pre-calculus final exam review · 2016-06-01 · final exam review answer key 8 6 4 2 2 4 6 8 15 10...

Final Exam Review Answer Key

8

6

4

2

2

4

6

8

15 10 5 5 10 15

f x( ) = 2∙ 3 x + 1

8

6

4

2

2

4

6

8

15 10 5 5 10 15

f x( ) = 2∙x 10

x2 2∙x 15

8

6

4

2

2

4

6

8

15 10 5 5 10 15

f x( ) = x2 + x 2

4∙x2 16

Pre-Calculus Final Exam Review Answer Key 2015 – 2016

CHAPTER 1

1. ( , 2) ( 2,2) (2, ) 2. [ 5,5] 3. [2,4) (4, ]

4. 1 3

3( )

4

xf x

D: ( , ) 5. 1 2( ) 10f x x D: [0, )

6. 1 8 3( )

7

xf x

D: ( , )

7. parent: ( )f x x 8. parent: 3( )f x x 9. 𝑓(−5) = −6 , 𝑓(3) = 2 , 𝑓(4) = 10

CHAPTER 2

1. left end: ( ) as f x x 2. left end: ( ) as f x x

right end: ( ) as f x x right end: ( ) as f x x

3. 2 2( ) ( 1) ( 3)f x x x 4.

3 2( ) ( 1) ( 2)f x x x

5. A. B. ±1

2, ±1, ±2, ±4, ±8, ±16 C. 𝑥 = −4, 𝑥 = −1, 𝑥 = 1, 𝑥 = 2

+ – I

2

0

2

0

4

2

0

6. VA: 2x 7. VA: 3x

HA: 1

4y HA: 0y

Hole: 3

2,16

HOLE: 1

5,4

x-int: (1,0)

𝑓(𝑥) = 2(𝑥 + 4)(𝑥 + 1)(𝑥 − 1)(𝑥 − 2)

Linear factorization:


8

6

4

2

2

4

6

8

15 10 5 5 10 15

f x( ) = 3x + 2 1

8

6

4

2

2

4

6

8

15 10 5 5 10 15

f x( ) = ex 3 + 2

8

6

4

2

2

4

6

8

15 10 5 5 10 15

f x( ) = 2 + ln x 2( )

ln 3( )

8

6

4

2

2

4

6

8

15 10 5 5 10 15

f x( ) = 1 ln x + 2( )

CHAPTER 3

1. 2 2. 1 3. 1

2

4. 3 5. not possible 6. 1

4

7. -2 8. 0 9. 3

10. parent: ( ) 3xf x 11. parent: ( ) xf x e

Asym: 1y Asym: 2y

int : (0,8)y 3

1int : 0, 2 (0,2.05)y

e

horizontal shift: left 2 units horizontal shift: right 3 units

vertical shift: down 1 vertical shift up 2

D: ( , ) R: ( 1, ) D: ( , ) R: (2, )

12. 3( ) logf x x 13. ( ) lnf x x

Asym: 2x Asym: 2x

int :y none y-int: (0,.31)

horizontal shift: right 2 units horizontal shift: left 2 units

vertical shift: up 2 vertical shift up 1

D: (2, ) R: ( , ) D: ( 2, ) R: ( , )

14. 1

4log log log 4log2

a a a ay x w z 15. 1

ln ln( 2)2

x x 16. 233ln ln( 1)

2x x

17. 2

logb

z

y 18.

3ln

( 1)

x

x y 19.

123ln ( 1) ( 1)x x x

20. 2

2 2ln

( 1) ( 1)

x

x x 21. 0a


8

6

4

2

2

4

6

π 2π

3

π

3

π

3

2π

3

π 4π

3

5π

3

2π 7π

3

8π

3

3π 10π

3

f x( ) = 4∙sin x + π

2( )

22. 9

14x 23.

ln.5 3; 1.1534

2x x

24. 2ln 5 2ln 4

ln 4 3ln 5x

; 1.7407x 25. 0, ln 2x x 26. 2n

27. 60

7x 28. 4x 29. 3x ; { 1x is extraneous}

CHAPTER 4

1. third quadrant 2. second quadrant

3. C: 7

30

; S:

11

15

4. C: 33 ; S: 123

5. 171.53 inches/sec 6. 0 .7 radians per second

9.75 miles per hour

7. 85.03 cm 8. 2

2

9. 3 10. 2

11. 3

cos2

; 1

sin2

12. 17

sec8

13. cot 5 14. 345

19

15. 12

sin37

; 35

cos37

; 37

csc12

; 37

sec35

; 35

cot12

16. 0.36 radians 17. 3

18. 21

19.


8

6

4

2

2

4

6

π 2π

3

π

3

π

3

2π

3

π 4π

3

5π

3

2π 7π

3

8π

3

3π 10π

3

f x( ) = 0.5∙cos 2∙x π( ) 1

3.5

3

2.5

2

1.5

1

0.5

0.5

1

1.5

2

2.5

3

3.5

π 2π

3

π

3

π

3

2π

3

π 4π

3

5π

3

2π 7π

3

8π

3

3π 10π

3

f x( ) = 1

sin 0.5∙x + π

4( )

20.

