PREPARED BY: Michael Brinegar
5/17
Table of Contents
Introduction
Lesson 1 part 1
Lesson 1 part 2
Lesson 2
Lesson 3
Lesson 4
Lesson 5
Lesson 6
Lesson 7
Lesson 8
Lesson 9
Lesson 10
Lesson 11
Lesson 12
Lesson 13
Lesson 14
Lesson 15
Lesson 16
Lesson 17
Lesson 18
Lesson 19
Lesson 20
Lesson 21
Lesson 22
Lesson 23
2
Page 1
Page 3
Page 8
Page 15
Page 24
Page 33
Page 44
Page 56
Page 66
Page 78
Page 86
Page 95
Page 105
Page 114
Page 122
Page 133
Page 141
Page 148
Page 156
Page 163
Page 171
Page 181
Page 187
Page 196
Page 202
INTRODUCTION
The videos that go along with this study guide were recorded in an actual MAC 1140,
Pre-Calculus Algebra class. If you have any questions or comments about the videos or
this study guide, please contact Mr. Michael Brinegar at (850) 769-1551 ext. 2857 or by
email at [email protected] .
ROAD TO SUCCESS
Since this course is a distance education course, the method of instruction is
primarily a self-study one. You are expected to demonstrate sufficient selfdiscipline and self-motivation to complete all unit tests and the final exams within
the designated time.
To be successful, you should follow these steps:
1.
Begin the lesson by looking at the objectives listed here in the study guide.
The study guide follows the section numbers of the text book.
2.
Watch the video for the section you are studying. While watching the
video, follow the study guide, answer the questions in the study guide, work
the problems provided in the study guide, and take notes as if you were
sitting in a classroom. Stop or pause the video as needed to catch up or
copy something down.
3.
Once you have viewed the video, look at the textbook for further
examples, steps, or procedures, etc.
4.
Try the homework. The answers to all of the problems are at the end of
each section in the book. There are some homework problems explained
on the videos for each section. The number of homework problems
explained varies for each section.
5.
HOW TO STUDY FOR TESTS!!!
The best way to study for a test is to take your objectives for each
section and find corresponding problems from the homework. Take
these problems and make yourself a practice test. Without looking
at your notes, take the practice test. This will tell you where you are
weak and need to study further.
1
Lesson 1 Part 1
I.
Piecewise-Defined Functions
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
II.
Evaluate piecewise functions.
Graph piecewise functions.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Piecewise Function: is a function represented by
more than one expression.
Turn in your textbook to page 14.
Problem #2.
t + 4
g (t ) = 
8 − t
for t ≤ 1
for t > 1
3
Turn your text book to page 14. Problem #6
3x + 2 for x > 2
g (x ) = 
2
9 − x for − 4 < x ≤ 2
4
Turn in your textbook to page 15.
Problem #12.
 x 2 − 25
for x ≠ 5
 x−5

g (x ) = 
10
for x = 5


5
Turn in your textbook to page 15.
Problem #14.
 x3 − 8

q(x ) =  x − 2
− 5

for
x≠2
for
x=2
6
Turn in your textbook to page 14.
Problem 10.
− x − 3

h( x ) =  9 − x 2
2 x − 2

for
x < −3
for
−3≤ x <3
x≥3
for
7
SECTION 1.1 Part 2
I.
Piecewise-Defined Functions
OBJECTIVES
At the conclusion of this lesson you should be able to:
II.
1.
2.
Find values for greatest integer functions.
Write absolute value functions as piecewise functions.
3.
Graph absolute value functions.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Greatest Integer Function: is a piecewise function
that gives the largest integer less than or equal to x.
f ( x ) = x 
or
f ( x) = [[x ]]
[[x]]
Find the following values.
[[2.1]]
[[2.7]]
[[4.8]]
[[− 2.1]]
[[− 2.7]]
[[− 6.3]]
8
[[− 0.27]]
Greatest Integer Function—Step Function
x
3
3≤ x<4
2
2≤ x<3
1
1≤ x < 2
0
0 ≤ x <1
-1
−1 ≤ x < 0
-2
− 2 ≤ x < −1
-3
− 3 ≤ x < −2
f (x)
[[3]] = 3
[[3.289]] = 3
[[3.47]] = 3
[[3.6]] = 3
[[3.8]] = 3
[[2]] = 2
[[2.1]] = 2
[[2.48]] = 2
[[2.579]] = 2
[[2.9]] = 2
[[1]] = 1
[[1.08]] = 1
[[1.578]] = 1
[[1.7]] = 1
[[1.99]] = 1
[[0]] = 0
[[0.398]] = 0
[[0.6]] = 0
[[0.812]] = 0
[[0.987]] = 0
[[− 0.1]] = −1
[[− 0.4]] = −1
[[− 0.61]] = −1
[[− 0.789]] = −1
[[− 1]] = −1
[[− 1.2]] = −2
[[− 1.6]] = −2
[[− 1.725]] = −2
[[− 1.99]] = −2
[[− 2]] = −2
[[− 2.34]] = −3
[[− 2.45]] = −3
[[− 2.863]] = −3
[[− 2.971]] = −3
[[− 3]] = −3
9
f ( x) = [[x ]]
Find the values.
g ( x) =
1
[[x + 3]]
2
Find g(- 4.2 )
Find g(7.6)
g ( x) = [[x − 4]]
Find g(-4.2)
Find (7.6)
Now graph g ( x) = [[x − 4]]
10
WRITING ABSOLUTE VALUE IN PIECEWISE FORM
11
Turn to page 15 in your textbook. Problem #26
Write in piecewise form.
f ( x) = 3 x − 4 + x + 6
12
Turn to page 15 in your textbook. Problem #28
Write in piecewise form.
f ( x) = x + 1 − x − 5
13
Graph
f ( x) = x + 1 − x − 5
14
Lesson 2
I.
Synthetic Division
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
4.
5.
II.
Use synthetic division to divide polynomials.
Use the Remainder Theorem to evaluate polynomials.
Use the Factor Theorem to factor and build polynomials.
Find a polynomial with specified zeros.
Use the intermediate value theorem.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Quotient + Remainder
Divisor
Dividend
Dividend = (Divisor) ⋅ (Quotient ) + Re mainder
P(x ) = d (x ) ⋅ q(x ) + r (x )
Synthetic Division condenses the long
division process. The divisor must be linear.
15
Turn in your textbook to page 26.
Problem #2.
(3x
3
)
+ x 2 + 4x −1 ÷ (x + 5)
16
Turn in your textbook to page 26.
Problem #8.
(6 x
3
)
2

