MATH 135 – Sample Final Exam Review

MATH 135 – Sample Final Exam Review
Updated Spring 2012
This review is a collection of sample questions used by instructors of this course at
Missouri State University. It contains a sampling of problems representing the material
covered throughout the semester and may not contain every type of question on the final exam.
Any material listed on the lecture schedule and/or the assignment sheet may be on the final
exam. Please also be aware that a few questions on the final exam, while requiring knowledge
and understanding of the content covered in the course, may be presented in a form different
than the problems in the text.
Short Answer Section
1) True or False:
(a + b)
2) True or False:
4+ 3
= 2+ 3
2
2
= a 2 + b2
3) True or False: e x ⋅ e x = e 2 x
4) True or False: e x + e x = e 2 x
5) True or False: log 2 x + log 2 5 = log 2 (5 x )
6) True or False: ln x − ln 6 =
ln x
ln 6
7) What is the relationship of the lines y = 3x – 2 and y = 3x + 5?
8) What do you know about the lines y =
2
3
x and y = − x + 1?
3
2
9) Can a function have 2 y-intercepts? Explain.
10) Give the y-intercept for y = x 2 − x − 6 .
11) Give the x-intercept for y = x 2 − x − 6 .
12) Give the y-intercept for f ( x) = log 2 ( x + 2)
13) Give the x-intercept for f ( x) = log 2 ( x + 2)
14) Can a parabola of the form y = ax 2 + bx + c have an inverse function? Why or why not?
15) If f (2) = 5, f −1 (5) = _____
16) If the point ( 3, −10 ) lies on the graph of f ( x ) , give a point on f −1 ( x) .
17) If f ( x ) has domain {2,3, 4,5} and range {4,9,16, 25} , what is the domain of f −1 ( x) ?
18) Is x = 10 in the domain of f ( x) = 3 − x ? Explain.
19) What is the domain of y = x 2 − 4 ?
20) What is the range of y = x 2 − 4 ?
Use this graph to answer 21-23
y
21) Does the graph to the right represent a function?
22) Give the domain of the graph to the right.
23) Give the range of the graph to the right.
x
24) Sketch any one-to-one function.
25) Sketch any function that is not one-to-one.
26) How does the graph of y =| x − 3 | +5 compare to y =| x | ?
27) What is the minimum value of f ( x) = x 2 + 3? Explain.
x −8
cross the line x = 2? Explain.
x−2
2x −1
29) What are the asymptotes of y =
?
3x − 5
30) Does y = ax 2 + bx + c have a maximum or minimum value when a < 0 ?
28) Does the graph of y =
(
)
31) Write a quadratic equation in standard form ax 2 + bx + c = 0 whose solution set is {−2, 7} .
32) Write log 4 a + log 4 b as a single logarithm.
33) Evaluate log b b
1

