Precalculus
FINAL EXAMINATION REVIEW 2015
Unit F&G
FUNCTIONS AND THEIR GRAPHS
1) Determine the domain and range of the following functions. Identify whether the
odd, even or neither.
a)
f(x) = x3 – x
b)
f(x) = |x + 3| - 5
c)
h(x ) = 25 - x 2
d)
f (x ) =
10
x 2 + 2x
2)
Find the values of x such that f(x) = g(x), when f(x) = x2 + 2x + 1 and g(x) = 7x – 5.
3)
Find the domain of f(g(x)), where f(x) =
4)
Restrict the domain of the function f(x) = x2 + 6x + 2 so that its inverse exists.
5)
Write the equation of the inverse function, g(x), and verify that they are inverses
x + 3 and g(x) =
x
2
.
(f g ) (x ) = ( g f ) (x ) = x
a)
f(x) =
4
b)
x
f(x) =
h(x) =
16 - x 2
of each other,
x +3
x -2
6) Decide whether or not the each of the following functions represents a one-to-one
not, restrict the domains so that they do.
a)
function is
b)
g(x) = (x + 5)3
b)
t1 = -6
tn = 4 tn-1 - 2n
function. If
UNIT S&S SEQUENCES AND SERIES
1)
Write the first 5 terms of the given sequence.
a)
tn = 3n2 - 4
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Precalculus
2)
Identify if the given sequence is arithmetic, geometric, or neither. Write an explicit definition for
each.
a)
4, -5, -14, -23, …..
b)
5, -25, 125, -625, …..
c)
9, 16, 25, 36, …..
d)
27, 41, 55, 69, …..
f)
-2 2 -2 2
, ,
, , .....
3 5 7 9
e)
3)
4)
5)
5
5
,
,
5
,
5
32 48 72 108
, .....
Determine the indicated term using the appropriate formula.
a)
t48 for 19, 12, 5, …..
c)
t87 for an arithmetic sequence if t5 = 16 and t12 = 44
d)
t11 for a geometric sequence if t1 = 6 and t6 = 192
b)
t20 for 27, 9, 3, …..
Evaluate each of the following limits.
18n2 + 5
a)
lim (.798) n
b)
lim
c)
lim (7n + 3)
d)
lim tan
e)
1/3, -1/9, 1/27, -1/81, …
f)
1, -2, 4, -8, …
g)
lim
h)
æ
16 ö
lim ç 11 + ÷
n®¥ è
nø
n®¥
n®¥
n®¥
n
46
n®¥
n®¥
11 - 4n2 - 7n
1
n
Find the indicated sum using the appropriate formula.
a)
S47 for 20 + 27 + 34 + …
c)
S40 for arithmetic series where t1 = 5 and t3 = 11
d)
19 + 8 – 3 – 14 - ….. – 619
b)
e)
S12 for -4 + 12 - 36 + …
1 + 3 + 9 + 27 + ….. + 6561
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Precalculus
6)
7)
Determine if each of following infinite series converges or diverges. If the series converges, state
the sum.
1
1
a)
1+
c)
2 + 4 + 8 + 16 + …..
4
+
16
+ .....
b)
15
2
+
15
4
- .....
Determine the interval of convergence and the sum for
a)
1 + 2x + 4x + 8x + …..
c)
6 - 12 (x - 3) + 24 (x - 3)2 - …..
8)
2
3
b)
5x
4
+
5x2
8
+
5x3
16
+ .....
Evaluate each of the following.
8
a)
å k (4 - 7k)
5
b)
k=4
å (6 - 11n)
å 3m
2
- 17
m=2
85
c)
32
d)
n=12
9)
15 -
å (3k
2
- 8)(2k + 5)
k=1
Solve each of the following word problems.
a)
If you save $10 the first month, $14 the second month, $18 the third month, how much
money will you have saved after 5 years?
b)
Assuming each person has exactly 2 parents, how many relatives would you have if you were
to count back 12 generations? (Your parents are the first generation).
c)
How many 4-digit multiples of 17 are there?
d)
What is the sum of all three-digit numbers divisible by 24?
e)
Students started stacking newspapers. The first pile had 45 papers, the second had 68, and
the third had 91. The students kept stacking the papers until the last one had 482. How
many papers were there in the stacks?
f)
How would you rewrite 15.129129129….. as a rational number?
g)
A rubber ball is dropped from a height of 236 feet and rebounds three-fourths of the way.
Theoretically, how far will the ball travel before coming to rest?
h)
If an infinite series converges to 63 with a first term of 27, what is the common ratio?
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Precalculus
UNIT CS
CONIC SECTIONS
1)
Is (4, -8) on, inside, or outside the circle with equation (x - 2)2 + (y + 3)2 = 81?
2)
Identify the lengths of the minor and major axes of
3)
Identify the lengths of the transverse and conjugate axes of
4)
Find the slopes of the asymptotes for
5)
Given
6)
Solve the following systems.
7)
(y - 7)2
16
-
(x + 6)2
9
(x - 2)2
25
-
(x + 3)2
36
(y + 3)2
4
+
(y - 7)2
= 1.
100
(y - 11)2
9
-
(x + 2)2
16
= 1.
= 1.
= 1 , find the equations of the asymptotes.
a)
x2 - 2x - 4y + 1 = 0
y-x-2=0
b)
y = x2 - 2
y = 6 - x2
c)
y2 = 3x - 5
y-x+1=0
d)
y2 - 4y - 3x = 0
y=x+4
e)
x2 + 3y2 + x – 2y – 2 = 0
2x2 + 3y2 + 3x – 2y – 2 = 0
f)
y2 + 11y = 10y + 6x
x+y-3=0
Identify the conic section, write the equation in standard form, and identify the indicated values.