− x2 − 5x + 4 ÷  x − 
3

17
Remainder Theorem- If a polynomial P(x) is
, then the remainder r is
divided by
equal to P(c).
P(−1) =
P( x) = x 3 − 7 x 2 + 5 x − 6
18
Turn in your textbook to page 26.
Problem #16.
P ( x) = 4 x 3 − 7 x + 33
Find
 1
P − 
 2
Factor Theorem – A polynomial P(x) has a
factor
if and only if P(c) = 0.
If r = 0, then
is a factor of P(x).
19
Turn in your textbook to page 26.
Problem #20.
P(x ) = 2 x3 − 5 x 2 + x + 2
( x − 2)
Turn in your textbook to page 26.
Problem #24.
P( x ) = x 4 − 26 x 2 + 25
20
( x − 4)
If
is a factor of the polynomial, then c
is called a zero of the polynomial. Each zero
of the polynomial is an x-intercept on the
graph of the polynomial.
Turn in your textbook to page 27.
Problem #26.
x=3
P ( x) = 3x 3 + 5 x 2 − 6 x + 18
x=2
P( x) = x 3 − 7 x + 6
Problem #30
21
Intermediate Value Theorem – Let a and b be
If P(a) and
real numbers such that
P(b) have opposite signs, then P(a) has a real
zero in the interval
.
Turn in your textbook to page 27.
Problem #32.
P( x) = x 3 + 3 x 2 − 9 x − 13
22
[− 5,−4]
Turn in your textbook to page 27.
Problem #36.
P( x) = x 3 − 5 x 2 + 4
23
[4, 5]
Lesson 3
I.
The Zeroes of Polynomial Functions
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
4.
II.
Find a polynomial with the given degree and zeros.
Find possible rational zeros of a real polynomial function using the
rational zeros theorem.
Obtain more information on the zeros of real polynomials using
Descartes’ rule of signs.
Find zeros of a polynomial.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Find a polynomial given the following zeros
and degree.
Turn in your textbook to page 35. Problem #2
Degree 3
, x = 1, x=3 ,
24
x=-7
Turn in your textbook to page 35. Problem #8
Degree 4, x = - 6i , x = g ( x) = 2
Multiplicity – If
is a factor of a
polynomial function, x = c has multiplicity k.
Multiplicity is the number of occurrences of the
zero in a function.
25
Turn in your textbook to page 35. Problem #12
Degree 4,
x = -1 , x = 2 -
26
3
Rational Zeros Theorem – Given a polynomial P(x)
with integer coefficients. The rational zeros, if they
exist, of P(x) must be of the form
, where p is a factor of the constant term and q is a
factor of the leading coefficient. The rational number
must be in lowest terms.
Turn to page 35 in your textbook. Number #14
P( x) = x 3 − 13x + 12
Turn to page 35 in your textbook. Number #26
P( x) = 3x 4 − 11x 3 + 10 x − 4
27
Decartes’ Rule of Signs- Given a polynomial
P(x) written in descending or ascending order
with real coefficients and a nonzero constant
term, the number of positive zeros is the
same as the number of variations of sign in
P(x), or an even number less. The number of
negative zeros is the same as the number
, or an even
variations of sign in
number less.
28
Try this one.
P( x ) = x 3 − 13x + 12
Positive
Negative
Complex
Turn to page 35 in your textbook. Number #26
P( x ) = 3 x 4 − 11x 3 + 10 x − 4
Positive
Negative
29
Complex
Turn to page 35 in your textbook. Number #16
Find the zeros of the polynomial.
P( x ) = 2 x 3 − 3x 2 − 9 x + 10
30
Turn to page 35 in your textbook. Number #24
Find the zeros of the polynomial.
P(x ) = 14 x 3 − 26 x 2 + 9 x − 1
31
Turn to page 35 in your textbook. Number #32
Find the zeros of the polynomial.
P( x ) = 2 x 4 + 3x 3 + 15 = 11x 2 + 9 x
32
Lesson 4
I.
Graphing Polynomial Functions
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
4.
II.
Identify the graph of a polynomial function and determine its degree.
Describe the end-behavior of a polynomial graph.
Discuss the attributes of a polynomial graph with zeroes of multiplicity.
Graph polynomial functions using the 5 guidelines.
PROCEDURE
While watching the videos, follow this study guide and take notes in the study
guide as if you were sitting in a classroom. Stop or pause the video as needed to
catch up or copy something down.
A polynomial is a function written in the form
The coefficients are real numbers and the
exponents are whole numbers. The number
is the leading coefficient, and the number
is
the constant term. The degree of a polynomial
is n, the largest exponent.
33
Leading Term Test – Given a polynomial
If
, then the following can be used to
determine the behavior of the graph as
.
1. If n is even, the ends of the graph will point
in the same direction.
2. If n is odd, the ends of the graph will point in
the opposite direction.
34
A polynomial has at most
turning points,
also called relative extrema. You can find the
relative extrema using a calculator.
35
Use the leading coefficient test to determine the
end-behavior of the graph. Then determine the
maximum number of real zeros and turning
points of the function.
Turn to page 46 in your textbook. Problem #2.
P ( x) = −2 x 5 − x 3 + 3
Turn to page 46 in your textbook. Problem #4
P( x) = x 6 + 3x5 − 4 x 4
36
Multiplicity of Zeros - Given a polynomial
function p(x) with factors of the form
,
where c is a real number:
1. If k is odd, the graph will cross through the
x-axis at (c, 0).
2. If k is even, the graph will have a turning
point at (c, 0).
37
Find the zeros of the polynomial. Determine the
multiplicity of each zero and the behavior of the
graph at each zero.
Turn to page 46 in your textbook. Problem #6.
P( x) = (3 x + 5) 2 ( x − 2)5 ( x + 8)
38
Turn to page 46 in your textbook. Problem #12.
P( x) = x 4 + 4 x3 − 2 x 2 − 12 x + 9
39
Find the turning points using the
maximum/minimum function on the calculator.
Round your x and y coordinates to the nearest
hundredths place, if necessary.
Turn to page 46 in your textbook. Problem #16.
P( x) = 2 x3 + 3 x 2 − 12 x + 1
40
Guidelines for Graphing Polynomials
1. Use the Leading Term Test to determine the
end behavior of the graph.
