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, 1.6, 2.1, 2.2, 2.3, 4.1, and 4.2
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?
6
5
4
c) What is the dependent
variable?
3
d) What is the value of f
( 6 )? f(-2)?
e) For what values of x is
f (x) = 2
y
2
1
-2
x
-1
f) What is the domain of
the function?
g) What is the range of
the function?
2
1
3
4
5
6
-1
-2
2. Do the tables represent functions? How do you know?
a)
b)
x 5 4 2 1 0
y 2 6 8 9 6
x 3 5 7 3 5
y 2 6 8 9 6
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?
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?
5
Number of companies
(thousands)
b) What is the n-intercept of the linear model and
what does it mean?
n
Number of Companies on the Nasdaq Stock Market
between 1990 and 1999
4
3
2
1
t
0
2
4
6
8
Years since 1990
10
12
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 f (x) = 2x 2 + 4 , find the following.
a) f (−3)
c) f (5.2)
f (0)
b)
Module II – Sections 4.3, 4.4, 4.5, 5.3, 5.4, 5.5, and 5.7
7. Simplify each of the following and write without negative exponents.
⎛ y3 ⎞
a) ⎜
⎟
⎜⎝ 4 ⎟⎠
−2
b)
6x 2 y −3
x −1 y 4
(
−2
5
2
c) 5x 2x + x
)
d)
10
p −4
8. Simplify each expression using the laws of exponents. Write the answers with positive exponents.
a)
(−5x )(3x )
2
3
9. Let f (x) =
4
3
3
4
b)
⎛ m2 ⎞
c) ⎜ 3 ⎟
⎝ t ⎠
4x
5x
− 53
d)
(m n )
6 4
1
2
1 x
(4)
2
a) What is the y-intercept of the graph of f?
c) Find f(-2)
b) Does f represent growth or decay?
e) Find x when f (x) = 32
d) Find f(2)
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) log 6 (36)
b) ln(e12 )
13. Rewrite the log equations in exponential form.
a) log b t = k
14. Rewrite the exponential equations in log form.
a) p t = q
b) ln p = m
c) e p = t
b) 10 x = y
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 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?
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
Year
1910
1920
1930
1940
1950
1960
Population
36000
17000
10050
4500
2100
1100
Module III – Sections 7.2, 7.3, 7.5, 7.7, and 7.8
19. Given the graph of the equation: y = 5x 2 − 3x − 2
a) Which does the graph have, a maximum or a minimum?
b) Calculate the coordinates of the vertex by hand and using the Minimum feature on a calculator.
c) What is the y-intercept of the graph?
d) What are the x-intercepts of the graph?
17
49
20. Simplify the radical expressions:
a)
18
b)
21. Solve each of the equations:
a)
( x − 4) 2 = 6
b) (x + 2)2 = −3
c) x − 7x = −12
2
d) x − 6x + 9 = 0
2
c)
−25
e) −x − 4 = 2x
2
22. 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) = −16t 2 + 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?
23. 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.
e) In what years does model breakdown occur?
Year
1950
1960
1970
1980
1990
2000
Population
(thousands)
130
176
215
245
264
275
Module IV – Sections 8.7, 10.1 and 10.2
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.
25. Using an notation, find a formula of each sequence.
a) −7, −11, −15, −19, −23,...
b) −7, −14, −28, −56, −112,...
26. Find the 21st term of the sequence: 67, 72, 77, 82, 87,...
27. Find the term number n of the last term of the finite sequence: 1, 6, 11, 16, 21, ... 471
28. Find the 67th term of the sequence. Write your answer in scientific notation if necessary.
5, 15, 45, 135, 405,...
29. −2,470,629 is a term of the sequence; −3, −21, −147, −1029, −7203,...
Find the term number of that term.
30. 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,...
Dimensional Analysis
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.
31. a) 253qt / hr to qt / sec
2
32. a) 22ton / ft to kg / m
b) 32 yd / hr to in. / sec
2
b) 38m3 / sec to cm3 / min
Solutions:
1. a) It passes the vertical line test.
b) x
g) −1 ≤ y ≤ 4
f) −2 ≤ x ≤ 6
e) x = −1 , x = 4
2. a) Yes. Each x-value corresponds to one y-value.
3. a) 1993
d) f (6) = 1 , f (−2) = −1
c) y
b) No. x = 3 corresponds to two different y-values.
b) 5. There were 5000 companies on the NASDAQ Stock Market in 1990.
c) 10. According to the model, zero companies were on the NASDAQ Stock Market in 2000.
4. a) y = −x + 8
b) y = 3x + 6
6. a) f (−3) = 22
b) f (0) = 4
7. a)
16
y6
8. a) −15x
9. a)
b)
2
1
2
b)
6x 3
y7
5x
c) f (5.2) = 58.08
c) 10 x 3 + 5
4
1
5. y = 5.5x − 7
c)
4
b) growth
10. y = 7(0.659) x
c)
t
9
m
5
6
d) 10 p 4
d) m3 n 2
5
1
32
11. a) f (t) = 4000(1.06)t
12. a) 2
b) 12
13. a) b k = t
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
e) x = 3
d) 8
b) f (12) = 8048.79
b) e m = p
d) 2000
b) f (5) = 8505
16. a) f (t) = 35(3)t
c) t = 5.98
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) minimum
b) (0.3, −2.45)
20. a) 3 2
b)
c) (0, −2)
17
7
c) 5i
21. a) x = 4 ± 6 , ≈ 6.45, 1.55
22. a) 41ft
b) x = −2 ± i 3 , ≈ −2 ± 1.73i
e) −1± 1.73i
c) x = 3, 4
d) x = 3
b) 0 sec, 3.75 sec
c) 3.832 sec
23. a) quadratic
d) x = 1 , −0.4
b) f (t) = −0.0455x 2 + 5.1882x + 129.5357
c) 2007
d) 277318
e) before 1929 and after 2085
24. a) t = 4 p 2 , t = 64
25. a) an = −4n − 3
15
, M =5
r
b) an = −7(2)n −1
b) M =
26. a21 = 167
27. n = 95
28. a67 ≈ 1.545 × 10 32
29. n = 8
30. f (n) = −4n + 11
31. a)
32.
253qt
1hr
1min
qt
•
•
= .07
1hr 60 min 60sec
hr
b)
32 yd 3 ft 12in. 1hr
1min
in.
•
•
•
•
≈ 0.32
1hr 1yd 1 ft 60 min 60sec
sec
a)
22ton 2000lb
1kg
1 ft 2
39.37 2 in2
kg
•
•
•
•
≈ 214789.19 2
2
2
2
2
1ton
2.205lb 12 in
1 ft
1m
m
b)
3
38m3 1003 cm3 60sec
9 cm
•
•
=
2.28
×
10
1sec
1min
min
1m 3