4.4 Evaluating Logarithms and change of Base Theorem
* common Logarithms
for all positive numbers x
Note:
If
if
if
* Natural Logarithm
for all positive numbers x
* Change of base Theorem
for any positive real numbers x, a and b
Ex: Evaluate
1.
2.
3.
4.
5.
7.7 Inverse trigonometric equations:
Ex: Solve
Ex: solve
Let
=
A is in 1st quadrant
Ex. solve
let
A is in 1st quadrant
Ex: solve
it has no solution
Ex: Solve.
Chapter 7 Trigonometric identity and equations (2) Page 1
let
A is in 1st quadrant
EX: solve
let
7.6 Trigonometric equations:
A Chart for
special angles
Solving by linear methods:
Ex: solve
over
or
Solving by Quadratic methods:
Ex: Solve
over
or
No solution
Solving by trigonometric identities:
Ex: Solve
over
Math002 Page 1
Check.
False
True
Ex: So
9.1 systems of linear equations
Linear system
A set of equations is called a system and the solutions of the system of equations must satisfy
every equation in the system If each equation in a system is linear then the system is a system
of linear equati
7.5 Inverse circular functions
Inverse function
1. In a 11 function each x is related to only one y and each y is related to only one x
value
2. If f is a 11 function then f has an inverse function
Domain ( f ) = Range (
) Range ( f )=Domain
3. The grap
9.2 Matrix solutions of linear systems
The Gauss Jordan method
Taking the coefficients and constants and arranging them in a matrix as follows
coefficients
constants
this is called the augmented matrix of the system
_ Matrix row transformations
for any a
9.3 Determinant solution of Linear systems
Determinants:
Every nxn matrix A is associated with a real number called the determinant
of A written
1. Determinant of 2X2 matrix
If A =
Then
Ex: Let
find
2. Determinant of a 33 matrix
Method 1
If
+
+.
+.


t
9.5 Nonlinear systems of equations
1. .solving a system of two equations with two variables
with one of the equations is Linear
Ex: solve
solution:
take the linear equation and solve for x or for y
replace it in equation 1 which is the equation
that we di
9.8 Matrix Inverses

The identity matrix
Ex: Let
show that
Solution:
Multiplicative inverses:
If A is an nxn matrix and
there is a matrix A such that
is called the inverse of matrix A
Finding the inverse of a matrix A
1. Form the matrix
2. Perform the
9.7 Properties of Matrices
a matrix (plural matrices) is a rectangular array of numbers, symbols, or
expressions, arranged in rows and columns.
The individual items in a matrix are called its elements or entries
The size of a matrix is the number of rows
10.1 Parabolas
Definition:
AParabola is a set of points whose distance from a fixed point called (focus) equals its
distance from a fixed line called (directrix)
FM = MD
Screen clipping taken: 7/24/2012 10:46 PM
Ex: Find the vertex, focus, axis, directrix
Reduction formula
* Functions of the form
this function can be written in the form
Where
Ex: Rewrite
in the form
Solution,
Ex: Graph
x
Chapter 7 Trigonometric identity and equations (2) Page 1
x
x+
y
0
2
0
2
0
Ex: find the phase shift of
phase shift =
Ex
10.2 Ellipses
Definition: An ellipse is the set of points in a plane the sum of whose distances from two
fixed points called (foci) is constant
horizontal ellipse
vertical ellipse
An ellipse centered at
With vertical
major axis of length 2a has equation
v
103 Hyperbolas
In the graph of each hyperbola considered so far, the center is the origin and the asymptotes pass
through the origin. This feature holds in general; the asymptotes of any hyperbola pass through the
center of the hyperbola. Like an ellipse
7.4 Double angle and half angle Identities
Double angle Identities
Ex: Given that
and
Find:
X is in 2nd quadrant
8
Chapter 7 Trigonometric identity and equations (2) Page 1
17
Ex: Find the values of the six trigonometric functions of
if
and
Ex: find the v
Chapter 5
5.1
Angles:
An angle consists of two rays or two segments in a plane with a common end point
moves anticlockwise
moves clockwise
*The degree measure of an angle
1 rotation =360 degrees
* Types of angles:
Definition:
1. Complement angles: their s
4.1 Inverse Functions
Onetoone Functions:
A function
i.e. each xvalue corresponds to only one yvalue, and each yvalue correspond to
only one Xvalue
Note: A onetoone function is either increasing or decreasing
Ex: Decide whether each function is 1
4.3 Logarithmic Functions
Definition:
For all real numbers y and all positive numbers
a and x where
if and only if
argument
base
EX: Solve each equation
1.