2. Find the zeros of the polynomial and plot the
x-intercepts.
3. Find the turning points, using the relative
minimum and relative maximum feature in your
calculator.
4. Find p(0) and plot the y-intercept.
5. Plot additional points if necessary to finish
the smooth continuous curve.
41
Follow the 5 guidelines to graph the polynomial.
Be sure to connect the points with a smooth and
continuous curve.
Turn to page 46 in your textbook. Problem #20
P( x) = − x3 − 2 x 2 + 5 x + 6
42
Turn to page 46 in your textbook. Problem #28
P( x) = x 4 − 2 x 2 + 1
43
Lesson 5
I.
Rational Functions
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
4.
5.
II.
Locate the vertical asymptotes and find the domain of a rational
function.
Apply the concept of “roots of multiplicity” to rational functions and
graphs.
Find horizontal asymptotes of rational functions.
Graph rational functions.
Use a piecewise function to repair the hole in the rational function.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
A rational function is defined as
where p(x) and q(x) are polynomials.
44
Domain of the rational function includes all x
values such that q(x) 0.
Vertical asymptote is the line x = c provided
45
Removable discontinuity – occurs when the
numerator and denominator share a common
factor x – c. The factor
cancels out,
creating a hole in the graph at x = c.
Try this one.
Try this one.
f ( x) =
f ( x) =
x −1
( x + 2)( x − 4)
x+3
( x + 3)( x − 4)
46
Turn to page 62 in your textbook. Problem #8
f ( x) =
x+3
x2 − 9
Horizontal asymptote is the line y = b provided
.
47
Case 1. Degree of the numerator is less than the
degree of denominator, the horizontal asymptote
is automatically y = 0.
Case 2. If the degree of the numerator is equal to
the degree of denominator, the horizontal
asymptote is the ratio of the leading coefficients.
48
Case 3. If the degree of the numerator is greater
than the degree of the denominator, there is no
horizontal asymptote. Using long division, an
oblique (or slant) asymptote can be found.
Turn to page 62 in your textbook. Problem #18.
x2 − x − 2
f ( x) =
x −1
49
Turn to page 62 in your textbook. Problem #24.
3x 3 + 5 x 2 + 12 x − 4
f ( x) =
x2 + 2x + 3
50
Guidelines for Graphing Rational Functions
1. Find all asymptotes. Draw these asymptotes
on the graph using dotted lines.
2. Determine if the graph crosses the horizontal
or oblique asymptote. Find the coordinates of the
crossing point, if applicable.
3. Find the x and y intercepts.
4. Plot at least 3 points in between each
asymptote.
5. Draw a smooth curve.
51
Try this one. f ( x) =
5
x−3
52
Turn to page 62 in your textbook. Problem #28
x2 −1
f ( x) =
x
53
Turn to page 62 in your textbook. Problem #37
x 3 − 3x + 2
f ( x) =
x2 − 9
54
Use a piecewise function to repair the hole in the
rational function.
Turn to page 63 in your textbook. Problem #42.
x2 − 4x
f ( x) =
x
55
Lesson 6
I.
Inequalities
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
II.
Solve and graph polynomial inequalities.
Solve and graph rational inequalities.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Lesson 6 things to remember.
56
Steps to solve polynomial inequalities:
1. Set one side of the inequality equal
to zero.
2. Find the zeros of the polynomial.
3. Draw a number line to represent
the x-axis.
4. Pick test values in between each
zero to determine if the inequality is
true or false on the interval. End
behavior and multiplicity of the
zeros can be used to determine
which interval or true or false.
5. The solution set is all the x-values
for which the inequality is true.
Shade the true intervals and write
the solution in interval notation.
57
Turn in your textbook to page 74.
Problem #8
x 2 < −5 x − 3
58
Turn in your textbook to page 74.
Problem #16
x 3 + 3x 2 ≤ x + 3
59
Turn in your textbook to page 74.
Problem #18
x 4 − 6 x 3 > −8 x 2 − 6 x + 9
60
1.
2.
3.
4.
5.
Steps to solve rational inequalities.
Set one side of the inequality equal
to zero.
Find the zeros of the numerator and
the denominator.
Draw a number line to represent
the x-axis. Place the zeros on the
number line in the correct order.
Pick test values in between each
zero to determine if the inequality is
true or false on the interval.
The solution set is all the x-values
for which the inequality is true.
Shade the true intervals and write
the solution in interval notation.
Note: The denominator zeros will
never be included.
61
Turn in your textbook to page 74.
Problem #24
x2
≥0
x −1
62
Turn in your textbook to page 74.
Problem #32
x +1
>1
2x − 1
63
Turn in your textbook to page 74.
Problem #40
x−4 x+2
≤
x + 3 x −1
64
Turn in your textbook to page 74.
Problem #44
2
5
>
x2 − 4x + 3 x2 − 9
65
LESSON 7
I.
Exponential and Logarithmic Functions
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
4.
II.
Use properties of exponential functions.
Graph exponential functions.
Use properties of logarithmic functions.
Graph logarithmic functions.
PROCEDURE
While watching the video, follow this study guide and take notes on the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Exponential Function:
,
, and x is a real
Where
number.
66
Graph
g ( x ) = 3x
Find the inverse.
f ( x) = b x
67
Logarithmic Function
.
Is equivalent to
Y represents the exponent on b that
yields x. The value x must be greater
than 0.
A logarithm is an exponent!
When you evaluate a logarithmic
expression, the answer is an exponent.
68
Graph
The two functions should be inverses.
69
Turn in your textbook to page 84. Problem #3.
70
f ( x) = 10 2 x
Turn in your textbook to page 84. Problem #6.
f (x) = log(2x −1) + 3
71
f ( x) = a x + c
________________________________________________
f ( x) = a x − c
________________________________________________
f ( x) = a x − c
