Class Notes for
MATH 2
Precalculus
Fall 2012
Prepared by
Stephanie Sorenson
Table of Contents
1.2 Graphs of Equations ......................................................................................................................................... 1
1.4 Functions .......................................................................................................................................................... 9
1.5 Analyzing Graphs of Functions ....................................................................................................................... 14
1.6 A Library of Parent Functions ......................................................................................................................... 21
1.7 Transformations of Functions ........................................................................................................................ 24
1.8 Combinations of Functions: Composite Functions......................................................................................... 30
1.9 Inverse Functions ........................................................................................................................................... 33
2.1 Quadratic Functions and Models ................................................................................................................... 37
2.2 Polynomial Functions of Higher Degree ......................................................................................................... 43
2.3 Polynomial and Synthetic Division ................................................................................................................. 49
2.4 Complex Numbers .......................................................................................................................................... 53
2.5 Zeros of Polynomial Functions ....................................................................................................................... 57
2.6 Rational Functions .......................................................................................................................................... 63
3.1 Exponential Functions and Their Graphs ....................................................................................................... 70
3.2 Logarithmic Functions and Their Graphs ....................................................................................................... 76
3.3 Properties of Logarithms ................................................................................................................................ 81
3.4 Exponential and Logarithmic Equations ......................................................................................................... 83
3.5 Exponential and Logarithmic Models ............................................................................................................. 85
4.5 Graphs of Sine and Cosine Functions ............................................................................................................. 89
4.6 Graphs of Other Trigonometric Functions ..................................................................................................... 93
4.7 Inverse Trigonometric Functions.................................................................................................................... 98
5.1 Using Fundamental Identities ...................................................................................................................... 104
5.2 Verifying Trigonometric Identities ............................................................................................................... 108
5.3 Solving Trigonometric Equations.................................................................................................................. 111
5.4 Sum and Difference Formulas ...................................................................................................................... 114
5.5 Multiple-Angle and Product-to-Sum Formulas ............................................................................................ 117
6.3 Vectors in the Plane ..................................................................................................................................... 123
6.4 Vectors and Dot Products ............................................................................................................................ 129
6.5 Trigonometric Form of a Complex Number ................................................................................................. 132
7.1 Linear and Nonlinear Systems of Equations................................................................................................. 139
7.2 Two-Variable Linear Systems ....................................................................................................................... 141
7.3 Multivariable Linear Systems ....................................................................................................................... 144
7.4 Partial Fractions............................................................................................................................................ 145
7.5 Systems of Inequalities................................................................................................................................. 151
8.1 Matrices and Systems of Equations ............................................................................................................. 154
8.2 Operations with Matrices............................................................................................................................. 164
8.4 The Determinant of a Square Matrix ........................................................................................................... 170
8.3 The Inverse of a Square Matrix .................................................................................................................... 174
8.5 Applications of Matrices and Determinants ................................................................................................ 180
9.1 Sequences and Series ................................................................................................................................... 184
9.2 Arithmetic Sequences and Partial Sums....................................................................................................... 191
9.3 Geometric Sequences and Series ................................................................................................................. 196
9.4 Mathematical Induction ............................................................................................................................... 202
9.5 The Binomial Theorem ................................................................................................................................. 205
10.2 Introduction to Conics: Parabolas .............................................................................................................. 208
10.3 Ellipses ........................................................................................................................................................ 214
10.4 Hyperbolas ................................................................................................................................................. 219
10.6 Parametric Equations ................................................................................................................................. 224
10.7 Polar Coordinates ....................................................................................................................................... 228
10.8 Graphs of Polar Equations .......................................................................................................................... 232
Section 1.2
1.2 Graphs of Equations
Example 1
Sketch the graph of
by plotting points.
Intercepts of a Graph
The -intercepts of the graph of an equation are the points at which the graph intersects or touches
the -axis.
The -intercepts of the graph of an equation are the points at which the graph intersects or touches
the -axis.
Identify the - and - intercepts of the graph sketched in Example 1:
-intercept(s):
-intercept(s):
Finding Intercepts
1. To find -intercepts, _______________________________________________________
2. To find -intercepts, _______________________________________________________
1
2
Section 1.2
Example 2
Find the - and - intercepts of the graph of the equation.
(a)
(b)
(c)
Section 1.2
Symmetry
Knowing the symmetry of a graph before attempting to sketch it is helpful, because then you need
only half as many solution points to sketch the graph.
There are 3 basic types of symmetry a graph may have:
___________________
Type of
symmetry
-axis
is on the graph,
___________________
Algebraic Tests for Symmetry
Replacing ________ with ________ yields an
______________ is also on the graph.
equivalent equation.
Whenever
Replacing ________ with ________ yields an
is on the graph,
______________ is also on the graph.
Whenever
Origin
___________________
Graphical Tests for Symmetry
Whenever
-axis
-axis symmetry
-axis symmetry
origin symmetry
is on the graph,
______________ is also on the graph.
equivalent equation.
Replacing ________ with ________ and
________ with ________ yields an equivalent
equation.
3
4
Section 1.2
Example 3
Use the algebraic tests to check for symmetry with respect to both axes and the origin.
(a)
(b)
Section 1.2
Use the algebraic tests to check for symmetry with respect to both axes and the origin.
(c)
Recall the following formula:
Distance Formula
The distance between two points
and
is given by the formula
5
6
Section 1.2
Circles
Center:
Point on circle:
Use the distance formula to derive the equation of a circle of radius and center
:
Standard Form of the Equation of a Circle
The point
lies on the circle of radius and center
For a circle with center at the origin,
if and only if
, the standard form simplifies to:
Section 1.2
Example 4
Find the center and radius of the circle, and sketch its graph.
Example 5
Write the standard form of the equation of the circle with the given characteristics.
(a)
Center:
(b)
Center:
; Radius: 3
; Solution point:
7
8
Section 1.2
Write the standard form of the equation of the circle with the given characteristics.
(c)
Endpoints of a diameter:
Section 1.4
9
1.4 Functions
Definition of Function
A function from a set to a set is a rule of correspondence (or relation) that assigns to each
element in the set exactly one element in the set .
The set of inputs, , is the __________________ of the function .
The set of outputs, , is the __________________ of the function .
A function may be represented as a set mapping, as a set of ordered pairs, graphically, or as an equation. In
algebra, it is most common to represent functions by equations. Can you see why?
Set Mapping
Set of Ordered Pairs
Graphically
Equation
10
Section 1.4
Example 1
Determine whether the relation represents
as a function of . Circle your answer.
Function
Function
Function
Not a Function
Not a Function
Not a Function
Example 2
Determine whether the relation represents
as a function of .
(a)
(b)
Note: If
is a function of , then
is the independent variable and
is the dependent variable.
Function Notation
When an equation is used to represent a function, it is convenient to name the function so that it can be
referenced easily. For example, the equation
describes as a function of . Suppose we give the
function the name “ ”. Then we can use the following function notation.
Input
The symbol
Output
Equation
represents the -value of the function
at . So we can write
.
Section 1.4
Example 3
Let
. Evaluate
at each specified value of the independent variable and simplify.
(a)
(b)
(c)
A function defined by two or more equations over a specified domain is called a piecewise-defined
function.
Example 4
Evaluate at each specified value of the independent variable and simplify.
(a)
(b)
(c)
(d)
As previously mentioned, the domain of a function is the set of all input values. The domain can be
described explicitly or it can be implied by the expression used to define the function.
The implied domain of a function is ____________________________________________________
_________________________________________________________________________________.
11
12
Section 1.4
Example 5
Find the domain of the function.
(a)
(b)
(c)
(d)
Section 1.4
Difference Quotients
The expression
is called a difference quotient and is important in calculus.
The difference quotient can also take on the following form:
Example 6
Find the difference quotient and simplify your answer.
13
14
Section 1.5
1.5 Analyzing Graphs of Functions
Interval Notation for Sets:
Interval
Notation
Inequality
xb
(, b)
xb
(, b]
xa
( a,  )
Graph
)
b
]
b
(
a
[a, )
xa
[
a
a xb
(a, b)
a xb
( a, b]
a xb
[ a, b)
(
)
a
b
(
]
a
b
[
)
a
a xb
[ a, b]
b
]
[
a
All real
numbers
( , )
xa
(, a)  (a, )
b
)(
a
Note: The union symbol “ ” is used to indicate the union of disjoint sets.
Example: To represent “
or
” in interval notation, we write
Section 1.5
A closed dot
An open dot
indicates a point is included in the graph of a function.
indicates a point is excluded from the graph of a function.
The use of dots at the extreme left and right points of a graph indicates that the graph does not
extend beyond these points. If no such dots are shown, assume that the graph extends beyond
these points.
Example 1
Use the graph of the function
to find the following. Use interval notation when appropriate.
4
3
2
1
-6
-4
-2
2
-1
-2
-3
-4
(a) Domain of
(b) Range of f
(c)
(d)
(e) Interval(s) for which
4
6
15
16
Section 1.5
Vertical Line Test
Intersects
in one point
Intersects in more
than one point
Fails the test
Not a Function
Passes the test
Function
If a vertical line intersects a graph in more than one point,
then the graph is not the graph of a function.
Example 2
Determine whether each graph is that of a function.
(a)
(b)
Zeros of a Function
If the graph of a function of
of the function.
The zeros of a function
of
has an -intercept at
, then
are the -values for which
Example 3
Find the zeros of the function algebraically.
(a)
is a zero
.
(b)
Section 1.5
Increasing, Decreasing, and Constant Functions
A function f is increasing on an interval if, for any
implies
and
.
in the interval,
A function f is decreasing on an interval if, for any
implies
and
in the interval,
A function f is constant on an interval if, for any
.
Example 4
Determine the intervals over which the function
.
and
in the interval,
is increasing, decreasing, or constant.
17
18
Section 1.5
Example 5
(a) Find the domain and range of g .
(b) Determine the intervals of increasing, decreasing, or constant.
(c) Determine the interval(s) for which
.
Section 1.5
Even and Odd Functions
A function f is even if, for every number x in its domain, the number
is also in the domain and
The graph of an even function is symmetric
with respect to the y-axis.
A function f is odd if, for every number x in its domain, the number
is also in the domain and
The graph of an odd function is symmetric
with respect to the origin.
To determine whether a function is even, odd, or possibly neither,
replace by – in the formula.
Example 6
Determine whether the function is even, odd, or neither. Then describe the symmetry.
(a)
19
20
Section 1.5
Determine whether the function is even, odd, or neither. Then describe the symmetry.
(b)
(c)
Section 1.6
21
1.6 A Library of Parent Functions
Quick Overview of Lines
Slope Formula:
Point-Slope Form:
Slope-Intercept Form:
Vertical Lines:
Horizontal Lines:
A linear function is any function that can be expressed in the form
The graph of a linear function is a line with slope
and -intercept
Example 1
Write the linear function
and
for which
. The domain is all real numbers.
. Then sketch the graph.
22
Section 1.6
Library of Parent Functions
Constant Function
Identity Function
Absolute Value Function
Quadratic Function
Cubic Function
Reciprocal Function
Square Root Function
Cube Root Function
Section 1.6
When functions are defined by more than one equation, they are called piecewise-defined functions.
Example: Absolute Value Function
To graph a piecewise-defined function, sketch each “piece” separately. Remember to use open and
closed dots appropriately.
Example 2
Graph the following piecewise-defined function.
23
24
Section 1.7
1.7 Transformations of Functions
Vertical shift
Horizontal shift
units upward
units to the right
Vertical shift
Horizontal shift
units downward
units to the left
Vertical and Horizontal Shifts
Let be a positive real number.
Vertical and horizontal shifts in the graph of
are represented as follows.
1. Vertical shift units upward:
2. Vertical shift units downward:
3. Vertical shift units to the right:
4. Vertical shift units to the left:
Section 1.7
Reflection in the x-axis
Reflection in the y-axis
Reflections in the Coordinate Axes
Reflections in the coordinate axes of the graph of
are represented as follows.
1. Reflection in the x-axis:
2. Reflection in the y-axis:
**********************************************************************************
*Note: Always graph a function working with the expression inside to outside.
2nd
1st
3rd
1. Shift left 2 units
2. Reflect across x-axis