Circle:
Ellipse:
Hyperbola:
Parabola:
center, radius
center, vertices, EPMA, foci
center, vertices, EPCA, foci
vertex, focus, EPFC, directrix, axis of symmetry
a)
x2 - 6x - 4y + 5 = 0
b)
2y2 - 5y + 2x2 + 7x - 1= 0
c)
5x2 + 4y2 - 30x + 8y + 29 = 0
d)
9x2 - 16y2 + 54x + 64y - 127 = 0
e)
x- 4y2 - 12y - 10 = 0
f)
16x2 + 16y2 + 16x + 24y - 51 = 0
g)
x2 + 6x + 8y - 7 = 0
h)
9x2 + 5y2 - 54x + 10y + 41 = 0
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Precalculus
8)
Write the equation of the conic section satisfying the given conditions in standard form.
a)
Ellipse with vertices (-3, 7) and (-3, -1) and eccentricity 1/2.
b)
Circle with diameter with endpoints (-7, 3) and (3, 9).
c)
Parabola with focal chord of length 12 and directrix parallel to y-axis. The focus is at (3, 4),
and “p” is positive.
d)
Hyperbola having foci at (2, 7) and (2, -3) and one vertex at (2, -1).
e)
Parabola with focus at (3, 1) and directrix y - 7 = 0.
UNIT CP COUNTING PRINCIPLES
1) Draw the Venn diagram.
a)
A Ç B
b)
A  B’
c)
B
2) In a recent cooking class, students were deciding what to add to their cookie dough to make it
more interesting. Of the 80 students in the class, 40 wanted to try chocolate chips, and 48 wanted to use
peanut butter chips. Twenty-two were willing to use either type of chip. How many students:
3)
4)
a)
wanted neither chip?
c)
wanted chocolate chips only?
b)
did not want peanut butter chips?
In a survey of 75 Math Analysis students, 45 said they liked trigonometry, and 38 said they liked
sequences and series. Nine students did not enjoy either of the topics. How many students:
a)
liked both topics?
b)
liked trigonometry only?
c)
did not like trigonometry?
d)
liked sequences or series only?
Kelsey is volunteering at the hospital. She has a cart with 300 containers of chocolate pudding,
pea soup, and pork and beans. A little accident happens that causes the contents of these
containers shift. After the accident, Beth determines that 73 have chocolate pudding and pea soup,
99 have pea soup and pork and beans, and 78 have pork and beans and chocolate pudding. Fortyseven containers show evidence of all three substances. In total, 168 have chocolate pudding, 160
have pea soup, and 159 have pork and beans. How many containers:
a)
are empty?
b)
have only chocolate pudding?
c)
have beans or soup?
d)
have pudding or soup but no beans?
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Precalculus
5)
How many different arrangements can be formed on a shelf with space for three books if there are
6 different books available?
6)
In how many ways can a coach assign positions on a baseball team of 9 people if only 3 are
qualified to be pitcher and only 2 are qualified to be catcher but all are able to play remaining
positions?
7)
How many different signals consisting of one or more flags can be hung from a set of five
different flags?
8)
Three different dice are thrown. In how many different ways can they land?
9)
How many positive 4-digit numbers can be form using the digits 2, 3, 4, 5, 6, 7 if:
a)
b)
c)
no digit is repeated?
no repetition and the number is even?
no repetition and the number is odd?
10) How many committees of 7 can be chosen from 8 Democrats and 5 Republicans if each committee
must have:
a)
b)
c)
3 Democrats and 4 Republicans?
2 Democrats and 5 Republicans?
at least five Republicans?
11) How many different arrangements of the following are there?
a)
MATHEMATICS
b)
ANTIBODY
12) A recent math test had 2 parts. Students had to answer 10 of 12 questions in Part I and 8 of 11
questions in Part II. How many different sets of questions could the students have answered?
13) A test has 15 multiple-choice questions, each with 5 possible answers. In how many unique ways
could answers to this test be made?
14) At an upcoming graduation party there are 20 prizes to be awarded to 20 of the 151 people
attending. The prizes consist of 7 identical clock radios, 6 identical microwaves, 5 identical TVs,
and 2 identical CD players. In how many unique ways could the prizes be awarded?
15) A college ID consists of your first initial, last initial, and 2 non-zero digits. How many such id’s
can be constructed?
16) How many 5-card hands are there containing 5 cards of the same suit?
17) How many ways can 12 people be seated at a round table?
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Precalculus
18) You are planning a trip to Canada and AAA sends you 7 different possible routes from NJ to
Buffalo and then 4 different routes from Buffalo to Canada. If you decide not to drive the same
roads twice, how many different trips can you make?
19) Using Pascal’s Triangle, expand each of the following.
a)
(2x + 7)3
b)
(3a - 2b)4
20) Find the indicated term for each of the following.
a)
6th for (x + 3)11
c)
5th for (2x - 8)7
b)
3rd for (a - 4b)8
21) An urn has 8 red, 5 blue, and 4 yellow balls. What is the probability of drawing three balls out so
that:
a)
b)
c)
22)
there is one of each color?
there are at least 2 reds?
there are no reds?
Two dice are rolled.
a)
b)
c)
d)
e)
f)
What is the sample space?
What is the probability that the sum is 3?
What is the probability that the sum is odd?
What is the probability that the first die is either 1 or 3?
What is the probability that the sum is even given that the first die is a 6?
What is the probability of the roll not being doubles given the sum is 8?
23) Five cards are dealt from a deck of 52.
a)
b)
c)
d)
How many card hands are possible?
What is the probability that the 5 cards are all diamonds?
What is the probability that the cards are 3 queens and 2 jacks?
What is the probability of having and ace, a king, a queen, a jack, and a 10 of any suit?
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