________________________________________________
f ( x) = a x + c
________________________________________________
f ( x) = −a x
________________________________________________
f ( x) = a − x
________________________________________________
_______________________________________________
f ( x) = ca x
_______________________________________________
72
f ( x) = log a ( x + c )
________________________________________________
f ( x) = log a ( x − c )
________________________________________________
f ( x) = log a (x ) + c
________________________________________________
f ( x) = log a (x ) − c
________________________________________________
f ( x) = − log a ( x )
________________________________________________
f ( x) = log a (− x )
________________________________________________
__________________________________________
f ( x) = c log a x
__________________________________________
73
Try this one.
f ( x) = −5 x −3 + 2
Try this one.
g ( x) = log(− x) + 4
74
Natural Base: e
75
Turn in your textbook to page 84. Problem #8.
f ( x) = 3 x − 2 + 1
Turn in your textbook to page 84.
Problem #12
f ( x) = e − x − 2
76
f ( x) = 2 − log 3 ( x + 5)
Turn in your textbook to page 84.
Problem #18
f ( x) = 3 ln( x + 2)
Turn in your textbook to page 84.
Problem #20
77
Lesson 8 Properties of Logarithms
I.
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
II.
Apply basic properties of logarithms.
Apply the product, quotient, and power properties of logarithms.
Apply the change-of-base formula.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Basic Properties of Logarithms
log b 1 = 0
logb b = 1
logb b x = x
b logb x = x
78
Product Rule for Logarithms
79
Turn in your textbook to page 93.
Condense as a single implies
term.
Problem #16.
log 50 + log 20
Turn in your textbook to page 93. Expand as a sum or difference.
Problem #30.
log 2 8( x + 1)
80
Quotient Rule for Logarithms
Turn in your textbook to page 93. Expand as a sum or difference.
Problem #32.
log
x+4
10 y
81
Power Rule for Logarithms
82
Turn in your textbook to page 93. Condense as a single term.
Problem #24.
x 2 − 25
ln
x −5
Turn in your textbook to page 93.
Problem #28.
3 log( x + 3) + 4 log( x + 1) + 2 log( x − 7)
83
Turn in your textbook to page 93. Expand as a sum or difference.
Problem #36.
log 3 x( x + 4)
Turn in your textbook to page 93.
Problem #42.
( x − 6)( x + 2)
log
( x + 4)( x − 2)
84
Change of Base Formula
Turn in your textbook to page 93. Use the change of base formula to evaluate.
Problem #14.
log5 51
Use the change of base formula to evaluate.
log3 10 ⋅ log 3
85
LESSON 9
I.
Exponential and Logarithmic Equations
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
II.
Solve exponential and logarithmic equations.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Turn in your textbook to page 101. Write the following in exponential form.
Problem #2.
log x = y
Problem #4.
ln t = s
86
Turn in your textbook to page 101. Write in logarithmic form.
Problem #6.
10 x = 6
Problem #8.
et = d
Property: If
, then
Turn in your textbook to page 101. Problem #12.
87
e4
2
2
=
e
⋅
e
e 2− x
Turn in your textbook to page 101.
Problem #10.
27 2 x + 4 = 9 4 x
Steps for solving Exponential
Equations where bases don’t match:
1. Isolate the exponential expression.
2. Take a logarithm on both sides of
the equation.
3. Solve for x.
4. Check your solution in the original
equation.
88
Turn in your textbook to page 101.
Problem #16.
(
)
1 x−2
10
= 7.256
2
89
Turn in your textbook to page 101.
Problem #20.
45e3 x + 100 = 1990
90
Steps for Solving Logarithmic
Equations
1. Isolate the logarithmic expression.
Condense Logarithms if necessary.
2. Change the equation to exponential
form.
3. Solve for x.
4. Check your solution in the original
equation.
_______________________________
Property:
If
91
Turn in your textbook to page 101.
Problem #22.
ln( x − 1) + ln 6 = ln(3 x)
Turn in your textbook to page 101.
Problem #25.
log 3 x + 10 = 7
92
Turn in your textbook to page 101.
Problem #28.
log( x − 15) = 2 − log x
93
Turn in your textbook to page 101.
Problem #30.
ln 21 = 1 + ln( x − 2)
94
LESSON 10
I.
Introduction to Matrices
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
4.
5.
6.
II.
State the size of a matrix and identify its entries.
Form the augmented matrix of a system of equations.
Solve a system of equations using row operations.
Solve a system of equations using technology.
Recognize inconsistent and dependent systems.
Use Gaussian and Gauss-Jordan elimination.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Matrix – rectangular arrangement of
numbers called entries.
Diagonal entries are from the upper left to
the lower right.
95
Augmented matrix contains the
coefficients for the variables and also the
constant terms.
96
Order of the matrix is
, where m is
the number of rows (horizontal entries)
and n is the number of columns (vertical
entries).
The entries,
are the numbers in the
matrix. We can label an entry using the
row and
column.
97
Row-Echelon Form
1. If a row does not consist entirely of
0’s,then the first nonzero element in the
row is a 1.
2. For any two successive nonzero rows,
the leading 1 in the lower row is farther to
the right than the leading 1 in the higher
row.
3. All the rows consisting of entirely of
0’s are at the bottom of the matrix.
98
Elementary Row Operations.
1. Any two rows in a matrix can be
interchanged.
2. The elements of any row can be
multiplied by a nonzero constant.
3. Any two rows can be added together,
and the sum used to replace one of the
rows.
The process of using the row operations
to get the matrix into Row-Echelon Form
(REF) is called Gaussian elimination.
99
Turn in your textbook to page 114.
Problem #8.
x + 2 y = 6

