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PREREQUISITE SKILLS FROM ALGEBRA II
The toolkit of functions: recognize the equations, graphs and applications to modeling of a variety of functions.
1.
2.
3.
4.
State the Fundamental Theorem of Algebra. (Two approaches: number of roots; number of factors.)
State the definition of number i. State the definition of a complex number.
What is the complex conjugate of 3 – 4i ?
Determine the nature of the remaining roots of the polynomial P(x) = 2x3 -9x2 + 27 if x = 3 is one of its roots.
5.
Is
6.
7.
Simplify (a + bi)(a – bi).
True or False:
1
( x + ) a factor of the polynomial P( x) = 4 x 3 ! 5 x ! 2 ?
2
x 2 =| x |
a.
For all real numbers x,
b.
c.
It is possible for a cubic polynomial to have roots 3, -3 and 2 + i
d.
e.
!9 = 3i
i19 = !1
4 = ±2
P( x) = x3 + 2 x 2 ! 3 x + 20 given that -4 is one of its roots.
9. Find a quadratic equation with roots 1 ± 3i . Express the answer in standard form.
8.
Find the exact values of the zeroes of
10. Find a cubic equation with roots
1 ± 3i and !2 . Express the answer in standard form.
11. Solve 9x2 + 45 = 0
Simplify:
!8 " !2
12.
!9
+ !1
4
13.
14.
(!8) " (!2)
15.
(2 + 3i ) ! ("4 + 4i )
17.
1+ i
i
19.
1+ i 3
1! i 3
16.
(2 + 3i ) ! (!4 + 4i )
18.
1! i
(1 + i ) 2
f ( x) = ! x 2 + x ! 3 if the initial value of x is x = 2 + i.
1
21. Find the first and second iteration of f ( x ) = 2 if the initial value of x is x = – 3i.
x
20. Find the first and second iteration of
22. Use your calculator and the Fundamental Theorem of Algebra to find the exact values of all of the roots of
P( x) = x 4 ! 15 x 3 + 70 x 2 ! 70 x ! 156 .
23. The function C ( x ) = .03 x + .02 x + .2 x + .1 approximates the cost, in billions of dollars, to the American
economy of computer viruses with x = 0 representing the year 1990.
A. Approximate the cost of the computer viruses in the year 1999.
B. In what year did the cost of the computer viruses to the economy reach 20 billion?
3
2
1.
y=
Give the domain and range of each of the rational functions and give the equations of their horizontal and vertical
asymptotes, if any:
6x
x!4
y=
x2 ! 9
x+3
y=
4
x +1
2
2.
If x varies directly as y and x = 4 when y = 28, find y when x = 11.
3.
If p varies inversely as t, and p = 2 when t = .25, find p when t = 7; find t when p = 6.
4.
Give a reasonable equation for each of the graphs below:
5.
True or false:
a.
b.
c.
d.
e.
f.
g.
x x3
+
is an example of a rational function
2 5
x + x3
f ( x) =
is an example of a rational function
2
x3
f ( x) =
is an example of a rational function
5! x
f ( x) =
The equation y = 4x is an example of direct variation
The equation y = 4xz is an example of a joint variation
Graphs of polynomials never have any breaks
Graphs of rational functions always have breaks
1.
State the definition of the logarithm to the base b of a positive number x.
2.
State the rules of exponents; state the rules of logarithms
True or False?
3.
1.
3log 2 x = log 2 x3
2.
(a !2 + b !2 ) !1 = a 2 + b 2
3.
log x ! log y =
4.
log b m ! n = log b m + n
5.
log 3 3 = 0
6.
((a + 2b) )
7.
log 80
= log10
log 8
8.
9.
log b x =
If
(!3) 2 = !3
2 !2
1
f ( x) = log 3 x , evaluate w/o calcualtor: f (1), f (3), f ( ), f ( 5 3), f (3!12 ), f (!3)
3
log x
log y
= a + 2b
log c b
log c x
4.
On the same set of axes, graph
y = log 3 x and
5.
Graph:
y = log 3 9 x (Hint: rewrite the log of a
product and use what you know about function
transformation.)
y = 3 . Comment on their relationship.
x
!1"
6. Give coordinates of any three points on the graph of y = # $
%5&
x
Express in scientific notation:
7.
.000000331
9.
(3.4 "10 )" (6.1"10 )
!6
8
8.
18,898,000,000
10.
