calculators will not be allowed on this part of the final exam ......math 190 g final exam review 1...

Math 190 G Final Exam Review

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DIRECTIONS PART 1: • Calculators will not be allowed on this part of the final exam. • Follow all directions including any rounding instructions. • When appropriate, answer should include correct units. • When specified, you must show work to receive credit for your answers

1. Solve the following inequalities. Write your answers using interval notation.

a) 3 7 4 1x x

b) 2 1 6x

c) 5 4x

d)2

20

2 3

x

x x

2. Put the following equation in standard form by completing the square and state the

center and the radius:

2 2 6 8 10 0x y x y

3. Find the domains of the following functions. Write your answers in interval notation.

a) 2( ) 6 5f x x x

b) 2

4( )

13 30

xf x

x x

4. Find the inverse of the function2

( )1

xf x

x

.Find the domain of 1( )f x and the range

of ( )f x .

5. Find the complete factorization of , using the fact that

is a root of f (x).

6. Solve a) 2 22 8x x b) 2 1 19t te e c) 15 2x xe

7. Solve 2

3 3log 3log 14y y


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8. Graph the following functions. Identify any intercepts and asymptotes a. y = ex

b. y = ln(x)

9. Graph 2 periods of the following functions. Label key points and asymptotes a. y = cos x


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b. y = sin x

c. y = tan x.

10. If 4

sin5

, and P is in the third quadrant, find

) tan )sin 2 )cos2

a b c

11. Given the information, find the values of all the other trigonometric functions:


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a) 1

cot3

x , 3

2x

b) 2

sin5

x , 2

x

12. Find all the values of , with [0, 2π) for which 1

sin2

.

13. Solve the following for , with [0, 2π): cos(2 ) sin 0 .

14. Sketch the points on the unit circle corresponding to the following three angles and find

the values of the six trigonometric functions at each angle:

2 7 7) ) )

3 4 6a b c

15. Use a sum or difference identity or a half-angle formula to find the following:

a) cos12

b) 5

cos8

16. a. Explain why cos[cos-1(π)] is undefined whereas cos-1[cos(π)] is defined.

b. Explain why cos-1[cos(π)] = π whereas cos-1[cos(2π)] ≠ 2π.

17. Convert the following to rectangular coordinates:

a) (4, 5π/4)

b) (1/2, 2π/3)

18. Convert the following to polar coordinates:

a) (-2, -2)

b) (7, 0)

19. Find polar equations for

a) x² + y² = 1

b) 3x + 8y = 5


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DIRECTIONS PART 2: (Note: The focus is on the application of the concepts.) •Calculators will be allowed on this part of the final exam. •Follow all directions including any rounding instructions. •When appropriate, answer should include correct units. •When specified, you must show work to receive credit for your answers.

1. a. Find all asymptotes of 22 5 7

( ) .2

x xf x

x

then graph the function.

b. Find all asymptotes of 2

3( )

2 8

xf x

x x

then graph the function.


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2. The height above ground of a toy rocket launched upward from the top of a

building is giving by 2( ) 16 96 256S t t t

a) What is the height of the building?

b) What is the maximum height attained by the rocket? c) Find the time when the rocket strikes the ground.

3. A company is planning to manufacture wheelchairs. The cost, C, in dollars, of

producing x wheelchairs is C(x) = 500,000 + 400x.

a. Write the average cost function, .C

b. Find and interpret (1000)C and (10,000).C

c. What is the horizontal asymptote for the graph of ?C Describe what this

means for the company.

4. The function ( ) 13.4ln 11.6f x x= - models the temperature increase, f(x), in an

enclosed vehicle after x minutes when the outside air temperature is between

72F and 96F. Use the function to find the temperature increase, to the

nearest degree, after 30 minutes.

5. In 2000, the population of Africa was 807 million and by 2011 it had grown to

1052 million. Use the exponential growth model 0 ,ktA A e in which t is the

number of years after 2000, to find the exponential growth function that models

the data. By which year will Africa’s population reach 2000 million, or two

billion?

6. The number of minutes needed to solve an Exercise Set of variation problems

varies directly as the number of problems and inversely as the number of

people working to solve the problems. It takes 4 people 32 minutes to solve 16

problems. How many minutes will it take 8 people to solve 24 problems?


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7. Find the amplitude, period, vertical and horizontal shift for the following functions.

Sketch their graphs over one interval.

a) sin(2 ) 23

f x x

b) cos(3 )g x x

c. 2cos 1.y x


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8. The distance across a lake, a, is unknown. To find this distance, a surveyor

took the measurements shown in the figure. What is the distance across the

lake?