21. Asymptotes: 6 3

x n

; range: ( , 3] [1, ) key pt.: ,13

key pt. ( )0, 3

22. Asymptotes: 3

22

x n

; range: ( , 1] [1, ) key pt.: , 12

key pt.

5,1

2


10

8

6

4

2

2

4

6

π 2π

3

π

3

π

3

2π

3

π 4π

3

5π

3

2π 7π

3

8π

3

3π 10π

3

q1 x( ) = 1 + 1

tan3∙x

4( )

8

6

4

2

2

4

6

8

2π 5π

3

4π

3

π 2π

3

π

3

π

3

2π

3

π 4π

3

5π

3

2π

t1 x( ) = 3∙tanx

3( )

23. Asymptotes: 4

3

nx

; Three Key points: ( )

2,2 ,1 , 1

3 3

24. Asymptotes: 3

32

x n

; Three Key points: ( )3 3

, 3 , 0,0 , ,34 4

25. 2sin 2 22

y x

26. 2cos(2 ) 2y x

27. 3tan 13

xy

28. 3cot 1

3 2

x

29. 9.19 feet 30. 1800; 1922.40b c angle of elevation = 20.56

31. 662.411 meters/second 32. 42.25N E

33. 27.76 feet using angle 22.9 34. 1999.92 feet


π

2π

3

π

3

π

3

2π

3

π

15 10 5 5 10 15

f x( ) = tan-1 x 2( )

5π

3

4π

3

π

2π

3

π

3

π

3

15 10 5 5 10 15

f x( ) = cos-1 x + 2( ) + π

4

π

2π

3

π

3

π

3

2π

3

π

15 10 5 5 10 15

q x( ) = sin-1x

2( )

35. 12.74 miles south 36. 3

4.89 miles east

37. 4

38. 0.7

39. 35 40. 2

41. 4

42.

7

24

43.

44. 45.


[cos sin sin cos ] [sin sin cos cos ]

[ 1 sin 0 cos ] [1 sin 0 cos ]

sin sin

cos( ) sin( ) 0

0

RHS

RHS

RHS

2

2 2

2

2sin 2sin cos cos

sin co

(sin

s 2sin cos

1

cos ) 1 sin 2

2sin cos

1 sin 2

x x x RHS

x x x x RHS

x x RHS

x x

x

x

2 2

2 2

(1 cos ) sin

sin (1 cos ) sin (1 cos )

1 2cos cos sin

sin (1 cos )

1 2cos 1

sin (1 cos )

2 2cos

sin (1 cos )

2(1 cos )

sin (1 cos )

2

sin

1 cos sin2csc

sin 1 cos

2csc

x xRHS

x x x x

x x xRHS

x x

xRHS

x x

xRHS

x x

xRHS

x x

RHSx

x

x xx

x x

2

2 3

2 3

3 3

3

3

sin(2 )

sin 2 cos cos 2 sin

2sin cos cos (1 2sin ) sin

2sin cos sin 2sin

2sin (1 sin ) sin 2sin

2sin

si

2sin sin 2sin

3sin

n(3 ) 3sin 4si

in

n

4s

RHS

RHS

RHS

RHS

RHS

RHS

2 2

2

2

2

2 2

2 2

3

cos (sin 1)

co

cos s

s (1 sin )

cos (cos )

co

cos

s

inx

x x

x x

x x

x

x

x

CHAPTER 4 – Part II

1. 50.1c 2. 19.87 or 2.42c c 3. 123 , 11.4, and 9.1A b c 4. a) 86.51 b) 4.54

5. 165.4ft 6. 15.762ft 7. 25.75m 8. 14.7 9. 41.4 , 74.2 , and 64.4A B C

10. a=7.06, B=16.52°, C=121.48° 11. a) 73.18un2 b) 256.64 un

2

Chapter 5

1. 2.

3. a)2sin 1 sec

1 cos sec

x x

x x

4.

( )

( )

2sin 1 cosRHS

1 cos 1 cos

x x

x x

2 2

2

sin sin cosRHS

1 cos

x x x

x

2 2

2

sin sin cosRHS

sin

x x x

x

2 2

2 2

sin sin cosRHS

sin sin

x x x

x x

cos

1 RHS1

x

1

1 RHSsec x

sec 1 1 sec

sec sec

x x

x x

5. 6.


2

2 2

2

2 2

2

sin (1 cot )

sin (cs

sin si c

c

n

)

1

otx

x

x

x

x

x

x

1

17

4

17

14

sin

cos

tan

x

x

x

17

1

17

4

csc

sec

cot 4

x

x

x

7. 8.

9. 6 2

4

10,

4 30

121 11.

120 119 120sin 2 ,cos2 , tan 2

169 169 119x x x

12. 1

tan2 3

x 13.

3 73sin

2 73

x 14.

2 3 2 3sin ,cos , tan 2 3

12 2 12 2 12

15. 10 10 3 39

56

16.

9

8 17.

3x n

18.

24

72

4

x n

x n

19.