3. Shift up 3 units
Exception: Apply a horizontal reflection before a horizontal shift.
(But only after you’ve factored out the negative in front of !!)
1st
1. Reflect across y-axis
2. Shift right 2 units
2nd
25
26
Section 1.7
Vertical stretch (each
Horizontal stretch (each
value is multiplied by )
value is multiplied by )
Vertical shrink (each
value is multiplied by
)
Horizontal shrink (each
value is multiplied by
)
Vertical and Horizontal Stretches and Shrinks
Let be a positive real number.
Vertical and horizontal stretches and shrinks in the graph of
1. Vertical stretch (each
2. Vertical shrink (each
value is multiplied by )
value is multiplied by )
3. Horizontal stretch (each
4. Horizontal shrink (each
value is multiplied by
value is multiplied by
)
)
are represented as follows.
Section 1.7
Example 1
Use the graph of
(a)
Example 2
Use the graph of
(a)
to sketch each graph. List which transformations are performed.
(b)
to write an equation for the function whose graph is shown.
27
28
Section 1.7
Use the graph of
to write an equation for the function whose graph is shown.
(b)
Example 3
(a)
(b)
(c)
(d)
Identify the parent function .
Describe the sequence of transformations from
Sketch the graph of .
Use function notation to write in terms of .
to .
Section 1.7
Example 4
(a)
(b)
(c)
(d)
Identify the parent function .
Describe the sequence of transformations from
Sketch the graph of .
Use function notation to write in terms of .
to .
Example 5
Write an equation for the function that is described by the given characteristics:
The shape of
, but shifted nine units to the right and reflected in both the -axis and -axis.
29
30
Section 1.8
1.8 Combinations of Functions: Composite Functions
Arithmetic Combinations of Functions
Let
and
be two functions with overlapping domains. Then, for all common to both domains,
the sum, difference, product, and quotient of and are defined as follows.
1. Sum:
2. Difference:
3. Product:
4. Quotient:
Example 1
Given
and
a)
b)
c)
d)
and specify domain
e) Evaluate
,
, find the following:
Section 1.8
Composition of Functions
The composition of the function
The domain of
Example 2
Given
is the set of all
and
with the function
.
in the domain of such that
, find the following:
a)
b)
c)
Example 3
Given
and
, find
and state the domain.
is
is in the domain of .
31
32
Section 1.8
Example 4
Given
Example 5
Find two functions
and
and
, find
such that
and state the domain.
.
Section 1.9
1.9 Inverse Functions
One-to-One Functions
A function
is one-to-one if each output value corresponds to exactly one input value.
Example 1
Is the function one-to-one?
(a)
(b)
The ____________________________________________ may be used to determine
if a function is one-to-one.
33
34
Section 1.9
The inverse function of a function , denoted
, is found by
interchanging the first and second coordinates.
A function
has an inverse function if and only if
is ___________________________.
Domain of
Range of
Since the inverse functions and
have the effect of “undoing” each other, when you form the
composition you obtain the identify function:
This is how we verify two functions are inverses!
Example 2
Use the table of values for
to complete the table for
34
.
Section 1.9
Example 3
(a) Show (or verify) that
and
are inverse functions algebraically.
(b) Sketch the graphs of
and
on the same coordinate grid. What do you notice?
How are the graphs of a function
and its inverse function
related?
35
36
Section 1.9
Finding an Inverse Function Algebraically
1.
2.
3.
Example 4
Find the inverse function of
and state the domain and range of
and
.
Example 5
Determine whether the function has an inverse function. If it does, find the inverse function.
(a)
(b)
36
Section 2.1
2.1 Quadratic Functions and Models
Definition
Let be a nonnegative integer and let
function given by
is called a polynomial function of
be real numbers with
. The
with degree .
Special Cases of Polynomial Functions:
Degree
Type
0
1
2
3
4
We will focus on the quadratic function in this section.
Definition
Let
and be real numbers with
. The function given by
is called a quadratic function.
The shape of the graph of a quadratic function is called a ______________________.
If the leading coefficient is positive (
If the leading coefficient is negative (
), the graph opens __________________.
), the graph opens __________________.
All parabolas are symmetric with respect to a line called the ____________________________.
The highest (or lowest) point on the graph is called the _________________ of the parabola.
37
38
Section 2.1
Example 1
Match the quadratic function with its graph.
(a)
(b)
(c)
+3
(d)
(i)
(ii)
(iii)
(iv)
Section 2.1
Example 2
Sketch the graph of
intercept(s).
. Identify the vertex, axis of symmetry and -
39
40
Section 2.1
Standard Form of a Quadratic Function
Example 3
Sketch the graph of
. Identify the vertex, axis of symmetry and -intercept(s).
Section 2.1
Example 4
Write the standard form of the equation of the parabola whose vertex is
the point
.
and passes through
Example 5
Find a quadratic function whose graph has -intercepts
and
.
41
42
Section 2.1
In general, if
, then completing the square yields the standard form:
(Try it at home!)
From this standard form, we get a formula for the vertex of the parabola:
Many applications involve finding the maximum or minimum value of a quadratic function. The
maximum or minimum value occurs at the vertex.
Example 6
Find two positive real numbers whose product is a maximum if the sum of the first and three times
the second is 42.
Section 2.2
2.2 Polynomial Functions of Higher Degree
The graphs of polynomial functions are continuous with smooth rounded turns. This means there are
no jumps, holes, or sharp turns!
Identify which of the following could be the graph of a polynomial function.
The polynomial functions that have the simplest graph are of the form
(
is an integer)
Polynomial functions of this form are called _____________________________________.
If
is even, the graph of
If
is odd, the graph of
is similar to the graph of __________________.
is similar to the graph of __________________.
The greater the value of , the ___________________ the graph near the origin.
Example 1
Sketch the graph of each polynomial function.
(a)
(b)
43
44
Section 2.2
The Leading Coefficient Test (for determining “End Behavior”)
As moves without bound to the left or to the right, the graph of the polynomial function
eventually rises or falls in the following manner.
1. When
is odd:
Leading coefficient is positive
Leading coefficient is negative
The graph ____________ to the left
The graph ____________ to the left
and ____________ to the right.
and ____________ to the right.
2. When
is even:
Leading coefficient is positive
Leading coefficient is negative
The graph ____________ to the left
The graph ____________ to the left
and ____________ to the right.
and ____________ to the right.
Section 2.2
The following function notation is often used to indicate end behavior of a graph:
Falls to the left:
Falls to the right:
_____________________________________
_____________________________________
Rises to the left:
Rises to the right:
_____________________________________
_____________________________________
Example 2
Describe the right-hand and left-hand behavior of the graph
If
is a polynomial function and
.
is a real number, the following are equivalent (TFAE):
1.
2.
3.
4.
Consider the graph of
below. Observe the number of turning points, where
the graph “crosses” the -axis, and where the graph “touches” the -axis.
Factored form
:
45
46
Section 2.2
For a polynomial function
1.
of degree ,
has __________________________ real zeros
2. The graph of
has ______________________________ turning points.
Repeated Zeros
A factor
yields a repeated zero
of multiplicity .
1. If the multiplicity
is odd, the graph _______________ the -axis at
2. If the multiplicity
is even, the graph _______________ the -axis at
Note: If the multiplicity
, the graph will also _________________ at
How to Sketch the Graph of a Polynomial Function
1.
2.
3.
.
.
Section 2.2
Example 3
Sketch the graph of the function by applying the Leading Coefficient Test, finding the zeros of the
polynomial, and plotting any additional points as necessary.
(a)
(b)
47
48
Section 2.2
The Intermediate Value Theorem
Let and be real numbers such that
. If is a polynomial function such that
then, in the interval
, takes on every value between
and
.
Example 4
Use the Intermediate Value Theorem to approximate the real zero of
given the following table of values.
,
Section 2.3
2.3 Polynomial and Synthetic Division
Example 1
Use long division to divide
.
Example 2
Use long division to divide
.
If you need to review the long division process, also refer to Examples 1, 2 and 3 in the book.
49
50
Section 2.3
Synthetic Division is a consolidated algorithm (a.k.a. short-cut) for dividing a polynomial by
(where is any constant).
Illustrative Example
Divide
by
.
Solution:
Example 3
Use synthetic division to divide
-
-
-
-
.
Example 4
Consider
(a) Divide
by
.
. Then compare your answer with the function value
(b) Divide
by
. Then compare your answer with the function value
.
.
Section 2.3
From Example 4, we observe the following:
The Remainder Theorem
If a polynomial
is divided by
, the remainder is _____________.
Example 5
Use the Remainder Theorem and synthetic division to evaluate
Now, suppose
when
is a zero of
is divided by
if
, so that __________________. Then by the remainder theorem,
, the remainder is ________________. But this implies
_______________________________________.
The Factor Theorem
A polynomial
has a factor
Example 6
Determine if
is a factor of
if and only if __________________.
.
51
52
Section 2.3
Example 7
Use synthetic division to show that
and
are solutions of the equation
Then use the result to factor the polynomial completely. List all real solutions of the equation.
Section 2.4
2.4 Complex Numbers
History of Numbers as Solutions to Equations
Integers
rational numbers
irrational numbers
imaginary numbers
Historical Note: Imaginary numbers were first conceived and defined by the Italian
mathematician Gerolamo Cardano (1501-1576), who called them “fictitious”. We now
know the applications extend to engineering, physics, and applied mathematics.
Cardano was a friend of Leonardo da Vinci.
Definition
The imaginary unit is the number whose square is
. That is,
Definition
Complex numbers are numbers of the form ______________, where
The number
The number
is called the _____________________________.
is called the _____________________________.*
(*Other books define the imaginary part to be , not
*Note:
Write
instead of
Powers of
, and
instead of
.
)
and
are real numbers.
53
54
Section 2.4
Example 1
Evaluate.
Equality of Complex Numbers
if and only if
and
Sum or Difference of Complex Numbers
Example 2
Product of Complex Numbers
To find the product of two complex numbers
,
follow the usual rules for multiplying two binomials: FOIL
Example 3
Section 2.4
Definition
and
are called ______________________________________________.
Example 4
Multiply the number by its complex conjugate.
We use conjugates to write the quotient of complex numbers in standard from…
Example 5
Write the quotient in standard form (
).
Definition
If is a positive real number, the principal square root of
, denoted by
, is defined as
55
56
Section 2.4
Example 6
Write the complex number in standard form.
Example 7
Perform the indicated operation and write the result in standard form.
Example 8
Solve using the quadratic formula.
Section 2.5
2.5 Zeros of Polynomial Functions
The Fundamental Theorem of Algebra (FTA)
If
is a polynomial with positive degree, then _____________________________________
_____________________________________________________________________________.
Historical Note: The FTA was proved by arguably one of the greatest
mathematicians/physicists of all time, Carl Friedrich Gauss (1777-1855). The proof is
beyond the scope of this class.
Now observe the following:
If
is a polynomial of degree
, then by the FTA,
_________________________________________________.
Then by Factor Theorem, _________________________________________________.
Thus,
By the FTA again, _________________________________________________.
Then by Factor Theorem, _________________________________________________.
Thus,
We can continue this process until
is completely factored…
Linear Factorization Theorem
If
is an th-degree polynomial, then ___________________________________________.
ie. A polynomial of degree
be repeated)
has exactly
Example 1
List all the zeros of the function.
zeros. (The zeros may be real or complex, and they may
57
58
Section 2.5
Descartes’ Rule of Signs
Suppose
is a polynomial with real coefficients.
1) If the formula for
has variations in sign, there are either
positive real zeros of .
2) If the formula for
has variations in sign, there are either
negative real zeros of .
, or etc.
, or etc.
*A variation in sign means that two consecutive coefficients have opposite signs.
Example 2
Describe the possible real zeros of each function.
(a)
(b)
Rational Zero Test
Suppose
is a polynomial with integer coefficients. Then
the possible rational zeros are all numbers of the form
.
Example 3
List the possible rational zeros of the function.
where
is a factor of
and
is a factor of
Section 2.5
Example 4
Find all the rational zeros of the function.
59
60
Section 2.5
Example 5
Find all solutions of the polynomial equation.
Section 2.5
Definitions
A quadratic factor with no real zeros is said to be irreducible over the reals.
Example:
A quadratic factor with no rational zeros is said to be irreducible over the rationals.
Example:
Example 6
Write
(a) as the product of factors that are irreducible over the rationals.
(b) as the product of linear and quadratic factors that are irreducible over the reals.
(c) in completely factored form.
The Conjugate Pairs Theorem
If the polynomial function
has real coefficients, and
zero.
is a zero (
Example 7
Find a polynomial function with real coefficients that has the given zeros:
), then
and
is also a
61
62
Section 2.5
Example 8
Find all the zeros of
polynomial as a product of linear factors.
given that
is a zero. Then write the
Section 2.6
2.6 Rational Functions
Definition:
A rational function can be written in the form
where
and
are polynomials and
.
Example 1
Find the domain of
and discuss the behavior of
using limit notation.
Domain:
−100
Limits:
100
63
64
Section 2.6
Definition of Vertical and Horizontal Asymptotes
1. The line
as
is a vertical asymptote of the graph of
or
, either from the right or from the left.
2. The line
as
if
is a horizontal asymptote of the graph of
or
if
.
How to Find Vertical Asymptotes and Holes
1. Write the rational function
in lowest terms by factoring and canceling common factors.
2. The graph of
will have holes at any zeros which were common to both numerator and
denominator and thus canceled out.
3. The graph of
will have vertical asymptotes at the zeros of the simplified denominator.
Example 2
Find any vertical asymptotes and/ or holes of the graph:
(a)
(b)
(c)
Section 2.6
How to Find Horizontal/Slant Asymptotes:
Let
be the rational function given by
1. If
, the line
2. If
, the line
3. If
(the -axis) is a horizontal asymptote.
(the ratio of leading coefficients) is a horizontal asymptote.
is exactly one more than
, there is a slant asymptote which is found using long division:
Example:
Since as
4. If
is two or more than
the the slant asymptote is
, there is no horizontal or slant asymptote.
Example 3
Find any horizontal/slant asymptotes of the graphs:
(a)
(b)
(c)
.
65
66
Section 2.6
How to Graph a Rational Function
Let
, where
and
are polynomials.
1. Find and sketch the horizontal/slant asymptote, if any.
2. Simplify , if possible (factor and cancel common factors). Write
for the zeros of any
common factors that were canceled. There will be holes in the graph at these values.
3. Find and sketch the vertical asymptote(s), if any.
4. Find and plot the - and - intercepts.
5. Use the - intercepts and vertical asymptotes to divide the -axis into intervals.
Evaluate
at one test point in each interval, emphasizing if the value is or .
Plot these points on the graph.
6. Use smooth curves to complete the sketch of the graph. Use the following guidelines:
 The graph must remain above the -axis for intervals where the test value was ,
and below the - axis for intervals where the text value was .
 The graph must never cross a vertical asymptote, but should go to
near it.
 The graph must approach any horizontal or slant asymptote as
.
 Don’t forget to indicate any holes on the graph!
Section 2.6
Example 4
Graph
67
68
Section 2.6
Example 5
Graph
Section 2.6
Example 6
Graph
69
70
Section 3.1
3.1 Exponential Functions and Their Graphs
An exponential function
where
1. Graph
x
3
by plotting points.
f ( x)  2 x
1 1
23  3   0.125
2
8
2
1
0
1
2
23  8
3
2. Graph
by plotting points.
x
x
1
f ( x)   
2
3
1
3
  2 8
2
3
2
1
0
1
2
3
3
1 1
1
   3   0.125
8
2 2
,
with base
and
is denoted by
is any real number.
Section 3.1
The graph of f ( x )  a x
If a  1 , then f ( x )  a x is increasing.
If 0  a  1 , then f ( x )  a x is decreasing.
(1, a )
(0,1)
(0,1)
(1, a )
Base e
There is a certain irrational exponential base which occurs frequently in calculus applications,
the number e .
Definition:
The number e is defined as
 1
e  lim  1  
n 
 n
n
n
n
1