9. a. Use the Law of Sines to solve the following triangle:

45 , 60 , 6a

10 You hike 2.3 miles on a bearing of S 31 W. Then you turn 90 clockwise and

hike 3.5 miles on a bearing of N 59 W. At that time, what is your bearing, to

the nearest tenth of a degree, from your starting point?

11 Find the distance across the lake from A to C, to the nearest yard, using the

measurements shown in the figure.


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12 Two fire-lookout stations are 13 miles apart, with station B directly east of

station A. Both stations spot a fire. The bearing of the fire from station A is

N35E and the bearing of the fire from station B is N49W. How far, to the

nearest tenth of a mile, is the fire from station B?


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Math 190 Final Review Answer Key – Part 1

1. a.

7

8,

b. ]7,3[]1,5[

c. )9,1(

d. ,31,2

2. 35)4()3( 22 yx

Center: )4,3( ; radius: 35

3. a. ),5[]1,(

b. 10,3 x

4.

1

2)(1

x

xxf

Domain: 1x

Range: 1y

5. )1)(3)(43( xxx

6. a. 3,1x

b.

3

3ln2t

c

15ln

12ln

x

7. 9y

8. a. Horizontal Asymptote: 0y

y-intercept: )1,0(

Increasing curve through

e

1,1 , 1,0 , ),1( e

b. Vertical Asymptote: 0x

x-intercept: )0,1(

Increasing curve through

1,

1

e, )0,1( , )1,(e

9. a. Through: )1,0( ,

0,

2

, 1, ,

0,

2

3, )1,2( ,

0,

2

5, 1,3 ,

0,

2

7, 1,4

b. Through: 0,4,1,

2

7,0,3,1,

2,0,2,1,

2

3,0,,1,

2,0,0

c. Asymptotes:

2,

2,

2

xxx

Through:

1,

4

5,0,,1,

4

3,1,

4,0,0,1,

4


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10. a.

3

4tan

b.

25

242sin

c.

5

5

2cos

11. a.

3

1cot,10sec,

3

10csc,3tan,

10

10cos,

10

103sin

xxxxxx

b.

2

21cot,

21

215sec,

2

5csc,

21

212tan,

5

21cos,

5

2sin

xxxxxx

12. 330,210 or

6

7,

6

11

13. 330,210,90 or

2

,

6

7,

6

11

14. a.

3

3cot,2sec,

3

32csc,3tan,

2

1cos,

2

3sin

b. 1cot,2sec,2csc,1tan,

2

2cos,

2

2sin

c. 3cot,

3

32sec,2csc,

3

3tan,

2

3cos,

2

1sin

15 a.

2

32

12cos

or

4

62

12cos

b.

2

22

8

5cos

16 a. is not in the domain for .cos 1 x The domain is [-1,1].

b. The range for x1cos is ],0[ .

17. a. 22,22

b.

4

3,

4

1

18. a.

4

5,22

b. 0,7

19 a. 1r b.

sin8cos3

5

r


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Math 190 Final Review Answer Key – Part 2

1. a. Vertical Asymptote: 2x , Oblique Asymptote: 12 xy

b. Vertical Asymptotes: 2,4 xx , Horizontal Asymptote: 0y

2. a. 256 ft

b. 400 ft

c. 8t seconds

3. a.

x

xxC

400500000)(

b. 450$)10000(,900$)1000( CC

c. 400C , The average cost will approach, but never be below $400.

4. Temperature Increase of .34 F

5. teA 0241.807

38A years, so by 2038.

6. 24 minutes

7. a. Amplitude: 1 Period:

Phase Shift: 6

Vertical Shift: 2 Through:

2,

6

13,1,

12

23,2,

3

5,3,

12

17,2,

6,1,

12

11,2,

3

2,3,

12

5,2,

6

b. Amplitude: 1

Period: 3

2

Phase Shift: 3

No vertical shift.

Through:

1,

3

5,0,

2

3,1,

3

4,0,

6

7,1,,0,

6

5,1,

3

2,0,

2,1,

3

c. Amplitude: 2

Period: 2 Phase Shift: None Vertical Shift: 1

Through: 4,1,2

5,3,2,1,

2

3,1,,1,

2,3,0

8. 334 yards

9. a. 75,45,60

13,2,6 cba

10. b. WS 7.87

11. 193 yards

12. 10.7 miles

calculators will not be allowed on this part of the final exam ......math 190 g final exam review 1...

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