5

8

7

8

x n

x n

20.

5 7 11, , ,

6 6 6 6x

21. x = 0 22. 2

3x

23.

13 25 11 23 35, , , , ,

18 18 18 18 18 18x

CHAPTER 6

1. 14 3 20

3 5 5

2.

11 3

6 4

5 17

3. 24 14

6 5

4. (-2, 1)

5. ( )3,2, 1 6. 121 13

,32 4

7. Inconsistent 8. 7 9. -11 10. -108

11. -4320 12. ( )2, 5,4 13. 4 4

,3 3

14. (−24, −8) 15. (1, −2, 4)


10

8

6

4

2

2

4

15 10 5 5 10 15

12

10

8

6

4

2

2

4

15 10 5 5 10 15

8

6

4

2

2

4

6

8

15 10 5 5 10 15

8

6

4

2

2

4

6

15 10 5 5 10 15

CHAPTER 7

1. horizontal parabola: ( )2

( 8) 6x y 2. circle : 2

2 3( 3) 16

2x y

Vertex: (-8,6) Focus: 31

,64

center: 3

3,2

radius: 4

Axis of Sym: 6y Directrix: 33

4x

3. horizontal ellipse: 22 5

2( )( 2)

116 9

yx 4. circle: 2 2 40x y

center: ( )5

22, Foci: ( )( )5 5

2 22 7, 2 7, center: (0,0) radius = 2 10

co-vertices: ( )( )11 1

2 22, 2, vertices: ( )( )5 5

2 22, 6,

eccentricity: 7

0.6614


20 15 10 5 5 10 15 20

10

8

6

4

2

2

4

6

8

10

12

14

20 15 10 5 5 10 15 20

2

2

4

6

8

10

12

14

16

18

20

22

w x( ) = 3∙ x 3( )2 5

20 15 10 5 5 10 15 20

8

6

4

2

2

4

6

8

10

12

14

16

20 15 10 5 5 10 15 20

6

4

2

2

4

6

8

10

12

14

16

18

5. horizontal hyperbola: 2 2

125 16

y x 6. vertical ellipse:

2 2( 3) ( 2)1

9 25

x y

center: (0,0) Foci ( )( )0, 41 0, 41 center: (3,-2) Foci: (3,2) (3,-6)

vertices: (0,-5) (0,5) 5

4y x vertices: (3,3) (3,-7) co-vertices (6,-2) (0,-2)

eccentricity : 4

.85

7. vertical parabola: ( ) 215 ( 3)

3y x ` 8. horizontal hyperbola:

2 2( 1) ( 4)1

16 1

x y

vertex: (3,-5) Focus: 61

3,12

center: (1,-4) vertices: (5,-4) (-3,-4)

Foci: ( (1 17, 4)(1 17, 4)

directrix: 59

12y axis of sym: 3x asymptotes:

1 17

4 4y x

1 15

4 4y x

9. 2 2( 5) 89x y 10.

2 2( 9) ( 5) 98x y

11. 2 2( 5) 25x y 12. 21

2( 3) ( 7)y x

13. 23( 4)x y 14.

2 2( 7) ( 1)1

4 100

x y

15. 2 2( 5) ( 7)

1100 9

x y 16.

2 2( 4) ( 2)1

110 80

y x 17.

2 2( 3) ( 6)1

25 64

x y


CHAPTER 10

10.1: 1. 4 2. – 5 3. Divergent 4. Convergent 5. 280 6. 417.375

7. – 6 8. 0 9. 𝑎1 = −4, 𝑎𝑛 = 𝑎𝑛−1 + 2𝑛 − 1, 𝑛 ≥ 2 ; 𝑎𝑛 = 𝑛2 − 5

10. 𝑎1 =1

2, 𝑎𝑛 = 3𝑎𝑛−1, 𝑛 ≥ 2 ; 𝑎𝑛 =

1

2(3)𝑛−1

10.2: 1. 𝑑 = 22 ; −17, 5, 27, 49. 2. Explicit: 𝑎𝑛 = 39 − 14𝑛 ; Recursive: 𝑎𝑛 = 𝑎𝑛−1 − 14 , 𝑎1 = 25, 𝑛 ≥ 2

3. 𝑛 = 11 4. −5203 5. 10

10.3: 1. −32, −16, −8, −4, −2 𝑂𝑅 − 32, 16, −8, 4, −2 2. 186 3. 2

9

4. 𝑟 = 2; 16𝑥 − 24, 32𝑥 − 48, 64𝑥 − 96 5. 5

1

1

1( 2)

2

n

n

10.5: 1. 286,720 2. – 320 3. 240 + 1372𝑔 + 294𝑔2 + 28𝑔3 + 𝑔4

4. 64𝑝6 − 576𝑝5𝑞 + 2160𝑝4𝑞2 − 4320𝑝3𝑞3 + 4860𝑝2𝑞4 − 2916𝑝𝑞5 + 729𝑞6 5. 16.8%

pre-calculus final exam review · 2016-06-01 · final exam review answer key 8 6 4 2 2 4 6 8 15 10...

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