1340000
12 "10!4
Solve each equation for the variable given without the use of your calculator:
1
b
11.
y = log b
13.
y = log 4 2 2
15.
17.
12.
x = log 3 3121
14.
log b
log 2 ( x 2 + 8) = log 2 x + log 2 6
1
=3
27
16. log 3 | x + 1|= 2
log 4 ( x + 3) + log 4 ( x ! 3) = 2
18.
82 y =
( 2)
y !3
(common base)
Solve each equation for the variable given; you will need to use of your calculator:
19.
3x = 1334
20.
4.2 ! 3x = 134
21.
x = log8 8990
22.
log 3 x !1 = 2.33
Evaluate/simplify without the use of your calculator. Your answers should have no negative exponents and no
fractional exponents in the denominator.
2 !1
23.
(16 x 5 ) 2
26.
1
x2 / 3
29.
log 6 2 + log 6 3 + log 6 18 + log 6
32.
x ! 11
11 ! x
35.
1
+ x + x !2
x
(express with a common denominator)
" " 2 !1 #!3 # 2
"a #
% %
24. $ $ $
$$ $ & 3 %' % %%
' '
&&
25.
2 1
( + ) !2
5 3
a ! 28
28.
log 3 4 x ! 4 log 3 2 y
9log9 241
1
27.
1
3
4
30.
4!1 " (32 ! 3!2 ) !1
31.
33.
i 22
1
9 ! x2
34.
3x + 9
2
( x ! 6 x + 9)
36.
1
2 + x !1
37.
38. Give the domain and range of each of the following functions:
f ( x) = log 3 x
f ( x) = 5x
39. Give a reasonable equation for each of the graphs below:
1
!3
( x + 5) 2
f ( x) = log 3 ( x ! 4)
f ( x) =
2
2 ! 3i
1.
It was determined in laboratory tests that a piece of wood used in a handle of an ancient ax contained 29% of its
t
! 1 " 5730
original Carbon14. Use the equation P (t ) = 100 # $ %
to find the age of the ax.
&2'
2.
You make two investments of $ 8 000 each, one good one and one poor one. The money in the good investment
account grows at 12% interest compounded yearly, and the money in the bad investment diminishes at 19%
compounded yearly.
A. How much money do you have in the two accounts together after 4 years?
B. After how many years does the money in the first account grow to $ 16 000?
C. After how many years does the money in the second account shrink to $ 4 000?
3.
The value of a house appreciates at of 6% compounded continuously. If the house is worth $ 410 000 in 2000, how
much will it be worth in 2012?
4.
Find an equation in the form
y = a !10c! x (that is, find the constants a and c) for an exponential function that passes
through the points (3, 2) and (4, 9). Repeat the exercise with the alternative forms of the equation:
y = a ! ek !x ,
y = a ! bx
1.
Find the three arithmetic means between 6.1 and 2.5.
2.
Find the partial sum S61 of the arithmetic series: 2, -3, …
3.
Use the formulae to find the sum of the finite arithmetic series: -12 + (-8) + … + 92
4.
Write the expression
4
! (k
2
+ k + 1) in full and evaluate it.
k =1
1 1 1
1
+
+ +
3 27 81 243
5.
Express in sigma notation (∑):
6.
.
Give the first 4 terms of the sequence defined recursively as follows:
a1 = 1.5, an = an-1 - 3
7.
Is the sequence in problem 7 arithmetic, geometric or neither arithmetic nor geometric?
8.
Find the four geometric means between 36 and
9.
Give the first four terms of the sequence defined recursively as follows:
a1 = 1, a2 = -3, and an = -4an-1 + an-2.
17
10. Evaluate:
4
.
27
n !1
# 2 " (3)
(Hint: write the first terms and figure out what kind of series this is. Then use the formulae)
n =1
11. Find the first three iterates of the function f(x) = -2x2 + x if x 1 = -2 (That is, find x1, x2, x3).
12. Give an example of any infinite geometric series that has a finite sum. Compute this sum.
13. A. Find the first term of a geometric series if S7 = 86 and r = - 2.
B. Find a12 for this series.
14. Find a50 for an arithmetic series in which a1 = -6 and S50 = -5150
Pr and n Cr .
1.
Write the formulae for computing
2.
Two dice are rolled. Find each probability:
a. Both dice show an even number.
a. The first die rolled shows 3 and the other shows an even number.
b. One of the dice shows 3.
c. One die shows 3 and the other shows an even number.
d. The sum of the numbers shown is 11.
e. The sum of the numbers shown is more than 10 or is less than 6.
f. The two numbers shown are different
g. Each number shown is greater than 4
3.