1  n 


1
 1
1    2
 1
10
1

 1    2.59374246
 10 
1
10
1,000
1,000
1 

 1  1,000 


 2.716923932
1,000,000
1,000,000
1


 1  1,000,000 


 2.718280469
As n gets larger, the approximation for e gets better!
For all intents and purposes, we can think of e as approximately 2.718.
Since e  1 , the graph of f ( x )  e x is an increasing exponential function.
71
72
Section 3.1
Graphing Exponential Functions
There are two techniques that can be used to sketch the graph of an exponential function. The
following two examples illustrate these techniques.
Example 1
Construct a table of values. Then sketch the graph of
Example 2
Use transformations to sketch the graph of
Section 3.1
Compound Interest
Principal – Total amount borrowed (or invested)
Compound Interest – When interest due (or earned) at the end of a payment period is added to the
principal, so that the interest computed at the end of the next payment period is based on this new
higher principal amount (old principal + interest), interest is said to have been compounded.
Annually – 1 time per year
Semiannually – 2 times per year
Quarterly – 4 times per year
Monthly – 12 times per year
Daily – 365* times per year
*In practice, most banks use a 360-day “year”
Simple Interest Formula
If a principal of P dollars is borrowed (or invested) for a period of t years at an annual
interest rate r, expressed as a decimal, the interest I charged (or earned) is:
Suppose we invest $100 in a Money Market account at an annual interest rate 6% for 1 year…
1) If interest is compounded annually, then:
 At the end of 1 year, we will earn the following in simple interest:
I  Pr t = 100(0.06)(1) = $6
2) If, however, interest is compounded semiannually, then:
 After 6 months (1/2 year), we will earn the following in simple interest:
Compounding #1
1
I  Pr t = 100(0.06)   = $3
2
Assuming this earned interest remains in the Money Market account, the new higher principal
amount is now:
$100  $3 =$103

After another 6 months, we will earn the following in simple interest:
1
Compounding #2
I  Pr t = 103(0.06)   = $3.09
2
So TOTAL interest earned after 1 year is $3  $3.09 = $6.09
73
74
Section 3.1
So we earned more than 6% interest annually with semiannually compounding because the interest
earned after the first compounding (6 months) made the principal HIGHER for the second
compounding!!
In fact, since we started with $100 and earned a total of $6.09 in interest after semiannually
compounding, we effectively earned 6.09% simple interest after 1 year.
We call 6.09% the effective interest rate.
Compound Interest Formula
The balance A in an account after t years due to a principal P invested at an annual interest
rate r (in decimal form) compounded n times per year is:
What happens to the balance as the number of times interest is compounded per year gets larger
and larger?? In other words, what happens as n   ?
n
 1
Recall, e  lim  1   .
n 
 n
n
r

In general, e  lim  1   .
n 
 n
r
Hence,
t
nt
n

t
r
r 

A  P   1    P   1     P  e r 
 n
 n  
as n  
Continuous Compounding Formula
The balance A in an account after t years due to a principal P invested at an annual interest rate r (in
decimal form) compounded continuously is :
Section 3.1
Example 3
Find the balance
in an account after 10 years if
a) monthly
b) continuously
is invested at 3% interest compounded
75
76
Section 3.2
3.2 Logarithmic Functions and Their Graphs
Since the exponential function
function:
Step 1: Replace
is one-to-one, we can compute the inverse
by .
Step 2: Interchange x and y.
Step 3: Solve for y.
It would be helpful to have a function that would give the power of
such a function and give it the name… logarithmic function!
that produces x. We create
Note: Since
, x must be positive!
is positive for all y, we have the result that for
Logarithmic Function
If
and
, then for
we have the following:
if and only if
The function given by
is called the logarithmic function with base .
Memory Device:
Logarithmic form into EXPONENTIAL form
logb x Logarithmic
y
means
Form