a
34) Evaluate log a 
35) Evaluate log b 1
36) What is the relationship of f ( x) = log 2 x and g ( x) = 2 x ?
37) Between what two consecutive integers will you find log2 10?
38) Does the model A(t ) = 90e −.01t represent exponential growth or decay? Explain.
39) Write an equation to represent the following statement: “y varies jointly with x and z.”
40) Write an equation to represent the following statement: “m varies inversely with n.”
41) The imaginary number i has the property i 2 = _____
42) For the complex number 3 + 4i the real part is _____ and the imaginary part is ______.
43) If a system of linear equations has no solution, then the lines are _______.
44) If a system of linear equations has one solution, then the lines _______.
45) If a system of linear equations has infinitely many solutions, then the lines are _______.
ଶ
46) The average rate of change of ݂ሺ‫ݔ‬ሻ = − ଷ ‫ ݔ‬+ 7 is
.
47) The average rate of change of ݃ሺ‫ݔ‬ሻ = ‫ ݔ‬ଶ − 2‫ ݔ‬+ 3 from 2 to 5 is
End of Short Answer Section.
.
Remaining problems require work to support answers.
Give the domain of each function.
x −1
x −5
1) f ( x) = 3x + 2
2) h( x ) =
5) f ( x) = 4 x 2 + 5
6) f ( x ) =
5x
x2 − 1
3) f ( x) = 3 − x
7) g ( x ) =
3x + 1
x2 + 4
4) f ( x ) = log( x − 8)
8) g ( x) = 5 x − 1
Evaluate.
1
81
9) log 5 125
10) log 9
13) ln e 2 x
14) log 7 3 7
11) log 8 32
12) log(.01)
15) log 3 36 − log 3 4
16) log 6 4 + log 6 9
Perform the indicated operation and write each expression in the form ࢇ ± ࢈࢏.
17) (14 + 2i ) + (1 − 4i )
18) 8i − (5 − 9i )
20) (2 + 7i )(2 − 7i )
2
21) (5 − 4i )
19) −3i(7i − 4)
Solve each equation. Leave all answers exact. DO NOT ROUND.
( x + 3)
2
22) 22 x+1 = 8
23)
25) x − 13 x + 40 = 0
26) ( 3 − 8 p ) = −10 ( 3 − 8 p ) − 21
27) log 2 8 − log 5 x = log 5 4
29) x 2 − 4 x + 29 = 0
30) log 2 x = −3
2
= 25
2
24) x 2 = 2 x + 1
1
28) 2 y 5 − 5 y 5 − 3 = 0
31) 2(
5 −3 x )
=
1
16
32) log x 64 = 3
 8 
=3
 27 
33) log x 
Solve each equation. Leave all answers exact. DO NOT ROUND.
2
35) log 9 x = log 9 ( 7 x + 8 )
36) log 3 ( x + 1) − log 3 4 = 2
1
37)   = 17
 4
38) 7 x = 6 x+ 7
39) log x + log 6 = 2
40) 5e3 x −1 = 25
41) log x + log ( x − 3) = 1
34) log(5 x ) = log 4 + log( x − 3)
x
Solve each system of equations.
 x+ y =9
 2 x − 3 y = −2
2 x − 3 y = 5
5x + 4 y = 1
3 x − 12 y = 6
 2x − 8 y = 4
42) 
43) 
44) 
 2x + 6 y = 7
45) 
3 x + 9 y = 10
 x 2 − 2 y = 10
46) 
 3x − y = 9
 x2 + y2 = 9
47)  2
 x − y = −3
x + y +1 = 0

48)  2
2
 x + y + 6 y − x = −5
 x 2 − 3 x + y 2 + y = −2
 2
49)  x − x
+ y +1 = 0

y

 x2 + y2 = 6 y
50) 
2
 x = 3y
Solve each inequality, graph its solution on a number line, then leave the solution in interval notation.
51) 3x − 5 + 9 > 2
52) 3 x − 2 + 1 ≤ 4
53) x 2 + 3 x ≤ 0
Solve each inequality, graph its solution on a number line, then leave the solution in interval notation.
54) x 2 − 7 x > −10
55)
3x
≥4
x−2
56)
3x + 1
>0
x−7
Find the linear equation to represent each of the following conditions.
57) Write the equation of the line through (2, 4) with slope of −
4
.
7
58) Write the equation of the line through the points (3,-7) and (8,-4).
59) Write the equation of the line through (0,1) and perpendicular to 2 x − 3 y = 5 .
60) Write the equation of a line through (3,1) and parallel to 2 x − 3 y = 5 .
Find the linear equation to represent each of the following conditions.
61) Write the equation of a line parallel to y + 5 = 2 and passing through the point (-4,6).
62) Write the equation of a line perpendicular to y = 8 and through (-1,-5).
63) Write the equation of a line parallel to y = 8 and through (-1,-5).
Find each of the following given
f ( x) = x + 1 ;
2
 3 x if

