Math 095 Final Exam Review (updated 10/28/11)

This review is an attempt to provide a comprehensive review of our course concepts and problem types, but there is no guarantee the final will only include problems like in this review. This is a good starting point in your review for the final, but you should also study the textbook, your notes and homework. Module I – Sections 1.1, 1.6, 2.1, 2.2, 2.3

1. Consider the graph of the function f to the right.

a) How can you tell the graph represents a function?

b) What is the independent variable? c) What is the dependent variable? d) What is the value of

(6)f ? ( 2)f  ?

e) For what values of x is ( ) 2f x  ? f) What is the domain of the function? g) What is the range of the function?

- 2

- 1

2

6

5

4

3

1

654321- 1- 2

x

y

2. Do the tables represent functions? How do you know? a) b) 3. The graph at right represents a scattergram and a linear model for the number of companies on the NASDAQ1 stock

market between 1990 and 1999, where n represents the number of companies t years after 1990.

a) Using the linear model, in what year were there approximately 3500 companies?

b) What is the n-intercept of the linear model and what

does it mean?

c) What is the t-intercept and what does it mean? d) From the linear model, what would you predict the

number of companies to be in the year 1996?

x 3 5 7 3 5 y 2 6 8 9 6

x 5 4 2 1 0 y 2 6 8 9 6

Years since 1990 0 2 4 6 108 12

1

2

3

4

N um

b er

o f

co m

p an

ie s

(t h

o us

an ds

)

Number of Companies on the Nasdaq Stock Market between 1990 and 1999n

t

5

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4. Find a linear equation of the line that passes through the given pairs of points. a) (3, 5) and (7,1)

b) (4, 6) and (2, 0)

5. The average consumption of sugar in the U.S. increased from 26 pounds per person in 1986 to 136 pounds per person in 2006. Let p be the average number of pounds consumed t years after 1980. Find an equation of a linear model that describes the data.

6. If 2( ) 2 4f x x  , find the following. a) ( 3)f 

b) (0)f c) (5.2)f

Module 2 – Sections 4.1, 4.2, 4.3, 4.4, 4.5

7. Simplify each of the following and write without negative exponents.

a)

2 3

4

y 

       

b) 2 36 1 4

x y

x y





c)  252 25 xxx 

d) 4

10

p

8. Simplify each expression using the laws of exponents. Write the answers with positive exponents.

a)   2 43 35 3x x

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b)

3 44

5 x x

c)

3 52

3

m

t

      

d)   1 26 4m n

9. Let 1

( ) (4) 2

xf x 

a) What is the y-intercept of the graph of f ? b) Does f represent growth or decay? c) Find ( 2)f  d) Find (2)f e) Find x when ( ) 32f x 

10. Find an approximate equation xy ab of the exponential curve that contains the given set of points. (0, 7) and (3, 2).

11. Sue invested $4000 in an account that pays 6% interest compounded annually.

Let ( )f t represent the value of the account after t years. a) Write an equation for f. b) What is the account worth after 12 years?

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Module 3 – Sections 5.2, 5.3, 5.4, and 5.6

12. Find the value of each logarithm. a) 6log (36)

b) 12ln( )e 13. Rewrite the log equations in exponential form.

a) log ( )b t k b) ln( )p m

14. Rewrite the exponential equations in log form.

a) tp q

b) 10x y

c) pe t

15. Solve each of the equations. Round decimal answers to three places.

a) 23(4) 15x  b) 3 log( 2) 9x  

c) 2 3 45xe  

16. A population of 35 fruit flies triples every day. Let ( )f t be the number of flies after t days. a) Write an equation for the function, f, that models the fruit fly population growth. b) How many fruit flies are there after 5 days? c) How long will it take for the fruit fly population to reach 25000? Round decimal answers to two places.

17. The population of Smalltown decreased from 1910 to 1960, as shown in the table at the right.

Let ( )P t be the population of Smalltown t years after 1910. a) Use exponential regression to find an equation for P. Round decimal numbers to four places.

Year 1910 1920 1930 1940 1950 1960

Population 36000 17000 10050 4500 2100 1100

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b) What is the coefficient a in your model and what does it represent? c) Use your function to predict the year the population reaches 150.

Module 4 – Sections 7.2, 7.3, 7.5, 7.7, and 7.8

18. Given the graph of the equation: 25 3 2y x x   a) Which does the graph have, a maximum or a minimum? b) Calculate the coordinates of the vertex by hand and using the Maximum/Minimum feature on a calculator. c) What is the y-intercept of the graph? d) What are the x-intercepts of the graph?

19. Simplify the radical expressions:

a) 18

b) 17

49

c) 25

20. Solve each of the equations:

a) 2( 4) 6x  

b) 2( 2) 3x   

c) 2 7 12x x  

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d) 2 6 9 0x x  

e) 2 4 2x x  

21. A football player kicks a ball. The height of the ball, h(t) in feet, t seconds after it is kicked, is given by the equation 2( ) 16 60 5h t t t    .

a) What is the height of the ball after 3 seconds? b) At what time/s is the ball 5 feet off the ground c) How long does it take the ball to hit the ground? Round decimal answers to three places.

22. The population of Iceland (in thousands) from 1950 to 2000 is given in the table at the right.

a) What kind of equation fits the data best, quadratic or exponential? b) Use quadratic regression to find a model for the data where f(t) is the population t years

after 1950. Round decimal numbers to three places. c) Predict the year that maximum population is reached. d) To the nearest person, predict the maximum population. e) In what years does model breakdown occur?

Module 5 – Sections 8.5, 10,1, and 10.2 23. Translate the sentence into an equation. Use k for your constant of variation. P varies inversely as the square of r. 24. Write an equation, then find the requested value of the variable. a) If t varies directly as the square of p, and t = 36 when p = 3, find t when p = 4.

b) If M varies inversely as the square root of r, and M = 3 when r = 25, find M when r = 9.

Year 1950 1960 1970 1980 1990 2000

Population (thousands)

130 176 215 245 264 275

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25. Using na notation, find a formula of each sequence.

a. 3, 7, 11, 15, 19, ...     b. 20, 100, 500, 2500, 12500 26. Find the 29th term of the sequence: 42, 47, 52, 57, 62, ... 27. Find the term number n of the last term of the finite sequence: 7, 11, 15, 19, 23, ... 407 28. Find the 66th term of the sequence. Write your answer in scientific notation if necessary. 6, 18, 54, 162, 486,... 29. 100, 663, 296 is a term of the sequence 6, 24, 96, 384, 1536, ... Find the term number n of that term. 30. Find an equation of a function f such that (1), (2), (3), (4), (5), ...f f f f f is the sequence 4, 1, 2, 5, 8,   