Exponential
Form
Exponential form into LOGARITHMIC form
means
Section 3.2
Example 1
Write each equation in exponential form.
(a)
(b)
(c)
Example 2
Write each equation in logarithmic form.
(a)
(b)
(c)
If the base of a logarithmic function is 10, we call this the common logarithmic function:
*Note: If the base of a logarithm is not indicated, it is understood to be 10.
If the base of a logarithmic function is e , we call this the natural logarithmic function:
*Note: This function occurs so frequently in applications that it is given a special symbol, ln.
Example 3
Evaluate the function at the indicated value of
(a)
;
Example 4
Use a calculator to evaluate
at
without using a calculator.
(b)
;
. Round your result to three decimal places.
77
78
Section 3.2
Properties of logarithms
1.
2.
3.
4.
and
If
then
.
Example 5
Use the properties of logarithms to simplify the expression.
(a)
(b)
(c)
Graphs of Logarithmic Functions
Example 6
The graph of
is provided below. Sketch the graph of:
Step 1. Replace
with
.
Step 2. Rewrite in exponential form.
Step 3. Plot points and graph.
x
y
Section 3.2
Example 7
The graph of
is provided below. Sketch the graph of:
Step 1. Replace
with
.
Step 2. Rewrite in exponential form.
Step 3. Plot points and graph.
x
y
Note: All logarithmic functions
pass through the points _________and _________
and have the y-axis as a vertical asymptote.
If
, then
is increasing.
If
, then
is decreasing.
79
80
Section 3.2
We can apply transformations to graph logarithmic functions.
Example 8
Sketch the graph of
.
Also, find the domain, -intercept, and vertical asymptote.
Example 9
Sketch the graph of
Also, find the domain, -intercept, and vertical asymptote.
Section 3.3
3.3 Properties of Logarithms
Here are some more properties of logarithms (the derivations are in the book on page 278). Do they
remind you of rules of exponents?
1. Product Property
_______________________________
2. Quotient Property
_______________________________
3. Power Property
_______________________________
Example 1
Assume that x, y, and z are positive numbers. Use the properties of logarithms to expand the
expression as a sum, difference, and/or constant multiple of logarithms.
(a)
(b)
(c)
81
82
Section 3.3
Example 2
Assume that x, y, z, and b are positive numbers. Use the properties of logarithms to condense the
expression to the logarithm of a single quantity.
(a)
(b)
Most calculators only have ln and log keys. So how should we approximate
of a value on a
calculator when a is neither e nor 10?
Change-of-Base Formula
Derivation of Change-of-Base Formula:
Let
Change to exponential form:
“Take the
of both sides”:
Apply Power Property:
Solve for y :
Example 3
Use the Change-of-Base Formula and a calculator to evaluate the following logarithm. Round your
answer to three decimal places.
Section 3.4
3.4 Exponential and Logarithmic Equations
Strategies for Solving Exponential and Logarithmic Equations
1. If possible, rewrite the original equation in a form that allows the use of the One-to-One
Property of exponential or logarithmic functions.
Example 1
Solve:
Example 2
Solve:
2. To solve an exponential equation, “take the log or ln of both sides”. Note it may be necessary
to isolate the exponential term first. Then apply the Power Property of logarithms.
To solve a logarithmic equation, condense all logarithms into a single logarithm on one side of
the equation, and then rewrite the equation in exponential form. Remember to check a
logarithmic equation for extraneous solutions!!
Example 3
Solve:
Example 4
Solve:
83
84
Section 3.4
Example 5
Solve.
Section 3.5
3.5 Exponential and Logarithmic Models
Example 1
An initial investment of $10,000 in a savings account in which interest is compounded continuously
doubles in 12 years. What is the annual % rate?
Example 2
Determine the time necessary for $100 to triple if it is invested at 4.5% compounded monthly.
85
86
Section 3.5
Law of Exponential (or Uninhibited) Growth/Decay
Many natural phenomena have been found to follow the law that an amount
according to the models
(growth)
or
(decay)
where
is the growth or decay rate, and
is the initial quantity at time
varies with time
.
Doubling time = Time for a population to double.
Half-life = Time for a radio-active substance to decay to half the original amount.
Example 3
The radioactive isotope
decays exponentially with a half-life of 1599 years. After 1000 years,
only 1.5 g of a sample remain. What was the initial quantity?
Section 3.5
Example 4
The number
of bacteria in a culture is modeled by
where is the time in hours. If
to double in size.
when
, estimate the time required for the population
87
88
Section 3.5
Other Applications
Example 5
Find the exponential model
whose graph passes through the points
and
.
Example 6
Use the acidity model given by
where acidity (pH) is a measure of the hydrogen ion concentration
for a solution in which
.
of a solution, to compute
Section 4.5
4.5 Graphs of Sine and Cosine Functions
One complete revolution of the unit circle is 2 , after which values of sine and cosine begin to
repeat.
The sine and cosine functions are periodic functions with period ____________.
_______________
_______________
Symmetric about ______________
___________ Function
Symmetric about ______________
___________ Function
The amplitude of a periodic function is half the distance between the maximum and minimum values
of the function.
The functions
and
have amplitude _______ and period _______.
Explanation:
Notice that when a is negative, the graph is reflected in the x-axis.
89
90
Section 4.5
Example 1
Find the period and amplitude.
(a)
(b)
To sketch the graph of
1. Method #1 Rewrite by factoring
Use the horizontal shift
*
out of the parentheses.
units and period
to find an interval of one period.
-ORMethod #2 Solve the 3-part inequality
to find an interval of one period.
2. Divide the period-interval into four equal parts to locate the 5 key points on the graph.
3. Either evaluate the function at the 5 key points on the graph, or use your knowledge of the
parent function, reflections, the vertical shift units, and the amplitude
to complete the
sketch of the graph.
*The process above may also be used to graph any transformation of
.
Section 4.5
Example 2
Sketch the graph of
Example 3
Sketch the graph of
. (Include two full periods.)
. (Include two full periods.)
91
92
Section 4.5
Example 4
After exercising for a few minutes, a person has a respiratory cycle for which the velocity of air flow is
approximated by
, where is the time (in seconds).
(a) Find the time for one full respiratory cycle.
(b) Find the number of cycles per minute.
(c) Sketch the graph of the velocity function.
Section 4.6
93
4.6 Graphs of Other Trigonometric Functions
Let’s graph y  tan x :
sin x
cos x
1
0  undefined
y  tan x 
x
sin x
cos x
 / 2
1
0
 / 3
 3/2
1/ 2
 / 4
 2 /2
2 /2
 / 6
1 / 2
3/2
0
0
 /6
1/ 2
3/2
1
 /4
2 /2
2 /2
0.5
 /3
3/2
1/ 2
3
1
2.5
2
1
1.5

2
  
3 4 6
 /2
1
2 / 3
3/2
1 / 2
3 / 4
2 /2
 2 /2
1.5
5 / 6
1/ 2
 3/2
2

0
1
7 / 6
1 / 2
 3/2
5 / 4
 2 /2
 2 /2
4 / 3
 3/2
1 / 2
3 / 2
1
0
0
Domain:
Range:
Period:
Vertical Asymptotes:
Symmetry:
0.5
1
2.5
3
1
1
0
 undefined
  
6 4 3

2
2 3 5
3 4 6

7 5 4
6 4 3
3
2
94
Section 4.6
Let’s graph y  cot x :
x
sin x
cos x

0
1
5 / 6
1 / 2
 3/2
3 / 4
 2 /2
 2 /2
2 / 3
 3/2
1 / 2
 / 2
1
0
 / 3
 3/2
1/ 2
 / 4
 2 /2
2 /2
 / 6
1 / 2
3/2
0
0
 /6
1/ 2
3/2
 /4
2 /2
2 /2
 /3
3/2
1/ 2
 /2
1
2 / 3
3/2
1 / 2
3 / 4
2 /2
 2 /2
5 / 6
1/ 2
 3/2

0
1
3  1.7
3
1
2
1

5 3 2
3
6
4

2
  
3 4 6
1
1
2
0
Domain:
Range:
Period:
Vertical Asymptotes:
Symmetry:
cos x
sin x
1
0  undefined
y  cot x 
3
1
0
 undefined
  
6 4 3

2
2 3 5
3 4 6

Section 4.6
To sketch the graph of
4. Solve the 3-part inequality
two consecutive vertical asymptotes.
*
to find an interval of one period and establish
5. At the midpoint between two consecutive vertical asymptotes is an -intercept of the graph.
Plot this point as well as two other points.
6. Sketch at least one additional cycle to the left or right.
To sketch the graph of
1. Solve the 3-part inequality
two consecutive vertical asymptotes.
*
to find an interval of one period and establish
2. At the midpoint between two consecutive vertical asymptotes is an -intercept of the graph.
Plot this point as well as two other points.
3. Sketch at least one additional cycle to the left or right.
*Note that Steps 2 and 3 are the same for both graphs.
Example 1
Sketch the graph of
.
95
96
Section 4.6
Example 2
Sketch the graph of
.
Graphs of the Reciprocal Functions
To get the graphs of the reciprocal functions
and
and
then take the reciprocals of the - coordinates.
*Note that
whenever
2  3   
2
2
is undefined whenever
, we first sketch
and
is undefined
.
0