h( x) =  x − 2 if
 5
if

g ( x) = 3x − 5 ;
64) h(12)
65) h(0)
66) g (t + 1)
69) ( f + g )( x )
70) ( f − g )( x )
71)
( g o f ) (2)
67) f (3 x )
72)
( f o g ) ( x)
x<0
0 ≤ x ≤ 10
x > 10
68) ( f + h)( −1)
73) g −1 (2)
Find the inverse for each one-to-one function.
74) f ( x) = 2 x − 8
75) f ( x) =
2x +1
3
76) f ( x ) =
3x − 2
2x + 1
77) f ( x) =
5
x−3
Write the equation that relates the quantities given. DO NOT SOLVE.
78) Express the area, A, of a rectangle as a function of the width, x, if the length is twice the width of the rectangle.
79) A commissioned sales person earns $100 base pay plus $10 per item sold. Express her gross salary, G,
as a function of the number of items sold, x.
80) The illumination, I, produced on a surface by a source of light varies directly with the candlepower, c, of the source
and inversely with the square of the distance, d, between the source and the surface.
Solve each problem. Remember to identify variables and give the equation used to solve each problem
81) The difference between the squares of two numbers is 3. Twice the square of the first number increased by the
square of the second number is 9. Find the numbers.
82) The product of two numbers is 10 and the difference of their squares is 21. Find the numbers.
83) The illumination provided by a car’s headlight varies inversely with the square of the distance from the headlight. A
car’s headlight produces an illumination of 3.75 footcandles at a distance of 40 feet. What is the illumination when
the distance is 50 feet?
84) The electrical resistance R of a wire varies directly with its length l and inversely with the square of its diameter d. A
wire 100 feet long of diameter 0.01 inch has a resistance of 25 ohms. Find the resistance of a wire made of the same
material that has a diameter of 0.015 inch and is 50 feet long.
Solve each problem. Remember to identify variables and give the equation used to solve each problem
85) A pool measuring 10 meters by 20 meters is surrounded by a walkway of uniform width. If the area of the pool and
the walkway combined is 600 square meters, what is the width of the walkway?
86) Find the length and width of a rectangle whose perimeter is 36 feet and whose area is 77 square feet.
87) One pan pizza and two beef burritos provide 1980 calories. Two pan pizzas and one burrito provide 2670 calories.
Find the caloric content of each item.
88) A riverboat travels 46 km downstream in 2 hours. It travels 51 km upstream in 3 hours. Find the speed of the boat
in still water and find the speed of the stream.
89) A 1000-acre farm in Illinois is used to grow corn and soybeans. The cost per acre for raising corn is $65, and the cost
per acre for raising soybeans is $45. If $54,325 has been budgeted for costs and all the acreage is to be used, how
many acres should be allocated for soybeans?
Solve each problem. Remember to identify variables and give the equation used to solve each problem
90) Bronze which costs $9.10/kg is made by combining cooper which costs $8.90/kg with tin which costs $9.50/kg.
Find the number of kg of cooper and tin required to make 15.3 kg of bronze.
91) The net income, y, (given in millions of dollars) of Pet Products Unlimited from 2004 to 2006 is given by the equation
y = 9 x 2 + 15 x + 52 , where x represents the number of years after 2004. Assume this trend continues and predict
the year in which Pet Products Unlimited’s net income will be $598 million.
92) Suppose that the manufacturer of a gas clothes dryer has found that, when the unit price is p dollars, the revenue R
(in dollars) is R ( p ) = −4 p 2 + 4000 p . a) What unit price should be established for the dryer to maximize revenue?
b) What is the maximum revenue?
93) Since 1950, the growth in the world population in millions closely fits the exponential function A(t ) = 2600e0.018t ,
where t is the number of years since 1950. Estimate the population in the year 2018 to the nearest million.
Solve each problem. Remember to identify variables and give the equation used to solve each problem
94) The formula D = 6e −0.04 h can be used to find the number of milligrams D of a certain drug in a patient’s