2 x − y = −8
100
Turn in your textbook to page 114.
Problem #12
 x + y− z= 5

 x + 2 y − 3z = 9
 x − y + 3z = 3

101
Reduced Row-Echelon Form
Matrix has 1’s on the main diagonal and
0’s above and below.
102
Turn in your textbook to page 115.
Problem #21
 2x − 3y + 6z = 5

 x − y − 2z = 2
3 x − 4 y + 4 z = 7

103
Turn in your textbook to page 114.
Problem #18.
 5x − 2 y = − 3

2 x + 5 y = − 24
104
LESSON 11
I.
Algebra of Matrices
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
II.
Determine if two matrices are equal.
Add and subtract matrices.
Compute the product of two matrices.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Two matrices are said to be equivalent if
they have the same order
and
corresponding entries are equal.
105
Turn in your textbook to page 125.
Problem #3.
 5x
− 7

− 3 y
1
4
6 
Try this one.
=
 5
− 7

 8
1 
b 
r + 2
 x  5
 y =  2
   
 z  − 6
106
Matrix addition and subtraction is defined
as adding/subtracting corresponding
entries in each matrix where is the row
number and is the column number. You
cannot add or subtract matrices if the
orders are not equal.
Given the following matrices.
6
A=
0

− 1
1 
2
Find A + B
107
1

4

2
B=

2
− 2 − 

3 
Given the following matrices.
6
A=
0

− 1
1 
2
6
A=
0

− 1
1 
2
 1
C =  0 
− 3
Find A + C
Given the following matrices.
Find
A - B
108
1

4

2
B=

2
− 2 − 

3 
Additive Inverse (opposite matrix)
The Additive Inverse of a matrix can be
found by replacing each entry with its
opposite.
Turn in your textbook to page 125. Find the additive inverse for matrix B.
Problem #5.
3 1.4
− 5
B=

6.3 − 1.1 0 
Scalar multiplication.
 1

A =  21
−
 4
1
8
− 3

8 
109
Matrix Multiplication
has order
Given matrix A=
and matrix
has order
The product AB =
is an
matrix, in which the entries across the
rows of A are multiplied with the
entries down the columns of B. These
products are added to form the entries of
matrix C.
The product
is defined only if the
number of columns in is equal to the
number of rows in
110
Given the following matrices.
Find
6
A=
0

− 1
1 
2
AB
111
1

4

2
B=

2
− 2 − 

3 
Turn in your textbook to page 125.
Problem #9.
 7
B=
− 10
Find
20
0 
0
D=
1
−9
3
4 − 7 
BD
Turn in your textbook to page 125.
Problem #9.
 7
B=
− 10
Find
20
0 
0
D=
1
DB
112
−9
3
4 − 7 
Properties of Matrices. Page 123
Turn in your textbook to page 126. Problem #26. Write as a matrix equation.
− 5 x + y = − 13
6 x + 2 y = 22
113
LESSON 12
I.
Matrix Inverse
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
II.
Recognize the identity matrix for multiplication.
Find the inverse of a square matrix.
Solve systems using matrix equations.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
A square matrix that has order
whose entries on the main diagonal are 1
and 0 elsewhere is called the identity
matrix, .
1
0

0
1 
1 0 0
0 1 0 


0 0 1 
114
A special property of the identity matrix
includes multiplication. If matrix
multiplication is possible, then
 2
A=
 1
1
2
1
I =
0
0
1 
115
Let A and B be square matrices with
order
. If
and
then matrix is called the inverse of
matrix . Matrix is renamed
.
5
A=
2
6
− 3
1
9
B=
2
 27
116
2
9 

5
−
27 
,
Steps for finding an inverse matrix:
1. Form an augmented matrix with the
identity on the right side.
2. Perform row operations to get the
matrix in reduced row-echelon form
(RREF). In other words, use GaussJordan elimination. This puts the identity
on the left side and the inverse on the
right.
117
Turn in your textbook to page 134.
5
A=
2
Problem #2
118
6
− 3
3
 −2
A =  5 − 7
 1
−1
Turn in your textbook to page 135.
− 5
12
2 
Problem #5
If a matrix does not have an inverse , the
matrix is called singular.
If a matrix has an inverse, the matrix is
called nonsingular (or invertible matrix).
119
Solving a System of Equations using the Inverse Matrix
A−1 ⋅ B = X
Turn in your textbook to page 134.
Problem #10



− 3 x + 4 y = −4
2x − y = 6
120
Turn in your textbook to page 134.
Problem #16
=2
 x+ y

2 z =5
 3x +
2 x + 3 y − 3 z = 9

121
LESSON 13
I.
Determinants
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
4.
5.
II.
Find the minor and cofactor of a matrix.
Evaluate a determinant by cofactors and column rotation.
Use Cramer’s rule to solve systems of equations.
Find the area of a triangle using determinants.
Find the determinant of a square matrix.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
The determinant of a matrix is a value that
can be computed from the entries of a square
matrix.
Try this one.
Try this one.
 2
A=
 7
 4
B=
 0
− 2
1 
− 1
− 1
122
The determinant can be used to find the
inverse of a 2 x 2 matrix.
a
A=
c
Try this one.
b
d 
 4
B=
 −2
A−1 =
1  d
ad − bc − c
3
1
123
− b
a 
A minor
of an entry is found by using
the determinant of the matrix formed by
deleting the
row and
column of that
entry.
 1 3 − 1
A =  0
4
6
 0 − 2 − 7 
124
A cofactor of an entry is found by
multiplying the minor by a power of
 1 3 − 1
A =  0
4
6
 0 − 2 − 7 
125
.
Determinant of a 3 x 3 matrix. Find the determinant by expanding the minors.
 1 3 − 1
A =  0
4
6
 0 − 2 − 7 
126
Column Rotation Method for finding the
determinant.
 1 3 − 1
A =  0
4
6
 0 − 2 − 7 
127
Cramer’s Rule
a11 x + a12 y = c1

a21 x + a22 y = c2
c1
x=
c a22
Dx
= 2
D a11 a12
a21 a22
a11
y=
a12
Dy
D
=
c1
a21 c2
a11 a12
a21
a22
128
Cramer’s Rule for a 2 x 2 matrix.
5x + 3 y = 7
2x + 5 y = 1
129
Turn in your textbook to page 144.
Problem #15
−z= 8
3 x