2

3
2
2
2  3 
2


2
0

2

3
2
2
Section 4.6
Example 3
Sketch the graph of
Example 4
Sketch the graph of
.
.
97
98
Section 4.7
4.7 Inverse Trigonometric Functions
Sketch a quick graph of
. Does the function have an inverse function? ____________
However, if we restrict the domain to the interval ___________________, then
the restricted domain and thus we can define the inverse function!
Recall, in general,
is 1-1 on
if and only if _________________.
Inverse Sine Function
_________________________
Restriction: ________________________
Domain of
Range of
=
=
The graph of the inverse sine function is obtained by reflecting the restricted sine function about the
line
.
Sketch the graph of
Section 4.7
Example 1
If possible, find the exact value.
a)
b)
c)
********************************************************************************
Sketch a quick graph of
. Does the function have an inverse function? ___________
However, if we restrict the domain to the interval ___________________, then
on the restricted domain and thus we can define the inverse function!
Inverse Cosine Function
_________________________
Restriction: ________________________
Domain of
Range of
=
=
is 1-1
99
100
Section 4.7
The graph of the inverse cosine function is obtained by reflecting the restricted cosine function about
the line
.
Sketch the graph of
Example 2
If possible, find the exact value.
a)
b)
c)
Section 4.7
********************************************************************************
Sketch a quick graph of
. Does the function have an inverse function? ___________
However, if we restrict the domain to the interval ___________________, then
on the restricted domain and thus we can define the inverse function!
is 1-1
Inverse Tangent Function
_________________________
Restriction: ________________________
Domain of
Range of
=
=
The graph of the inverse tangent function is obtained by reflecting the restricted tangent function
about the line
.
Sketch the graph of
101
102
Section 4.7
Example 3
If possible, find the exact value.
a)
b)
Example 4
Use a calculator to approximate the value (if possible).
a)
b)
Example 5
Write as a function of .
5
c)
Section 4.7
Example 6
Evaluate the expression.
a)
b)
c)
Example 7
Find the exact value of the expression.
a)
b)
Example 8
Write an algebraic expression that is equivalent to the expression.
c)
103
104
Section 5.1
5.1 Using Fundamental Identities
Fundamental Trigonometric Identities
Reciprocal Identities
Quotient Identities
Pythagorean Identities
Divide by
:
Divide by
:
Cofunction Identities
Even/Odd Identities
Section 5.1
Example 1
Given the values
and
Method #1) Draw triangle
Method #2) Use the identities
, evaluate all six trig functions.
105
106
Section 5.1
Example 2
Simplify:
a)
b)
c)
Example 3
Factor, then simplify.
a)
b)
Section 5.1
Example 4
Perform the indicated operation, then simplify.
a)
b)
Example 5
Use the trig. substitution to write the algebraic expression as a trig function of , where
,
107
108
Section 5.2
5.2 Verifying Trigonometric Identities
Guidelines for Verifying Trig Identities:
1. Work with one side of the equation at a time. Start with the more complicated side. Keep in
mind the goal is to simplify this side to look like the other side of the equation!
2. Look for opportunities to factor an expression, find a common denominator to add fractions,
or create a monomial denominator by multiplying numerator and denominator by the same
quantity.
3. Use the reciprocal/quotient identities to convert all terms to sines and cosines if this helps
simplify the expression.
4. If both sides are complicated, work each side separately to obtain a common 3 rd expression.
Example 1
Verify the identity:
Example 2
Verify the identity:
Section 5.2
Example 3
Verify the identity:
109
110
Section 5.2
Example 4
Verify the identity:
Section 5.3
5.3 Solving Trigonometric Equations
When solving a trigonometric equation, your primary goal is to isolate the trigonometric function
involved in the equation.
It may be necessary to do one or more of the following:
1) arrange all terms on one side of the equation and factor
2) rewrite the equation with a single trig function using the identities
3) square both sides and remember to check your answer for extraneous solutions
********************************************************************************
Example 1
Solve the equation
Example 2
Solve the equation
111
112
Section 5.3
Example 3
Find all solutions of the equation in the interval
.
Example 4
Find all solutions of the equation in the interval
.
Section 5.3
To solve equations involving trigonometric functions of multiple angles of the forms
, first solve the equation for . Then isolate .
Example 5
Solve the equation
Example 6
Solve the equation
and
113
114
Section 5.4
5.4 Sum and Difference Formulas
Sum and Difference Formulas*:
*Proofs are on page 424 in the book.
Example 1
Find the exact value of
a)
b)
Section 5.4
Example 2
Find the exact value of
Example 3
Find the exact value of
Quadrant IV.)
given that
and
. (Both
and
are in
115
116
Section 5.4
Example 4
Verify the cofunction identity
Section 5.5
5.5 Multiple-Angle and Product-to-Sum Formulas
Double-Angle Formulas:
Derivation of Double-Angle Formula for Sine:
Example 1
Find
using the figure.
(The others are derived similarly.)
117
118
Section 5.5
Example 2
Solve in the interval
Example 3
Use the following to find
.
.
Section 5.5
Power-Reducing Formulas:
Derivation of Power-Reducing Formula for Sine:
Example 4
Rewrite
(The others are derived similarly.)
in terms of the first power of cosine.
119
120
Section 5.5
Half-Angle Formulas*:
*To derive, replace
Example 5
ind
Example 6
ind
given
with in the power reducing formulas.
The sign depends on the
quadrant in which lies.
Section 5.5
Product-to-Sum Formulas*:
*may be verified using the sum and difference formulas in the preceding section
Example 7
Write the product as a sum or difference.
Sum-to-Product Formulas*:
*Proofs are on page 426 in the book. Slightly complicated substitution is required.
121
122
Section 5.5
Example 8
Write the sum as a product.
Example 9
Verify the identity.
Section 6.3
6.3 Vectors in the Plane
Quantities such as force and velocity involve both magnitude and direction. We represent such
quantities by a vector:
Terminal point (Tip)
Initial point (Tail)
Its magnitude (or length) is denoted by
and can be found using the distance formula.
Vectors have no fixed orientation in the plane. They may be translated up down left or right. All that
defines a vector is its direction (angle) and magnitude (length).
Equal
Same Magnitude
Same Direction
Not Equal
Different Magnitudes
Not Equal
Different Directions
When we position a vector in the plane such that its initial point (or tail) is at the origin, we say that the
vector is in standard position. (Many books call such a vector a position vector).
The component form of a vector is:
where is the horizontal component
and is the vertical component.
Note: A vector with initial point
and terminal point
has component form
and magnitude
If
, then
is called a unit vector. If
, then
is called the zero vector and denoted .
123
124
Section 6.3
Example 1
Find the component form and the magnitude of the vector
.
The negative (or opposite) of
but opposite direction.
The scalar multiple of
the magnitude of .
times
with initial point
is the vector
is the vector
The sum of
and
is the vector
the resultant vector, which is formed by placing the initial point of
This is represented using the parallelogram law:
Note: The difference of
and
is the vector
and terminal point
which has the same magnitude,
which has magnitude
times
, often referred to as
at the terminal point of (“tip to tail”).
Section 6.3
Example 2
Let
and
. Find and sketch the following:
(a)
Let ,
(b)
(c)
Properties of Vector Addition and Scalar Multiplication
be vectors and let and be scalars. Then the following properties are true.
and
1.
2.
3.
4.
5.
6.
7.
8.
and
9.
If we divide a vector by its magnitude
, the resulting vector has a magnitude of 1 and the same
direction as . The vector is called a unit vector in the direction of .
Example 3
Find a unit vector in the direction of
.
125
126
Section 6.3
The unit vectors
and
are called the standard unit vectors and are denoted by
and
Sketch and :
In fact, any vector may be represented as a linear combination of the standard unit vectors as follows:
Example 4
Sketch the vector
Example 5
Let
and
. Then write the component form of .
. Find
.
Section 6.3
If
is a unit vector in standard position, then it’s tip lies on the unit circle. Label the diagram:
The angle
If
127
is the direction angle of the vector .
is any vector that makes an angle with the positive -axis, then
times , where is the unit vector with direction angle .
Note: If
has direction angle , then
.
is the scalar multiple of
128
Section 6.3
Example 6
Find the magnitude and direction angle of the vector:
a)
b)
Example 7
Find the component form of
given its magnitude and the angle it makes with the positive -axis.
and
Example 8
Find the component form of the sum of
and
with direction angles
,
,
and
.
Section 6.4
6.4 Vectors and Dot Products
Definition
The dot product of
Example 1
Find the dot product of
and
is
and
.
Properties of the Dot Product
1.
2.
3.
*There are other properties listed in the book, but we will only study these ones.
Proof of Property 1:
Proof of Property 2:
Proof of Property 3:
Example 2
Use the dot product to find the magnitude of
.
129
130
Section 6.4
The Angle Between Two Vectors
If is the angle (
) between two nonzero vectors
and , then
Or alternatively,
*The proof is on page 492 in the book and uses Law of Cosines.
Example 3
Find the angle between
and
.
Example 4
Use vectors to find the interior angles of the triangle with the given vertices:
Section 6.4
Note, if the angle between
Definition
The vectors
and
Example 5
Determine whether
and
is a right angle (or
), then
are orthogonal if
and
are orthogonal, parallel, or neither.
*Note: Two vectors are parallel if one vector can be written as a scalar multiple of the other.
a)
b)
and
and
131
132
Section 6.5
6.5 Trigonometric Form of a Complex Number
The Complex Plane
We may represent any complex number
follows:
as the point
Plot:
(a)
(b)
Definition
The absolute value of the complex number
Example 1
Find the absolute value of
.
is
in the complex plane as
Section 6.5
Trigonometric Form of a Complex Number
Find expressions for
and
in terms of and .
The trigonometric (or polar) form of the complex number
The number is the ____________________of , and
is
is called an _____________________of .
Note: The trigonometric form is not unique since there are infinitely many choices for . However
normally is given in radians and restricted to the interval
.
133
134
Section 6.5
Example 2
Write the complex number in trigonometric form.
a)
b)
Example 3
Write the complex number
in standard form.
Section 6.5
Suppose we are given two complex numbers:
and
Find the product.
Product and Quotient of Two Complex Numbers
Let
and
be complex numbers.
In words, to find the product, _______________________________________________________.
In words, to find the quotient, _______________________________________________________.
135
136
Section 6.5
Example 4
Perform the operation and leave the result in trigonometric form.
Powers of Complex Numbers
Let’s apply the product rule to find powers of .
Section 6.5
DeMoivre’s Theorem
If
is a complex number and
is a positive integer, then
Example 5
Use DeMoivre’s Theorem to find
The complex number
Example 6
Find all the fourth roots of 1.
. Write the result in standard form.
is an th root of the complex number if
137
138
Section 6.5
In general…
th Roots of Unity
The distinct th roots of 1 are called the th roots of unity.
The possible values for
here are the th roots of unity.
********************************************************************************
To find the th roots of unity:
Example 7
Find all the cube roots of unity.
Section 7.1
7.1 Linear and Nonlinear Systems of Equations
We can use substitution to solve both linear and nonlinear systems. Graphically, the solutions are the
points of intersection of the two graphs.
Example 1
Solve by substitution.
Sometimes it is convenient to solve a system graphically when it is difficult to solve by substitution.
Example 2
Solve by graphing.
139
140
Section 7.1
Break-Even Analysis
The total cost
of producing
The total revenue
units is given by the equation
from selling
units is given by the equation
Both of these equations describe a line. The break-even point occurs when __________________,
that is revenue cost, which corresponds to the point of intersection of the two graphs.
*Note: Profit
Revenue
Cost
Example 3
A restaurant invests $5000 to produce a new food item that will sell for $3.49. Each item can be
produced for $2.16.
(a) How many items must be sold to break even?
(b) How many items must be sold to make a profit of $8500?
Section 7.2
7.2 Two-Variable Linear Systems
A system of two linear equations containing two variables represents a pair of lines.
The points of intersection are the solutions of the system.
Intersect at exactly one point
Parallel
They are the same line
One solution
No solution
Infinite number of solutions
Consistent system
Independent system
Inconsistent system
Consistent system
Dependent system
We can use elimination to solve linear systems fast and efficiently.
Example 1
Solve
Example 2
Solve
141
142
Section 7.2
Example 3
Solve
Applications:
Wind Problems:
Speed of plane with the wind Plane airspeed Wind speed
Speed of plane against the wind Plane airspeed Wind speed
Example 4
An airplane flying into a headwind travels 250 miles in 2 hours and 25 minutes. The return flight
takes only 2 hours. Find the airspeed of the plane and the speed of the wind.
Section 7.2
In a free market, the demand for a product is related to the price of the product. As price decreases,
demand increases and thus the amount producers are able or willing to supply decreases.
The equilibrium point is the price
equations.
and number of units that satisfy both the demand and supply
Price per
unit,
Supply
Equilibrium
Point
Demand
Number of units,
Example 5
Demand equation:
Supply equation:
Find the equilibrium point.
143
144
Section 7.3
7.3 Multivariable Linear Systems
The following is an example of a multivariable linear system:
Refer to Sections 8.1, 8.3 and 8.5 for “matrix methods” used to solve multivariable linear systems.
Section 7.4
7.4 Partial Fractions
We call the sum on the RHS a partial fraction decomposition of the rational expression on the LHS.
How to Decompose a Rational Expression into Partial Fractions:
Step 1
If the rational expression is improper (ie. the degree of numerator the degree of denominator) first
divide using long division. Then proceed to decompose the proper rational expression.
Step 2
Completely factor the denominator into linear and irreducible (non-factorable) quadratic factors.
Step 3
For each linear factor of the form
, include in the decomposition
Step 4
For each irreducible quadratic factor of the form
, include in the decomposition
Step 5
Proceed to find the coefficients in the numerators as follows:
If the basic equation contains only linear factors…
1. Substitute the zeros of the distinct linear factors into the basic equation.
2. For repeated linear factors, use the coefficients determined in the previous step to rewrite
the basic equation. Then substitute other convenient values of and solve for the
remaining coefficients.
If the basic equation contains any quadratic factors…
1. Expand the basic equation.
2. Collect terms according to power of .
3. Equate the coefficients of like terms to obtain equations involving
4. Use a system of linear equations to solve for
and so on.
145
146
Section 7.4
In the following example, we will practice Steps 2 through 4 only of the decomposition process.
Once we have mastered “setting up” partial fraction decomposition problems, we will be ready to
move on to Examples 2-5 where we will complete the entire process.
Example 1
Write the form of the partial fraction decomposition of the rational expression. Do not solve for the
constants.
Section 7.4
Example 2
(Distinct Linear Factors)
Write the partial fraction decomposition of the rational expression.
147
148
Section 7.4
Example 3
(Repeated Linear Factors)
Write the partial fraction decomposition of the improper rational expression.
Section 7.4
Example 4
(Distinct Linear & Quadratic Factors)
Write the partial fraction decomposition of the rational expression.
149
150
Section 7.4
Example 5
(Repeated Quadratic Factors)
Write the partial fraction decomposition of the rational expression.
Section 7.5
7.5 Systems of Inequalities
How to Sketch the Graph of an Inequality:
1) Graph the boundary line by replacing the inequality with an equal sign. Use a dashed or solid
line for the boundary as follows:
2) Shade the correct region.
(One method for shading: Test a point not on the line in the original inequality. If a true
statement results, shade the region containing the test point. If a false statement results,
shade the OTHER region.)
Example 1
Sketch the graph of the inequality
151
152
Section 7.5
How to Sketch the Graph of a System of Inequalities:
1) Sketch the graph of each inequality separately.
2) The solution region is the region that is common to every graph in the system.
(Note: If there is no common region, the system of inequalities has no solution)
3) Find and label the vertices of the solution region and indicate each with either an open or
closed dot as appropriate. – WARNING: There is a mistake in many of the answers to HW
problems in the back of the book. The author of the answer key forgot to include “open dots”
when a vertex should not be part of the solution.
(Hint: To find the vertices of the solution region, solve the systems of corresponding
equations obtained by taking pairs of equations representing the boundary of the
solution region.)
Example 2
Sketch the graph (and label the vertices) of the solution set of the system.
Section 7.5
Example 3
Sketch the graph (and label the vertices) of the solution set of the system.
153
154
Section 8.1
8.1 Matrices and Systems of Equations
If
and
are positive integers, an
matrix is a rectangular array
in which each entry,
, of the matrix is a number. A matrix having rows and columns is said to
be of order
. Matrices are usually denoted by capital letters. The plural of matrix is matrices.
If a matrix has the same number of rows as columns, then the matrix is square of order
For a square matrix, the entries
are the main diagonal entries.
.
Circle the main diagonal entries on the square matrices given below:
square matrix:
square matrix:
A matrix that has only one row is called a row matrix, and a matrix that has only one column is called
a column matrix.
Example 1
Determine the order of each matrix.
Section 8.1
Consider the following system of equations:
System:
A matrix derived from a system of linear equations is the augmented matrix of the system.
Augmented Matrix:
The matrix derived from the coefficients of the system (but not including the constant terms) is the
coefficient matrix of the system.
Coefficient Matrix:
Example 2
Write the system of equations corresponding to the augmented matrix.
155
156
Section 8.1
Row-Echelon Form and Back Substitution
Consider the following two systems of linear equations. Write the corresponding augmented matrix
for each system.
Augmented Matrices:
The second system is said to be in row-echelon form, which means that the coefficient matrix has a
“stair-step” pattern with leading coefficients of 1.
A matrix is in row-echelon form when the following conditions are met:
1. The first nonzero entry in each row (called the lead entry) is 1.
2. Lead entries appear farther to the right as we move down the rows of the matrix.
3. Any rows containing only 0’s are at the bottom of the matrix
Examples:
It may be verified that both systems above have the same solution
,
, and
, which
can be written as the ordered triple
. Two systems of equations are equivalent if they have
the same solution set. Hence, the two systems given above are equivalent. After comparing the two
systems, it should be clear that it is easier to solve the system given in row-echelon form!
To solve a system that is not in row-echelon form, first convert it to an equivalent system that is in
row-echelon form by using the following operations.
Elementary Row Operations
Each of the following row operations performed on the augmented matrix of a system of linear
equations produces an equivalent system of linear equations.
1. Interchange two rows.
2. Multiply a row by a nonzero constant.
3. Add a multiple of a row to another row.
Section 8.1
Consider the system of equations:
1. Interchange two rows:
This corresponds to the following move involving the augmented matrix:
3 / 2 7 / 2 1 