bloodstream h hours after the drug has been given. When the number of milligrams reaches 2, the drug is given
again. What is the time (in hours) between injections?
95)
If Emery has $1800 to invest at 6% per year compounded monthly, how long will it be before he has $2700? If
the compounding is continuous, how long will it be? (round your answers to 3 decimal places)
96)
Cindy will require $20,000 in 4 years to return to college to get an MBA degree. How much money should she
ask her parents for now so that, if she invests it at 9% compounded continuously, she will have enough for
school?
97)
How long does it take $1700 to double if it is invested at 5% interest, compounded quarterly?
Solve each problem. Remember to identify variables and give the equation used to solve each problem
98)
The half-life of silicon-32 is 710 years. If 100 grams is present now, how much will be present in 600 years?
(Use the exponential growth/decay formula, round answer to 3 decimal places.)
99)
The logistic model ܲሺ‫ݐ‬ሻ =
ଵ଴଴଴
ଵାଷଶ.ଷଷ௘ షబ.రయవ೟
represents the population (in grams) of a bacterium after t hours.
Determine the carrying capacity of the environment and the initial population size. How long does it take for
the population to reach one-half the carrying capacity?
Graph each of the following functions. Label reference points and all asymptotes.
2x + 9
3
100) 5 y = 10 − 3 x
101) y =
102) y = −2 x 2 + 5
103) f ( x ) = −
5
2 − 3x
104) f ( x ) = e − x − 1
106) f ( x ) = ( x − 2 )
3
105) y = 3 x 2 + 6 x + 6
( x + 3) ( x − 4 )
2
107) f ( x ) = 2 − x 3
108)
f ( x) =
5x
( x − 1)( x + 4 )
Graph each of the following functions. Label reference points and all asymptotes.
109) f ( x) = log 3 ( x − 2)
110) f ( x) = 2 x +1
111) f ( x) = log 2 ( x ) + 3
113)
( x + 3)( x + 1)
f ( x) =
x 2 + 8 x + 16
112) y = 2 − ln ( x + 4 )
2
 x if x ≤ 2
f
(
x
)
=
3
114)
 4 if x > 2
Describe the transformation and sketch each of the following, given the graph of f(x) to the right.
y
115) f −1 ( x )
x
116) y = 2 f ( x ) − 3
117) y = − f ( x + 5)
118) g ( x) = f ( x − 2) + 1
Solutions
32) log 4 ( ab )
SHORT ANSWER SECTION
33)
1)
2)
3)
4)
5)
6)
7)
8)
9)
False
False
True
False
True
False
They are parallel
They are perpendicular
No, explanation
required
10) ( 0, −6 )
11) ( 3, 0 ) and ( −2, 0 )
12) ( 0,1)
13) ( −1, 0 )
14) No, explanation
required
15) f −1 (5) = 2
16) ( −10,3)
17) {4,9,16, 25}
18) No, explanation
required
19) ( −∞, ∞ )
20) [ −4, ∞ )
21) Yes
22) ( −∞, ∞ )
23) ( 0, ∞ )
24) Answers will vary
25) Answers will vary
26) It is shifted up 5 units
and to the right 3 units
27) Min = 3,
explanation required
28) No, explanation
required
2
5
29) HA: y = ; VA: x =
3
3
30) Maximum
31) x 2 − 5 x − 14 = 0
1
2
34) −1
35) 0
36) They are inverse
functions
37) 3 and 4
38) Decay,
explanation required
39) ‫ݖݔ݇ = ݕ‬
40) ݉ =
௞
௡
41) -1
42) 3,4
43) Parallel or Inconsistent
44) Intersect or are
Consistent
45) Are the same line or
Coincident
46) −
47) 5
ଶ
ଷ
REMAINDER OF THE REVIEW
REQUIRES WORK TO
SUPPORT ANSWERS
 2 
1)  − , ∞ 
 3 
2) All real numbers except
5
3)
4)
5)
( −∞,3]
(8,∞ )
( −∞, ∞ )
31) x = 3
32) x = 4
2
33) x =
3
34) ∅
35) x = −1, 8
36) x = 35
ln17
log17
37) x =
=
ln(.25) log(.25)
7 ln 6
7 ln 6
=
ln 7 − ln 6
7
ln  
6
7 log 6
7 log 6
=
=
log 7 − log 6
7
log  
6
38) x =
6) All reals except ±1
7)
8)
( −∞, ∞ )
( −∞, ∞ )
9) 3
10) −2
5
11)
3
12) −2
13) 2 x
1
14)
3
15) 2
16) 2
17) 15 − 2݅
18) −5 + 17݅
19) 21 + 12݅
20) 53
21) 9 − 40݅
22) x = 1
23) x = −8, 2
24) x = 1 ± 2
25) x = 25, 64
26)
27)
28)
29)
30)
3 5
x= ,
4 4
125
x=
= 31.25
4
1
y = − , 243
32
x = 2 ± 5i
1
x=
8
50
3
1 + ln 5
40) x =
3
41) x = 5
42) ሺ5,4ሻ
43) ሺ1, −1ሻ
44) Infinitely many
solutions
45) No solution
46) ሺ2, −3ሻ, ሺ4,3ሻ
47) ሺ0,3ሻ
39) x =
ହ
4
36
57) y = − x +
7
7
3
44
58) y = x −
5
5
3
59) y = − x + 1
2
2
60) y = x − 1
3
61) y = 6
62) x = −1
63) y = −5
64)
65)
66)
67)
68)
69)
70)
71)
72)
5
−2
3t − 2
9 x2 + 1
−1
x 2 + 3x − 4
x 2 − 3x + 6
10
9 x 2 − 30 x + 26
7
73)
3
x +8 1
74) f −1 ( x ) =
= x+4
2
2
3
1 3x − 1
75) f −1 ( x ) = x − =
2
2
2
76) f −1 ( x ) =
଻
48) ሺ0, −1ሻ, ቀ , − ቁ
ଶ
ଶ
−x − 2 x + 2
=
2x − 3 3 − 2x
49) ሺ1, −1ሻ
50) ሺ0,0ሻ, ሺ−3,3),(3,3)
77) f −1 ( x) = x 5 + 3
51)
79) G ( x ) = 100 + 10 x
( −∞, ∞ )
 1 5
52)  − , 
 3 3
[ −3, 0]
54) ( −∞, 2 ) ∪ ( 5, ∞ )
55) ( 2,8]
53)
1