 − y − z = −3
 x + 2 y + 5 z = 10

130
Area of a Triangle
Area =
det (T )
2
(x1 , y1 )
 x1
T =  x2
 x3
where
(x2 , y2 ) (x3 , y3 )
Turn in your textbook to page 145.
Problem #17
(− 7,3) (− 2,1) (5,5)
131
y1 1 
y2 1
y3 1
If the determinant of matrix T equals zero,
the area would equal zero as well. This
would indicate the points do not form a
triangle but a straight line. These points are
said to be collinear.
 x1
det (T ) =  x2
 x3
y1 1 
y 2 1 = 0
y3 1
Turn in your textbook to page 145.
Problem #19
132
LESSON 14
I.
Partial Fraction Decomposition
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
II.
Set up a decomposition template to help rewrite a rational
expression as the sum of its partial fractions.
Decompose a rational expression using convenient values.
Decompose a rational expression by equating coefficients and using
a system of equations.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
6
4
−
x−7 x+3
133
Case #1 Non-repeated Linear Factors Number 2 page 154.
2 x + 46
x 2 − 4 x − 21
134
Case #2 Repeated Linear Factors Number 8 page 154.
− 8 x 2 − 38 x − 41
x 3 + 9 x 2 + 27 x + 27
135
Case #3 Non-repeated Quadratic Factors Number 11 Page 154.
− x 3 + 5x − 3
x 4 + 5x 2 + 4
136
Case #4 Repeated Quadratic Factors
Number 19 page 154.
6x3 + 9x 2 + 7x − 3
(3x
2
)
+1
2
137
Case #5 Degree in the numerator is greater than the degree in the denominator.
Must use long division first. Number 17 page 154.
2 x 3 + 5 x 2 − 31x − 12
2x2 − 7x − 4
138
EXAMPLE (NOT ON THE VIDEO)
5 x 2 + 9 x − 56
A
B
c
=
+
+
(x − 4)(x − 2)(x + 1) x − 4 x − 2 x + 1
5 x 2 + 9 x − 56 = A(x − 2)( x + 1) + B( x − 4)( x + 1) + C ( x − 4)( x − 2)
(
) (
) (
5 x 2 + 9 x − 56 = A x 2 − x − 2 + B x 2 − 3 x − 4 + C x 2 − 6 x + 8
)
5 x 2 + 9 x − 56 = Ax 2 − Ax − 2 A + Bx 2 − 3Bx − 4 B + Cx 2 − 6Cx + 8C
5 x 2 + 9 x − 56 = Ax 2 + Bx 2 + Cx 2 − Ax − 3Bx − 6Cx − 2 A − 4 B + 8c
5 x 2 + 9 x − 56 = ( A + B + C )x 2 + (− A − 3B − 6C )x + (−2 A − 4 B + 8C )
1. A + B + C = 5
2. −A −3B − 6C = 9
3.−2A −4B + 8c = −56
1.
A+ B+ C =5
2.
− A − 3B − 6C = 9
4.
4. −2B −5C = 14
5. 2B + 20C = −74
15C= −60
2.
3.
C = −4
2.
3.
4. −2B −5(−4) = 14
−2B + 20 = 14
−2B = 6
B=3
− 2B − 5C = 14
− 2(− A − 3B − 6C) = −2(9)
− 2A − 4B + 8c = −56
2A + 6B + 12C = −18
− 2A − 4B + 8c = −56
5.
139
2B + 20C = −74
1.
A+ B+ C =5
A + 3 + (−4) = 5
A −1 = 5
A=6
6
3
−4
5x 2 + 9x − 56
=
+
+
(x − 4)(x − 2)(x + 1) x − 4 x − 2 x + 1
140
LESSON 15
I.
Factor Completely
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
II.
Factor polynomials containing binomials with rational exponents.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Count the number of terms.
For 2 terms, look for the following
options.
Difference of Squares
a 2 − b 2 = ( a − b)(a + b)
Difference of Cubes
a 3 − b 3 = (a − b)(a 2 + ab + b 2 )
Sum of Cubes
a 3 + b 3 = (a + b)(a 2 − ab + b 2 )
Note: The sum of squares is prime.
a2 + b2
Cannot factor.
141
For 3 terms, use trinomial techniques.
Perfect Square Trinomial
a ± 2ab + b = (a ± b )
2
2
2
FOIL – Trial and Error
For 4 terms, factor by grouping or
synthetic division.
15 X − 5 x
10
6
142
Turn to page 162 in your book. Number 8.
(
)
(
2(x + 1) x − 5 + 4 x(x + 1) x − 5
3
2
2
4
2
Turn to page 162 in your book. Number 12.
(x
2
+9
) (x + 6)
4
−
4
3
+ 2 x(x + 6)
−
143
1
3
(x
2
)
+9
)
3
Turn to page 162 in your book. Number 16.
(x − 2)
−5
1
3
−6  1 
4
(
)
(
)
(
)
(
)
2
( 2)
2
3
2
+
+
+
−
x
x
x
x
 
 
3
4
144
Turn to page 162 in your book. Number 18.
(
)
2
(
4 x x + 3 − 3x x + 3
6
x
4
2
2
145
2
)
3
Turn to page 162 in you book. Number 22.
(x
2
)
(
)
− 1 (2 x ) − x (4) x − 1 (2 x )
4
2
(x
2
)
−1
146
8
2
3
Turn to page 162 in your book. Number 24.
(x
2
)
(x

1
3
(
+ 4) 