2 1
 1
r1  r2
2. Multiply a row by a nonzero constant.
This corresponds to the following move involving the augmented matrix:
 1
2 1


3 / 2 7 / 2 1  R2  2r2
3. Add a multiple of a row to another row.
This corresponds to the following move involving the augmented matrix:
1 2 1


3 7 2 
R2  3r1  r2
After performing the above three row operations, we get an equivalent system in row-echelon
form. Now solve by back-substitution:
157
158
Section 8.1
To solve a system of linear equations using matrices, we perform elementary row operations on the
augmented matrix to obtain a matrix that is in row-echelon form. This method of solving a system of
equations is called Gaussian elimination with back-substitution:
Step 1:
Step 2:
Step 3:
Example 3
Write the augmented matrix of the system of linear equations.
Use elementary row operations to rewrite the augmented matrix in rowechelon form.
Write the system of linear equations corresponding to the matrix in rowechelon form, and use back-substitution to find the solution.
Use matrices and the method of Gaussian elimination to solve the system of equations.
 3x  3 y  3

8

4 x  2 y  3
Section 8.1
How to Solve a System of Linear Equations using Gaussian Elimination:
Step 1 Write the augmented matrix that represents the system.
Step 2 Perform row operations to get a leading 1 in the first row, first column entry.
1 a b i 


 c d e j
 f g h k 
Step 3 Perform row operations to get a column of 0’s below the first leading 1, while leaving the
first leading 1 unchanged.
1 a b i 


0 j k n 
0 l m o 
Step 4 Perform row operations to get a leading 1 in the second row, second column entry. (If this
entry is already 0, then get a leading 1 in the third column of this row.)
1 a b i 


0 1 p q 
0 l m o 
Step 5 Perform row operations to get a column of 0’s below the second leading 1, while leaving
the previous 1’s and 0’s unchanged .
1 a b i 


0 1 p q 
0 0 r s 
Step 6 Now repeat step 4, placing a 1 in the next row, but one column to the right. Continue
until the bottom row or the vertical bar is reached.
1 a b i 


Row Echelon Form
0 1 p q 
0 0 1 t 
Step 7 Write the system of equations and solve.
159
160
Section 8.1
Example 4
Use matrices and the method of Gaussian elimination to solve the system of equations.
 2 x  y  4

 2 y  4 z  0
 3x  2 z  11

Section 8.1
Example 5
Use matrices and the method of Gaussian elimination to solve the system of equations.
Example 6
Use matrices and the method of Gaussian elimination to solve the system of equations.
 2x  3y  z  0