56)  −∞, −  ∪ ( 7, ∞ )
3

78) A( x) = 2 x 2
80) ‫= ܫ‬
௞௖
ௗమ
81) 2 and 1, 2 and -1,
-2 and 1, -2 and -1
82) 5 and 2, -5 and -2
83) 2.4 footcandles
84)
ହ଴
ଽ
ohms
85) 5 meters
86) 11 ft and 7ft
87) Pan pizza 1120 calories
Beef burrito 430 calories
88) Boat 20 km/hr
Stream 3 km/hr
89) 533.75 acres
90) Copper 10.2 kg
Tin 5.1 kg
100)
91) 7 years;
in the year 2011
92) a) $500 b) $1,000,000
93) 8842 million
94) About 27.5 hours
95) Monthly about 6.775
yrs Continuously ≈
6.758 yrs
101)
y
About $13,954
Almost 14 years
About 55.668 grams
1000g; About 30g;
About 7.9 hours
96)
97)
98)
99)
102)
y
4
y
5
3
3
2
4
1
2
3
x
−2
−1
1
2
3
4
5
1
2
−1
x
1
−5
−2
−4
−3
−2
−1
1
x
−1
−3
−3
−2
−1
1
2
3
−1
−2
y
y
7
5
5
y
4
103)
6
4
104)
3
105)
5
2
3
4
1
2
x
−6
−5
−4
−3
−2
−1
1
2
3
4
5
3
6
−1
1
2
−2
x
−2
−1
1
2
3
4
5
1
−3
x
−1
−4
−5
−4
−3
−2
−1
1
2
3
−5
−2
9
y
8
7
y
5
6
5
106)
4
107)
4
108)
3
3
2
1
x
2
−9
−8
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
7
−1
−2
1
−3
x
−4
−4
−3
−2
−1
1
2
3
4
−5
−6
−1
−7
−8
−2
−9
y
y
3
y
5
5
2
4
4
3
1
109)
110)
x
1
2
3
4
5
6
3
7
111)
2
2
1
−1
1
x
−2
−2
−1
1
x
−6
−5
−4
−3
−2
−1
1
2
−1
−3
−1
−4
−2
−2
2
3
4
5
6
7
8
9
10
15
y
y
14
13
y
12
7
11
112)
113)
6
114)
10
9
8
5
7
6
4
5
4
3
x
3
2
1
2
x
−14
−13
−12
−11
−10
−9
−8
−7
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
−1
1
−2
x
−4
−3
−2
−1
1
2
3
−3
4
−4
−1
115)
Reflection over the line ‫ݔ = ݕ‬
116)
y
Vertical stretch by a factor of 2, Vertical shift
down 3 units. y
x
x
f −1 ( x )
117)
y = 2 f ( x) − 3
Horizontal shift left 5 units, Reflection on the x-axis.
y
118)
Horizontal shift right 2 units, Vertical shift up 1
y
unit.
x
x
y = − f ( x + 5)
g ( x ) = f ( x − 2) + 1