+ 4 + 2x x + 4
2
2
2
1
3
2
147
)
− 23
LESSON 16
I.
Circle and Ellipse
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
4.
5.
6.
II.
Given the standard form of a circle, find the center, radius, and
graph.
Given the center and radius of a circle, write the equation of the
circle in standard form.
Given the general form of the equation of a circle, complete the
square and get the standard form of the equation of a circle. Find
the center and radius.
Find the vertices and foci of an ellipse.
Find the equation of an ellipse.
Applications.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
148
Circles
Standard Form of a Circle
x  h2   y  k 2  r 2
Center:
Radius:
y 2  2y  20  10x  x2  5
149
ELLIPSES
Watch the video and take notes on the ellipse
and its formulas.
Ellipse definition.
Foci
Center
Standard form of the ellipse.
a
b
c
150
eccentricity
a > b
b > a
Length of an axis.
Formulas to find C.
151
Turn in your book to page 169. Number 2.
Turn in your book to page 169. Number 6.
Turn in your book to page 169. Number 10.
152
Turn in your book to page 169. Number 12.
153
Turn in your book to page 169. Number 20.
154
Turn in your book to page 168. Example 4.
155
Lesson 17
I.
Hyperbola
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
II.
Find the vertices, foci, and asymptotes of a hyperbola.
Find the equation of a hyperbola.
Applications.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
156
A hyperbola is the set of all points in a
plane for which the absolute value of
the difference from two fixed points is
constant. The two fixed points are the
foci.
The line segment connecting the two
vertices is called the transverse axis.
The other axis is called the conjugate
axis.
157
Horizontal Transverse Axis formulas
Vertical Transverse Axis formulas
158
Try this one.
2
9x 16y 2 144
159
Turn to page 178 in your textbook. Number 20.
7x 2  9y 2 14x  72y  200  0
160
Turn in your textbook to page 178. Number 12.
Turn in your textbook to page 178. Number 14.
161
Turn in your text book to page 178. Number 25.
162
LESSON 18
I.
Parabola
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
II.
Find the focus and directrix of a parabola.
Find the equation of a parabola in standard form.
Applications.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
163
A parabola is the set of all points in a
plane equidistant from a fixed line (called
a directrix) to a fixed point (called the
focus) not on the line.
164
Standard form of a parabola that opens up
or down.
Vertex: (h, k)
Focus: (h, k + p)
Directrix: y = k – p
If
it opens up.
If
it opens down.
Standard form of a parabola that opens
left or right.
Vertex: (h, k)
Focus: (h + p , k)
Directrix: x = h – p
If
it opens right.
If
it opens left.
165
A line segment that passes through the
focus of a parabola and has endpoints on
the parabola is called a focal chord.
166
Turn in your textbook to page 188. Number 6.
Find the equation of the parabola in standard form that satisfies the conditions.
Focus (1,1) , Directrix
y6
Turn in your textbook to page 188. Number 10.
Find the equation of the parabola in standard form that satisfies the conditions.
Vertex ( -1 , 4) , directrix is parallel to the y – axis, and ( 3, 0) is a point on the Parabola.
167
Turn in your textbook to page 188. Number 16.
x 2  6 x  12 y  33  0
168
Turn in your textbook to page 188. Number 20.
3 y 2  6 y  24 x  21  0
169
Turn in your textbook to page 188. Number 21.
170
LESSON 19
I.
Sequences and Series
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
4.
II.
Find the terms of a sequence given the nth term.
Look for a pattern in a sequence and try to determine a general term.
Convert between sigma notation and other notation for a series.
Construct the terms of a recursively defined sequence.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
A Sequence is a function whose domain is
the set of consecutive natural numbers
(positive integers beginning with 1).
171
Infinite sequence is a sequence whose
domain is D:{ 1, 2, 3, 4, ...}
Finite sequence is a sequence whose domain
is
D:{ 1, 2, 3, 4, …, n}
Turn in your textbook to page 204. Number 6.
(−1) n +1
an = 2
n +1
Find
a7
172
Turn in your textbook to page 204. Number 9.
an =
3
n+6
4
173
Turn in your textbook to page 204. Number 14.
Turn in your textbook to page 204. Number 18.
Find the general term of the sequence.
−
1 2
3 4
, , − , ,
4 5
2 3
Turn in your textbook to page 204. Number 21.
Find the general term of the sequence.
1, e, e 2 , e3 , 
174
Turn in your textbook to page 204. Number 26.
n
1
an =   − 5 ; S3
4
Partial Sum Definition:
Sn
175
Turn in your textbook to page 204. Number 27.
an = n + 8 ;
10
∑ (i + 8)
i =1
SUMMATION NOTATION
176
Turn in your textbook to page 204. Number 30.
(−1) k +1
∑
k =1 k ( k + 1)
4
Turn in your textbook to page 204. Number 32.
10
∑ (−1)
i
+1
i=2
177
Properties of Summation
Page 201 in your textbook.
n
I.
n
∑ (a ± b ) = ∑ a ± ∑ b
i
i =1
n
II.
n
∑ ca
i
i =1
i
i
i =1
i =1
i
n
= c ∑ ai
i =1
n
III.
∑ c = cn
i =1
m
IV.
n
n
∑a + ∑a = ∑a ,
i =1
i
i = m +1
i
i
1
178
1≤ m < n
Turn in your textbook to page 204. Number 40. Write in sigma notation.
an =
1
; S7
3
n
Turn in your textbook to page 204. Number 45. Write in sigma notation.
− 2 + 4 − 8 + 16 − 
Factorial
n!=
0!=
5!=
179
Try these 2.
3⋅ 8!
9!
(4 + 6)!
(−1 + 7)!
Turn in your textbook to page 204. Number 53.
an =
(n + 2)!
n!
180
LESSON 20
I.
Arithmetic Sequences
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
II.
Find the nth term of an arithmetic sequence.
Find a partial sum of an arithmetic sequence.
Applications.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Sequence - function whose domain is the set of
consecutive natural numbers, which are positive
integers beginning with 1
Arithmetic Sequence - sequence in which each term
after the first is found by adding the preceding term by
a constant, called the common difference