Members of a math department are to be chosen at random to attend a convention. If the department has 9 mend and
8 women, what is the probability that 2 women will attend the conference?
4.
A test has 7 multiple choice questions, each with 4 possible answers. What is the probability that in random
guessing on this quiz you will get all questions right?
5.
The odds against winning a race are 7:1. What is the probability of winning the race?
6.
The probability of rain today is 10%. What are the odds in favor of rain today?
7.
A club consists of 11 juniors and 6 seniors. Find the number of ways in which:
h. A committee of 4 people can be selected.
i. A committee of 3 juniors and 3 seniors can be selected.
j. A committee of 4 juniors can be selected.
8.
5 fair coins are tossed.
k. What is the probability that each coin shows tails?
l. What is the probability that at least one of the coins shows heads?
9.
In how many different ways can the letters in the word EXAMINATION be rearranged?
10. Simplify: A.
n
n !! (n + 1)
B.
n!
(n ! 2)!
C.
P(12, 6)
P(12,3) ! P(8, 2)
11. Solve for n using the formulae that we have studied:
a. C ( n,5) = C ( n, 7)
b.
P(n, 4) = 40 ! P(n " 1, 2)
12. A sandwich shop has 7 breads, 4 kinds of spreads, 8 different cheeses and 5 different kind of meat. If you order a
sandwich and select one of each option, how many sandwich possibilities are there?
13. How many 4 - letter “words” can be formed using the letters TROUBLE?
a. If the letters may be repeated
b. If the letter may not be repeated
14. How many 4 - digit even numbers less than 4000 can be made from the digits: 0, 1, 2, 3, 4, 6, 7, 8 if the digits may
be repeated? (No leading zeroes are allowed.)
15. You are playing poker and the dealer deals you 5 cards. Find each probability:
a. All 5 cards are black.
b. 3 cards are aces and two cards are face cards.
c. Exactly 4 cards are diamonds.
FORMULAE REFERENCE SHEET:
EXPONENTS:
Same base
b !b = b
m
n
Same exponent
(a ! b )
m+n
n
bm
= b m ! n provided b ! 0 (for now: m > n )
bn
If
= a !b
n
n
n
an
!a"
=
provided b ! 0
# $
bn
%b&
n
n
If a > 0, b > 0 , n ! 0 then a = b if and only if a = b
b ! 1, " 1, 0 then b x = b y if and only if x = y
Power of a power:
(b n ) m = b m!n
DEFINITIONS: ZERO , NEGATIVE, AND FRACTIONAL EXPONENTS:
In the definitions with m and n, m and n are natural numbers (positive integers):
b 0 = 1 , provided b ! 0
1
n
b !1 =
m
b!n =
1
, provided b ! 0
bn
m
1
m
! 1n "
m n
n
n
b = # b $ = b , or equivalently, b = (b ) = n b m
% &
provided n b is a real number (that is, b ! 0 if n is even)
m
n
b = b provided b is a real
b ! 0 if n is even)
n
n
1
, provided b ! 0
b
number (that is,
( )
LOGARITHMS.
Definition:
log b x = y if and only if b y = x , b > 0, b ! 1, x > 0
Rules of logarithms:
log b m ! n = log b m + log b n
log b
m
= log b m ! log b n
n
QUADRATICS:
log b m p = p ! log b m
log b m = log b n if and only if m = n
log b b = x
b logb x = x
x
log b x =
log c x
log c b
Standard form: y = ax 2 + bx + c
Vertex from: y = a(x - h)2 + k
x-intercept form: y = a(x – r1) (x - r2)
SEQUENCES AND SERIES:
an
Arithmetic
an = a1 + (n ! 1)d
Geometric
an = a1 " r n !1
Sn
Sn =
S (the sum of all the term)
n(a1 + an )
2
Sn =
Does not exist
a1 (1 ! r n )
1! r
If
| r |< 1, S =
a1
, otherwise does not exist
1! r
PROBABILITY AND COMBINATORICS:
n
Pr =
n!
, for example, 8 P3 = 8 ! 7 ! 6
(n ! r )!
n
Cr =
Pr
n!
8!7 !6
=
, for example, 8 P3 =
r ! (n ! r )!r !
3 ! 2 !1
n