coslim

x

x x

1. -5

2. 0

3. π

4. 1

5. ∞

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.

63

2

x

xy

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

Use the linear approximation of the function

at a = 0 to approximate the number .

1 2 3 4 5

20% 20% 20%20%20%

1. 3.02

2. 0.15

3. 7.44

4. 7.4

5. 2.25

xxf 9)(09.9

0.09

1 1 1( ) , (0)

62 9 ) 2 9 0)

( 0.09) (0) (0)

1( 0.09) 3 ( )( 0.09) 3.015

6

dx

f x fx

f f f dx

f