 x  2 y  z  5
 3x  4 y  z  1

161
162
Section 8.1
A modification of Gaussian elimination, called Gauss-Jordan elimination, uses row operations to
produce a matrix in reduced row-echelon form. With this method, we can obtain the solution of the
system from that matrix directly, and back substitution is not needed.
A matrix is in reduced row-echelon form when the following conditions are met:
1. It is in row echelon form.
2. Entries above each lead entry are also zero.
Examples:
To get a matrix in reduced row echelon form requires the extra step of getting 0’s above and not just
below leading 1’s.
Example 7
Use matrices and the method of Gauss-Jordan elimination to solve the system of equations.
Section 8.1
Example 8
Use matrices and the method of Gauss-Jordan elimination to solve the system of equations.
163
164
Section 8.2
8.2 Operations with Matrices
In Section 8.1, we used matrices to solve systems of linear equations. There is a rich mathematical
theory of matrices, and its applications are numerous. This section and the sections that follow
introduce some fundamentals of matrix theory.
Matrix Representation
A matrix can be denoted by:
1. an uppercase letter such as .
2. a representative element enclosed in brackets, such as
3. a rectangular array of numbers.
Two matrices
and
entries are equal, that is
.
are equal if they have the same order and their corresponding
.
Example 1
Find and
Definition of Matrix Addition
If
and
are matrices of order
, their sum is the
matrix given by
.
In other words, add two matrices by adding their corresponding entries. The sum of two
matrices of different orders is undefined.
Example 2
(a)
Section 8.2
(b)
In operations with matrices, numbers are usually referred to as scalars.
Definition of Scalar Multiplication
If
is an
matrix and is a scalar, the scalar multiple of
given by
In other words, multiply a matrix
Note: The symbol –
by a scalar by multiplying each entry in
represents the scalar product
. Thus,
Example 3
Suppose that
(a)
(b)
and
by is the
. Find the following:
by .
matrix
165
166
Section 8.2
Let
and
be
Properties of Matrix Addition and Scalar Multiplication
matrices and let and be scalars.
1.
2.
3.
4.
5.
6.
If
is an
matrix and
is the
zero matrix:
Example 4
Solve for in the matrix equation
zero matrix consisting entirely of zeros, then
and
zero matrix:
, where
and
.
Section 8.2
Definition of Matrix Multiplication
If
is an
matrix and
matrix
is an
where
matrix, the product
is an
.
In other words, the entry in the th row and th column of the product
is obtained by
multiplying the entries in the th row of by the corresponding entries in the th column of
then adding the results. We can think of this as a dot product of rows of by columns of :
Example 5
and
If possible, find each of the following products.
(a)
(b)
and
167
168
Section 8.2
Example 6
,
,
, and
If possible, find each of the following products and state the order of the result.
(a)
(b)
(c)
Section 8.2
Let
and
Properties of Matrix Multiplication
be matrices and let be a scalar.
1.
2.
3.
4.
Note: In general,
.
Definition of Identity Matrix
The
square matrix that consists of 1’s on its main diagonal and 0’s elsewhere is called the
identity matrix of order
and is denoted by . When the order is understood from the context,
you can denote the identity matrix simply by .
;
Example 7
If
, find the following:
(a)
(b)
The identity matrix has the property that
.
169
170
Section 8.4
8.4 The Determinant of a Square Matrix
Determinant of a
Matrix
The determinant of the matrix
is given by
*Note: The determinant of a square matrix is a real number.
Example 1
Find the determinant of each matrix.
(a)
(b)
Example 2
Solve for .
Section 8.4
To define the determinant of a square matrix of order
the concepts of minors and cofactors.
or higher, it is convenient to introduce
Minors and Cofactors of a Square Matrix
If is a square matrix, the minor
of the entry
is the determinant of the matrix obtained by
deleting the th row and th column of . The cofactor
of the entry
is equal to the minor
with the “correct sign” attached.
Sign pattern for cofactors (in
Example 2
Find all the minors and cofactors of
case):
171
172
Section 8.4
Determinant of a Square Matrix
If is a square matrix (of order
or greater), the determinant of is the sum of the entries in
any row (or column) of multiplied by their respective cofactors. For instance, expanding along the
first row yields
Applying this definition to find a determinant is called expanding by cofactors.
Example 3
Find the determinant of the matrix. Expand using the indicated row or column.
(a) Row 1
(b) Column 2
Section 8.4
Example 4
Find the determinant of the matrix.
173
174
Section 8.3
8.3 The Inverse of a Square Matrix
Definition of the Inverse of a Square Matrix
Let be an
matrix and let be the
Then
identity matrix. If there exists a matrix
is called the inverse of , and is denoted with the symbol
such that
.
***********************************************************************************
Note: Only square matrices can have an inverse. However, not all square matrices have an inverse.
If a matrix has an inverse, it is called invertible (or ______________________________).
Otherwise, if a matrix
does not have an inverse, it is called ______________________________.
Example 1
Show that is the inverse of A.
,
Section 8.3
Finding an Inverse Matrix
If is
, the elementary row operations performed on the augmented matrix
will
transform it into
. If it is not possible to row reduce to , then is not invertible and thus
does not have an inverse.
Refer to Example 2 (pg. 600) in the book for the rationale of this method.
Example 2
Find the inverse of the matrix (if it exists).
175
176
Section 8.3
Example 3
Find the inverse of the matrix (if it exists).
Finding the Inverse of a
Matrix
(Using the Formula)
If
is a
matrix given by
the inverse can be found using the formula
Note: If
, then
does not have an inverse.
What is the determinant of the matrix in Example 3?
Section 8.3
Example 4
Find the inverse of the matrix (if it exists). Use the formula.
Example 5
Find the inverse of the matrix (if it exists). Use the formula.
177
178
Section 8.3
Consider the system of linear equations:
The system can be represented by the matrix equation
, where is the coefficient matrix of
the system, and and are column matrices. The column matrix is called a constant matrix.
Verify that this matrix equation represents the system of linear equations.
Thus, solving the matrix equation
Solve for in the matrix equation:
for
will yield the solution to the system of equations.
Solving a System of Equations Using an Inverse Matrix
If is an invertible matrix, the system of linear equations represented by
solution given by
has a unique
Section 8.3
Example 6
Use the inverse matrix found in Example 2 to solve the system of linear equations.
179
180
Section 8.5
8.5 Applications of Matrices and Determinants
So far, we have studied 2 methods for solving a system of linear equations using matrices:
1) Gaussian (or Gauss-Jordan) Elimination performed on the augmented matrix
2) Using an Inverse Matrix to solve the corresponding matrix equation
In this section, we will study one more method, Cramer’s Rule, named after Gabriel Cramer (17041752).
Consider the system of 2 equations
We define the following determinants:
***********************************************************************************
Now, for a system of 3 equations
we have the following:
Section 8.5
If a system of linear equations in
, the solution of the system is
Cramer’s Rule
variables has a coefficient matrix
with a nonzero determinant
If the determinant of the coefficient matrix is zero (ie.
), the system has either no solution or
infinitely many solutions. In such a case, Cramer’s Rule does not apply, and another technique should
be used to solve the system.
Example 1
Use Cramer’s Rule to solve the system of linear equations.
181
182
Section 8.5
Example 2
Use Cramer’s Rule to solve the system of linear equations.
–
Hint:
Use the following information to save time on this problem.
Section 8.5
Another application of determinants is finding the area of a triangle.
The area of a triangle with vertices
where the appropriate sign
Example 3
Find the area of the triangle.
Area of a Triangle
,
, and
is
should be chosen to yield a positive area.
183
184
Section 9.1
9.1 Sequences and Series
Definition of Sequence
An infinite sequence is a function whose domain is the set of positive integers.
The function values
are the terms of the sequence. If the domain of the function consists of the first
only, the sequence is a finite sequence.
positive integers
Examples:
(finite sequence)
*
(infinite sequence)
*This last sequence is known as the Fibonacci sequence.
Note: On occasion it is convenient to begin subscripting a sequence with 0 instead of 1 so that the
terms of the sequence become
When this is the case, the domain includes 0.
Example 1
Write the first four terms of the sequence. (Assume that
(a)
begins with 1.)
(b)
Section 9.1
Example 2
Write an expression for the apparent th term of the sequence. (Assume that
(a)
(b)
(c)
begins with 1)
185
186
Section 9.1
A sequence can be defined recursively by giving one or more of the first few terms and a rule
showing how to obtain the next term from the previous term(s).
Example 3
Find the first five terms of the sequence defined recursively.
Definition of Factorial
If is a positive integer,
factorial is defined as
As a special case, zero factorial is defined as
Section 9.1
Example 4
Write the first five terms of the sequence. (Assume that
Note, based on the definition of factorial:
Example 5
Simplify the factorial expression.
begins with 0.)
187
188
Section 9.1
The sum of the first
Summation Notation (or Sigma Notation)
terms of a sequence is represented by
where is called the index of summation,
the lower limit of summation.
is the upper limit of summation, and in this case 1 is
Examples:
Note that the lower limit of a summation does not have to be 1. Also note that the index of
summation does not have to be the letter .
Example 6
Find each sum.
Section 9.1
Example 7
Use sigma notation to write the sum
Note: The representation of a sum using sigma notation is not unique. Variations in the upper and
lower limits of summation can produce quite different-looking summation notations for the same
sum.
Example 8
Find two representations for the following sum
(a) Use 1 as the lower limit of summation.
(b) Use 0 as the lower limit of summation.
189
190
Section 9.1
Definition of Series
Consider the infinite sequence
1. The sum of the first terms of the sequence is called a finite series (or the nth partial sum of
the sequence) and is denoted by
2. The sum of all the terms of the infinite sequence is called an infinite series and is denoted by
Example 9
Use a calculator to find the sum of the following finite series.
Based on your answer in Example 7, can you guess the following sum of the infinite series?
Section 9.2
9.2 Arithmetic Sequences and Partial Sums
Consider the sequence:
2, 5, 8, 11, 14, …
Compute the differences of consecutive terms:
Each term (except the first term) is found by adding the constant
_____ to the preceding term.
An arithmetic sequence is a sequence of the form
______
______
______
______
where there is a common difference
between consecutive terms.
The th term of an Arithmetic Sequence
Example 1
Write the first five terms of the arithmetic sequence with
the th term of the sequence.
and
. Then find a formula for
191
192
Section 9.2
Example 2
Find the 100th term of an arithmetic sequence with
and
.
Example 3
Write the first five terms of the arithmetic sequence defined recursively:
Then find the common difference and write the th term of the sequence as a function of .
Section 9.2
The Sum of a Finite Arithmetic Sequence (nth partial sum)
Let
represent the sum of the first
terms of an arithmetic sequence.
Then,
Note: This formula only works for arithmetic sequences!
Historical Note:
An elementary school teacher of a young Carl Friedrich Gauss (1777-1855) asked him to add all the
integers from 1 to 100. When Gauss returned with the correct answer after only a few moments, the
teacher could only look at him in astounded silence.
This is what it is speculated that Gauss did:
193
194
Section 9.2
Example 4
Find the 10th partial sum of the arithmetic sequence
Example 5
Find the partial sum.
Section 9.2
Example 6
Determine the seating capacity of a movie theater with 36 rows of seats if there are 15 seats in the
first row, 18 seats in the second row, 21 seats in the third row, and so on.
195
196
Section 9.3
9.3 Geometric Sequences and Series
Consider the sequence:
2,
,
,
,
, …
Compute the ratios of consecutive terms:
Each term (except the first term) is found by multiplying the preceding term by the constant
A geometric sequence is a sequence of the form
______
______
______
______
where there is a common ratio between consecutive terms.
The th term of a Geometric Sequence
_____.
Section 9.3
Example 1
Write the first five terms of a geometric sequence with
the th term of the sequence.
and
. Then find a formula for
Example 2
Write the first five terms of the geometric sequence. Determine the common ratio and write the th
term of the sequence as a function of .
197
198
Section 9.3
Example 3
Find the 6th term of a geometric sequence if
