Math 095 Final Exam Review - MLC Although this is a comprehensive review, you should also look over your old reviews from previous modules, the readings, and your notes. Round to the thousandth unless indicated otherwise. Module I – Sections 1.1, 1.2, 1.3, 1.4, 1.5, 2.1, and 2.2 1. Consider the graph of the function, f at the right. a) How can you tell that the graph represents a function? b) What is the independent variable? c) What is the dependent variable? d) What is the value of f ( 6 )? f(-2)? e) For what values of x is f (x) = 2 f) What is the domain of the function? g) What is the range of the function? 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 Nasdaq 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? 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.
Transcript
Math 095 Final Exam Review - MLC
Although this is a comprehensive review, you should also look over your old reviews from previous modules, the readings, and your notes. Round to the thousandth unless indicated otherwise. Module I – Sections 1.1, 1.2, 1.3, 1.4, 1.5, 2.1, and 2.2 1. Consider the graph of the function, f at the right.
a) How can you tell that the graph represents a function?
b) What is the independent variable?
c) What is the dependent variable?
d) What is the value of f ( 6 )? f(-2)?
e) For what values of x is f (x) = 2
f) What is the domain of the function?
g) What is the range of the function?
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 Nasdaq
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?
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.
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6. If f (x) = 2x2 + 4 , find the following. a) f (−3) b) f (0) c) f (5.2) 7. Simplify each of the following and write without negative exponents.
a)
y3
4
⎛
⎝⎜⎜
⎞
⎠⎟⎟
−2
b)
6x2y−3
x−1y4 c) 5x−2 2x5 + x2( ) d)
10p−4
8. Simplify each expression using the laws of exponents. Write the answers with positive exponents.
a) −5x
23( ) 3x
43( ) b)
4x34
5x c)
m2
t3
⎛⎝⎜
⎞⎠⎟
− 35
d)
m6n4( )12
Module II – Sections 2.3, 2.4, 2.5, 3.1, 3.2, and 3.3
9. Let f (x) = 1
2(4)x
a) What is the y-intercept of the graph of f? b) Does f represent growth or decay? c) Find f(-2) d) Find f(2) e) Find x when f (x) = 32
10. Find an approximate equation y = abx 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?
12. Find the value of each logarithm. a) log6(36) b) ln(e12 )
13. Rewrite the log equations in exponential form. a) logb t = k b) ln p = m
14. Rewrite the exponential equations in log form. a) pt = q b) 10x = y c) ep = t
15. Solve each of the equations. a) 3(4)x−2 = 15 b) 3log(x + 2) = 9 c) 5ln(x − 3) = 45 16. A population of 35 fruit flies triples every day. Let 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?
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 to two decimal places. 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.
18. Use the intersect feature on a graphing calculator to solve the equation. 3ln(x +5) = 5+ 2x
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Module III – Sections 4.1, 4.2, 4.3, 4.4, and 4.5 19. Find each of the products. Simplify the answers. a) 3x 2x2 + 5x − 4( ) b) (x + 3)(x − 7) c) (5x +1)(2x − 3)
20. Factor each of the expressions. a) x2 − 4x − 21 b) 10x2 −13x − 3 c) 16x2 − 49 21. Solve each of the equations by factoring. a) x2 − 21= 4x b) 10x2 = 13x + 3 22. Given the graph of the equation: y = 5x2 − 3x − 2
a) Calculate the vertex by hand. Show your work. b) What is the equation of the axis of symmetry? c) What is the y-intercept of the graph?
23. 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 h(t) = −16t2 + 60t +5 . 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?
24. Simplify the radical expressions: a) b)
1749
c) −25
25. Solve each of the equations: a) b) (x + 2)2 = −3
c) x2 − 7x = −12 d) x2 − 6x + 9 = 0 e) −x2 − 4 = 2x 26. 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. c) Predict the year that maximum population is reached. d) Predict the maximum population.
Module IV – Sections 5.1, 5.2, 5.3 and 5.4 27. 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. 28. Using an notation, find a formula of each sequence. a) −7, −11, −15, −19, −23,... b) −7, −14, −28, −56, −112,... 29. Find the 21st term of the sequence: 67, 72, 77, 82, 87,... 30. Find the term number n of the last term of the finite sequence: 1, 6, 11, 16, 21, ... 471
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31. Find the 67th term of the sequence. Write your answer in scientific notation if necessary. 5, 15, 45, 135, 405,...
32. −2,470,629 is a term of the sequence; −3, −21, −147, −1029, −7203,...
Find the term number of that term. 33. Find an equation of a function f such that f (1), f (2), f (3), f (4), f (5), ...
is the sequence 7, 3, −1, −5, −9,... For problems 34 and 35, use dimensional analysis to perform the following conversions. Show the procedure that you used, including all of your unit fractions. If an answer is not exact, round to two decimal places. 35. a) 253qt / hr to qt / sec b) 32yd / hr to in. / sec 35. a) 22ton / ft2 to kg / m2 b) 38m3 / sec to cm3 / min Solutions: 1. a) It passes the vertical line test. b) x c) y d) f (6) = 1, f (−2) = −1 e) x = −1, x = 4 f) −2 ≤ x ≤ 6 g) −1≤ y ≤ 4 2. a) Yes. Each x-value corresponds to one y-value. b) No. x = 3 corresponds to two different y-values. 3. a) 1993 b) (0,5). There were 5000 companies on the NASDAQ Stock Market in 1990. c) (10,0). According to the model, zero companies were on the NASDAQ Stock Market in 2000. d) 2000 4. a) y = −x +8 b) y = 3x + 6 5. y = 5.5t − 7 6. a) f (−3) = 22 b) f (0) = 4 c) f (5.2) = 58.08
7. a) b) c) 10x3 +5 d)
8. a) −15x2 b) c) d)
9. a)
0, 12
⎛⎝⎜
⎞⎠⎟
b) growth c) d) 8 e) x = 3
10. y = 7(0.659)x 11. a) f (t) = 4000(1.06)t b) f (12) = 8048.79
12. a) 2 b) 12 13. a) bk = t b) em = p
14. a)
log p (q) = t b) log( y) = x c) ln(t) = p
15. a) x ≈ 3.16 b) x = 998 c) x = 8106.08
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16. a) f (t) = 35(3)t b) f (5) = 8505 c) t = 5.98 days 17. a) p(t) = 36436.96(0.93)t b) 36436.96. The population of Smalltown was approximately 36437 people in 1910. c) 1985 or 86 18. x ≈ −0.1233 , −4.7815 19. a) 6x3 +15x2 −12x b) x2 − 4x − 21 c) 10x2 −13x − 3 20. a) (x + 3)(x − 7) b) (5x +1)(2x − 3) c) (4x − 7)(4x + 7)
21. a) x = −3, 7 b) x = − 1
5, 3
2 22. a) (0.3, − 2.45) b) x = 0.3 c) (0,− 2)
23. a) 41ft b) 0 sec, 3.75 sec c) 3.832 sec
24. a) b) c) 5i
25. a) x = 4 ± 6 , ≈ 6.45, 1.55 b) x = −2 ± i 3 , ≈ −2 ±1.73i c) x = 3, 4 d) x = 3 e) −1±1.73i 26. a) quadratic b) f (t) = −0.0455t2 +5.1882t +129.5357 c) 2007 d) 277318 e) before 1929 and after 2085
27. a) t = 4 p2 , t = 64 b) M =15r
, M = 5
28. a) an = −4n − 3 b) an = −7(2)n−1 29. a21 = 167 30. n = 95 31. a67 ≈ 1.545 ×10