1st Term:
2nd Term:
3rd Term:
4th Term:
For any natural number k,
What is the k + 1 term?
181
General Term of an Arithmetic Sequence
an = a1 + (n − 1)d
Try this one. Find the first 5 terms and the general term.
a1 =
3
4
d =−
1
4
182
Try this one. Find the general term then find the 20th term.
− 5, − 2, 1, 4,...
Try this one. Given
a16 = 39 a9 = 25
183
Find
a1
and
d
Try this one. Given − 4, 1, 6, ,116
What term is 116? How many terms are in the sequence?
184
S1 = a1
S 2 = a1 + a2 = a1 + (a1 + d ) = 2a1 + d
S3 = a1 + a2 + a3 = a1 + (a1 + d ) + (a1 + 2d ) = 3a1 + 3d
S 4 = a1 + a2 + a3 + a4 = a1 + (a1 + d ) + (a1 + 2d ) + (a1 + 3d ) = 4a1 + 6d
S5 = a1 + a2 + a3 + a4 + a5 = a1 + (a1 + d ) + (a1 + 2d ) + (a1 + 3d ) + (a1 + 4d ) = 5a1 + 10d
In general, the partial sum is given
by S n = a1 + (a1 + d ) + (a1 + 2d ) + ... + an = na1 +
S n = na1 +
n(n − 1)
d
2
n(n − 1)
n
d = [2a1 + (n − 1)d ]
2
2
=
n
[a1 + a1 + (n − 1)d ]
2
an
Partial Sum of an Arithmetic Series
n
S n = (a1 + an )
2
185
Note: 2a1 = a1+ a1
Remember the general term:
an = a1+ (n – 1)d
Try this one. Find the sum.
20
∑ 6i − 4
i =1
Try this one. Write the sum in sigma notation.
6 + 4 + 2 + 0 + (−2) + (−4)
Try this one. Write in sigma notation.
142 + 146 + 150 + ... + 198
186
LESSON 21
I.
Geometric Sequences
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
4.
II.
Determine whether a sequence is a geometric sequence.
Find the nth term of a geometric sequence.
Find the nth partial sum of a geometric sequence.
Find the value of an infinite geometric series.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Sequence - function whose domain is the set of consecutive natural numbers,
which are positive integers beginning with 1
an = ______________
Arithmetic Sequence - sequence in which each term after the first is found by adding the
preceding term by a constant, called the common difference
an = a1 + (n − 1)d
Geometric Sequence - sequence in which each term
after the first is found by multiplying the preceding
term by a constant, called the common ratio
1st Term:
2nd Term:
3rd Term:
4th Term:
For any natural number k,
What is the k + 1 term?
187
General Term of a Geometric Sequence
an = a1 ⋅ r
n −1
Turn in your textbook to page 225. Problem # 9.
7,
7
,
3
7
,
9
7
,
27
188
Turn in your textbook to page 225. Problem #14
a2 = 2
a9 =
1
64
189
Turn in your textbook to page 225. Problem #16.
a2 = 0.01
a8 = 1000
190
Turn in your textbook to page 225. Problem #18.
a1 = 2, r = −3, and
an = 1458
191
S1 = a1
S 2 = a1 + a2 = a1 + a1r
S 3 = a1 + a2 + a3 = a1 + a1r + a1r 2
In general, the partial sum is given by S n = a1 + a1r + a1r 2 + ... + a1r n−1 .
r S n = a1r + a1r 2 + a1r 3 + ... + a1r n
Subtract S n and r S n .
(
S n − rS n = a1 + a1r + a1r 2 + ... + a1r n−1 − a1r + a1r 2 + a1r 3 + ... + a1r n
S n − rS n = a1 − a1r
(
S n (1 − r ) = a1 1 − r n
Sn =
(
)
n
)
a1 1 − r n
1− r
)
Partial Sum of a Geometric Series
a1 1 − r n
Sn =
Note: r ≠ 1
1− r
(
)
Sum of an Infinite Geometric Series
For − 1 < r < 1 , the sum of an infinite geometric series
a1
S =
is given by the formula ∞ 1 − r .
If the infinite geometric series has a limit, we call the
series convergent. If the infinite geometric series
does not approach a specific number, we call the
series divergent.
192
Turn in your textbook to page 225. Problem #23. Find the sum of the series.
9
∑100(0.5)
k −1
k =1
∞
Try this one. Find the sum of the series.
 1
42
− 
∑
 2
k =1
193
k −1
Turn in your textbook to page 225. Problem #29. Find the sum of the series.
∞
∑ (0.1)
i −1
i =1
∞
5(3)
Try this one. Find the sum of the series. ∑
k −1
k =1
Turn in your textbook to page 226. Problem #32. Find the infinite sum.
0.1+ 0.01+ 0.001+ . . .
194
Turn in your textbook to page 226. Problem #43.
195
LESSON 22
I.
Mathematical Induction
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
II.
Use mathematical induction to prove a statement.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
Mathematical Induction
Quick review. General term review.
a n = 2n + 1
196
Sum of a series review.
S n = n(n + 2)
197
Prove
3 + 5 + 7 + 9 + ... + (2n +1) = n(n + 2 )
198
Prove
12 + 2 2 + 32 + ... + n 2 =
n(n + 1)(2n + 1)
6
199
Prove
4n < 2 n
for all n ≥ 5.
200
Prove
1 + 3 + 15 + ... + (2n − 1) = n 2
201
LESSON 23
I.
Binomial Theorem
OBJECTIVES
At the conclusion of this lesson you should be able to:
1.
2.
3.
II.
Evaluate a binomial coefficient.
Use Pascal’s triangle and the binomial theorem to expand and
simplify.
Find the ith term of a binomial expansion.
PROCEDURE
While watching the video, follow this study guide and take notes in the
study guide as if you were sitting in a classroom. Stop or pause the video as
needed to catch up or copy something down.
In combinatorics, the number of
combinations of n objects taken r at a
time is denoted by
.
If there are 7 students in a classroom and
4 are chosen at random, then 35 different
combinations are possible.
202
The combination formula for
produces the coefficients of the binomial
expansion.
The Binomial Theorem
For any binomial a + b and any natural
number n,
203
Turn in your textbook to page 248. Number 8.
(x
2
+ 6
)
6
204
Powers of i .
Page 246.
205
Turn in your textbook to page 248. Number 11.
(1 − 2i )5
206
Pascal’s Triangle
(a + b )0
(a + b )1
(a + b )2
(a + b )3
(a + b )4
(a + b )5
207
Finding a Specific Term of a Binomial
Expansion.
Turn in your textbook to page 248. Number 15.
(2 x + y )9
Find the 5th term.
Turn in your textbook to page 248. Number 18.
(x
3
)
−1
12
Find the 9th term
208
NOT ON THE VIDEO!
 x2 2 
 − 2 
 2 x 
Expand:
 4  x 2
 
 0  2
4

 4  x 2
 +  
1  2

4
x2
a=
2
3
  2   4  x 2
  − 2  +  
  x   2  2
2
b=−
2
x2
  2 
 4  x 2
  − 2  +  
 3  2
  x 
2
n=4
 2   4  2 
 − 2  +   − 2 
 x   4  x 
3
 x8 
 x 6  2 
 x 4  4 
 x 2  8 
(1)  + (4)  − 2  + (6)  4  + (4)  − 6  + (1) 168 
x 
 16 
 8  x 
 4  x 
 2  x 
x8
16 16
4
− x +6− 4 + 8
16
x
x
209
4