sequence are positive.
Method #1 (Brute Force)
Method #2 (Algebraically)
and
. Assume that the terms of the
Section 9.3
Let
Then for
The Sum of a Finite Geometric Sequence ( th Partial Sum)
represent the sum of the first terms of a geometric sequence.
,
Note: This formula only works for finite geometric sequences!
Example 4
Find the sum of each finite geometric sequence.
199
200
Section 9.3
The Sum of an Infinite Geometric Series
represent the sum of all the terms of an infinite geometric
Let
sequence.
Then for
*Note: If
,
, the series does not have a sum.
Note: This formula only works for infinite geometric sequences!
Example 5
Find the sum of each infinite geometric series.
Section 9.3
Recall, the balance in an account after years due to a principal
rate (in decimal form) compounded times per year is:
If a deposit is compounded monthly, then
(
years) is given by
invested at an annual interest
, so the balance in the account after
months
Example 6
A deposit of $50 is made at the beginning of each month* in an account that pays 8%, compounded
monthly. The balance in the account at the end of 4 years (or 48 months) is
Find .
*This type of savings plan is called an increasing annuity.
201
202
Section 9.4
9.4 Mathematical Induction
The Principle of Mathematical Induction
If a statement involving the positive integer has the following two properties:
1. The statement is true for
, and
2. If the statement is true for
, then it is true for
then the statement is true for all positive integers.
How to Prove a Statement using Mathematical Induction
1. Base Step: Show the statement is true for
.
2. Induction Step: Assume the statement is true for some integer
. Show that this
implies the statement is true for the next integer
.
Example 1
Use induction the prove that the following formula is true for every positive integer .
Base Step:
Induction Step:
Section 9.4
Example 2
Use induction the prove that the following formula is true for every positive integer .
Base Step:
Induction Step:
203
204
Section 9.4
Example 3
Use induction the prove that the following formula is true for every positive integer .
Base Step:
Induction Step:
Section 9.5
9.5 The Binomial Theorem
In this section, we learn how to expand binomials of the form
.
Observe:
Notes:
1) In each expansion there are ___________ terms.
2) In each expansion,
and
have symmetrical roles. The powers of
in successive terms, whereas the powers of
_______________ by ____
_______________ by ____.
3) The sum of the powers of each terms is _________.
4) The coefficients increase and then decrease in a symmetrical pattern as represented by
_______________________________________ illustrated below:
*Each entry (other than the 1’s) is the sum of the closest pair of numbers in the line above it.
The Binomial Theorem
If
is any positive number, then
where the coefficients, called binomial coefficients, are defined by
and may either be found using the nth row of Pascal’s Triangle or computed on a scientific calculator
using the
feature.
205
206
Section 9.5
Note: The Binomial Theorem may be expressed using sigma notation. Also, the symbol
place of
to denote binomial coefficients.
Hence, the following is an alternative form of the Binomial Theorem:
The Binomial Theorem (using Sigma Notation)
Example 1
Use the Binomial Theorem to expand and simplify the expression
(a)
(b)
is often used in
Section 9.5
Example 2
Use the Binomial Theorem to expand the complex number. Simplify your result.
207
208
Section 10.2
10.2 Introduction to Conics: Parabolas
A conic section (or conic) is the intersection of a plane and a double-napped cone.
There are 4 basic types:
Geometric properties of conics were discovered during the classical Greek period (600 to 300 B.C.)
When the plane passes through the vertex, the resulting figure is a degenerate conic. The result is
either a point, line or two intersecting lines.
General Equation of Conics
For example, a circle is defined as the collection (locus) of all points
fixed point
.
which is equivalent to
that are equidistant from a
Section 10.2
We have already discussed quadratic functions
that open up or down. They can be written in the form:
vertex.
whose graphs are parabolas
where
is the
We now discuss parabolas in a more general sense, where they may open up, down, left or right.
Definition of Parabola
A parabola is the set (locus) of all points
in a plane equidistant from a fixed line (called the
directrix) and a fixed point (called the focus) that is not on line . The vertex and axis (of
symmetry) are defined as before.
Sketch of a parabola using the definition:
209
210
Section 10.2
Activity
Let’s derive the equation of a parabola (using the definition)! For simplicity, we will derive the
equation of a parabola with:
Vertex,
Focus,
Directrix,
Directrix
1. The distance from
to
, denoted
2. The distance from
to the directrix
, is found using the distance formula:
is ___________________.
3. By definition of a parabola, the distances found in steps (1) and (2) are equal. Thus,
4. Now solve for
(in terms of
and ). Simplify your answer by FOILing.
Section 10.2
Standard Equation of a Parabola
with Vertex
:
Opens Up
or Down
Focus:
Directrix:
Axis of Symmetry: Vertical,
:
-axis
distance from vertex to focus
Opens Right
or Left
Focus:
Directrix:
Axis of Symmetry: Horizontal,
-axis
distance from vertex to directrix.
Example 1
Find the standard equation of the parabola with vertex at the origin and focus
Example 2
Find the vertex, focus, and directrix of the parabola
:
.
and sketch its graph.
211
212
Section 10.2
Translations can be applied to parabolas with vertex
to obtain parabolas with vertex
Standard Equation of a Parabola
with Vertex
:
Opens Up
or Down
:
Focus:
Directrix:
Axis of Symmetry: Vertical,
distance from vertex to focus
Opens Right
or Left
:
Focus:
Directrix:
Axis of Symmetry: Horizontal,
distance from vertex to directrix.
Example 3
Find the standard equation of the parabola with vertex at
and directrix
.
:
Section 10.2
Example 4
Find the vertex, focus, and directrix of the parabola and sketch its graph.
213
214
Section 10.3
10.3 Ellipses
An ellipse is the set of all points
in a plane such that the sum of the distances from
distinct fixed points and (called the foci) is constant.
to two
’
where:
1/2 the length of the major axis distance from the center
1/2 the length of the minor axis
distance from the center to a focus
to a vertex
Label , , and on the following ellipse:
’
Interesting Fact!
Any light or sound that starts at one focus of an ellipse will be reflected through the other.
Some examples:
Whispering galleries (Statuary Hall in Washington D.C. – originally chamber of House of Reps)
Lithotripsy – kidney stone at one focus is pulverized by shock waves originating at the other focus.
Section 10.3
Standard Equation of an Ellipse
with Center
Center:
, Foci:
, Vertices:
Major axis: Horizontal (along -axis)
Center:
, Foci:
, Vertices:
Major axis: Vertical (along -axis)
Example 1
Find the equation of the ellipse with center at the origin, major axis of length 10 along the -axis, and
minor axis of length 8.
215
216
Section 10.3
Example 2
Find the equation of the ellipse with center at the origin, focus at
, and vertex at
Standard Equation of an Ellipse
with Center
Center:
, Foci:
, Vertices:
Major axis: Horizontal (parallel to -axis)
Center:
, Foci:
, Vertices:
Major axis: Vertical (parallel to -axis)
.
Section 10.3
Example 3
Find the standard form of the equation of the ellipse with focus at
.
Example 4
Find the center, vertices and foci of the ellipse, and sketch its graph.
and vertices at
and
217
218
Section 10.3
Example 5
Find the center, vertices and foci of the ellipse, and sketch its graph.
Section 10.4
10.4 Hyperbolas
A hyperbola is the set of all points
in a plane such that the difference of the distances from
to two distinct fixed points and (called the foci) is a positive constant.
’
’
where:
distance from the center to a vertex
1/2 the length of the conjugate axis
distance from the center to a focus
Label , , and on the following hyperbola:
’
219
220
Section 10.4
Standard Equation of a Hyperbola
with Center
:
Center:
, Foci:
, Vertices:
Transverse axis: Horizontal (along -axis)
Asymptotes:
Center:
, Foci:
, Vertices:
Transverse axis: Vertical (along -axis)
Asymptotes:
’
’
Example 1
Find the standard equation of the hyperbola with center at the origin, vertices
foci
and
.
and
and
Section 10.4
Standard Equation of a Hyperbola
with Center
Center:
, Foci:
, Vertices:
Transverse axis: Horizontal (parallel to -axis)
Asymptotes:
Center:
, Foci:
, Vertices:
Transverse axis: Vertical (parallel to -axis)
Asymptotes:
’
’
Example 2
Find the standard equation of the hyperbola with vertices
.
and
and foci
and
221
222
Section 10.4
Example 3
Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola, and sketch its
graph.
Example 4
Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola.
a.
b.
c.
d.
Section 10.4
Example 5
Find the center, vertices and foci of the hyperbola, and sketch its graph.
223
224
Section 10.6
10.6 Parametric Equations
Consider the following path followed by a little ant on the ground who discovers a piece of cake and
then proceeds to bring a piece back to the ant hill:
Ant Hill
Piece of Cake
The ant follows the parabolic path
,
(Rectangular Equation)
However, this equation does not tell the whole story. Although it does tell you where the ant has
been, it doesn’t tell you when the ant was at a given point
on the path.
To determine this time, we can introduce a third variable , called a parameter. It is possible to write
both and as functions of to obtain the parametric equations
ill in the table to find the ant’s position at time (in minutes).
4
6
How long does it take the ant to reach the piece of cake from his starting point? ____________
How long does it take the ant to travel from the piece of cake to the ant hill? ____________
Section 10.6
225
If and are continuous functions of on an interval , the set of ordered pairs
is a
plane curve . The equations
and
are parametric equations for , and is the parameter. Plotting the points in the order of increasing
values of traces the curve in a specific direction. This is called the orientation of the curve.
Example 1
Consider the parametric equations
a) Sketch the curve represented by the parametric equations (indicate the orientation of the curve)
b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents
the curve.
Hint: Solve for in one equation. Then substitute in other equation & simplify.
c) Adjust the domain of the resulting rectangular equation, if necessary.
226
Section 10.6
The next example illustrates one of the most useful applications of parametric equations. If we let the
parameter be the angle , we may describe and graph circular and elliptical curves with ease.
To eliminate the parameter in equations involving trigonometric functions, try using the identity
Example 2
Consider the parametric equations
Fill in the table of values.
Eliminate the parameter and write the corresponding rectangular equation whose graph represents the
curve. Sketch the curve represented by the parametric equations (indicate the orientation of the curve).
Section 10.6
Example 3
Consider the parametric equations
Sketch the curve represented by the parametric equations (indicate the orientation of the curve).
Eliminate the parameter and write the corresponding rectangular equation whose graph represents the
curve.
227
228
Section 10.7
10.7 Polar Coordinates
Rectangular Coordinate System
-axis
Polar Coordinate System
-axis
Polar axis
Pole
Each point
in the plane can be assigned polar coordinates
as follows:
from
directed
distance
, counterclockwise
from polar axis to segment
directed angle
Polar axis
Example 1
Plot the point given in polar coordinates.
a.
b.
c.
to
Section 10.7
Example 2
Find three additional polar representations of the point
Because
using
.
lies on a circle of radius , it follows that:
,
Coordinate Conversion
The polar coordinates
,
are related to the rectangular coordinates
Polar-to-Rectangular
as follows.
Rectangular-to-Polar
Example 3
Convert the point
given in polar coordinates to rectangular coordinates.
229
230
Section 10.7
Example 4
Convert the point given in rectangular coordinates to polar coordinates.
a.
b.
To convert a rectangular equation to polar form, simply replace
Example 5
Convert the rectangular equation to polar form.
by
and
by
.
Section 10.7
To convert a polar equation to rectangular form requires a little more ingenuity.
Example 6
Convert the polar equation to rectangular form.
a.
b.
c.
d.
231
232
Section 10.8
10.8 Graphs of Polar Equations
Polar Equation Form:
Example 1
Sketch the graph of the polar equation
A table of values has been provided to save time.
4
3.5
2
0
Section 10.8
Polar Equation Form:
Example 2
Sketch the graph of the polar equation
Restriction on
A table of values has been provided to save time.
233
234
Section 10.8
Polar Equation Form:
Example 3
Sketch the graph of the polar equation
A table of values has been provided to save time.
1
0.5
0
0
1
Section 10.8
Polar Equation Form:
Example 4
Sketch the graph of the polar equation
A table of values has been provided to save time.
4
3.7
3
2
1
0.3
0
235
236
Section 10.8
Polar Equation Form:
Example 5
Sketch the graph of the polar equation
A table of values has been provided to save time.
0
3.5
5
3.5
0