2.
3.
Definition:
If
and
then
defines the logarithmic function with base a
Note:
The exponential fu
4.2 Exponential Functions:
Exponential properties
Ex: Evaluate
Ex: If
find
* Exponential Functions
1. If
,
Domain =
Range =
key
points
Math002 Page 1
1. If
,
Domain =
Range =
key
points
2. If
Domain=
Range =
key
points
Ex, Graph each function, give the do
5.3 Evaluating Trigonometric Functions:
* Cofunctions : Cofunction Identities:
In the adjacent figure write the missing angle in terms of A
its measure is:
A
So
these are called cofunctions
in the same way we can prove that
also
EX: Write each function in
5.2 Trigonometric Functions:
In aright angled triangle far an acute angle
(theta)
Hypotenuse
opposite
to
adjacent
to
Ex: Find the values of the six trigonometric functions of angle
passes through the point
1.
solution:
2.
\
solution;
Math002 Page 1
whose
6.2 The Unit Circle and Circular Functions
Ex: Find the exact value of
1.
2.
3.
Note:
4.
5.
6.
7.
Note:
Conterminal =
reference =
8.
Note:
conterminal =
reference =
9.
Math002 Page 1
Ex. If
is the reference angle of
is the reference angle of
find
solution
5.4: Solving right triangle
Definition:
Solving aright triangle means finding all unknown angles and sides in the tri angle .
Ex: Solve right triangle ABC; if
and
where C is the right angle
Solution:
we can compare the triangle with
the triangle with 30,
Chapter 6: the Circular Functions.
6.1: Radian Measure
Radian:
An angle with its vertex at the center of a circle that intercepts an arc on the
circle equal in length to the radius of the circle has a measure of. l radian
So 1 radian means the are length
6.3 Graphs of Sine and Cosine Functions
Periodic Function:
is a function
such that
for every real number x in the domain of
, every integer n, and some positive real number p
The least possible positive value of p is the period of the function
i.e: the fu
6.4 Translations of the graph of sine and Cosine functions
Review:
Ex: Graph
over one period
solution:
Ex: Graph
over one period
solution:
Math002 Page 1
over two periods
Ex: Graph
solution:
over two periods
Ex: Graph
!
Ex: Graph
over two periods
solution
Chapter 7: Trigonometric identity and equations
71  Fundamental Identities:
Reciprocal Identities
_Quotient Identities
Pythagorean identities
Negative angle identities
Note:
and
Ex: If
and x is in 4th quadrant find
solution:
Let
Chapter 7 Trigonometric
7.2 Verifying trigonometric identities
Ex: verify the following
1.
LHS =
LHS =
RHS=
LHS.
LHS=
RHS
Ex: Perform the indicated operation and simplify
Chapter 7 Trigonometric identity and equations (2) Page 1
Ex: Factor each of the following
Ex: Each expressi
Department of Mathematics & Statistics, KFUPM
Math 202 Syllabus (143)
Coordinator Dr. Bader Al Humaidi
Course Title:
Elements of Differential Equations
Textbook:
A First Course in Differential Equations by D.G. Zill, 10 th Ed.
Course Description:
First or
King Fahd University of Petroleum and Minerals
Department of Mathematics & Statistics
Math 202 Syllabus
20152016 (151)
Coordinator: Dr. Boubaker.S
Title:
Credit:
Textbook:
Description:
Elements of Differential Equations.
303
A First Course in Different
King Fahd University of Petroleum and Minerals
Department of Mathematics & Statistics
Math 202 Syllabus
20162017 (162)
Coordinator: Dr. Husain AlAttas
[email protected]
Title:
Elements of Differential Equations.
Credit:
303
Textbook:
A First Cours
King Fahd University of Petroleum and Minerals
Department of Mathematics & Statistics
Math 202 Syllabus
20152016 (152)
Coordinator: Dr. Husain AlAttas
[email protected]
: Elements of Differential Equations.
Title
Credit
: 303
Textbook
:
A First Co
King Fahd University of Petroleum and Minerals
Department of Mathematics & Statistics
Math 202 Syllabus
20162017 (161)
Coordinator: Dr. Khalid Alshammari
[email protected]
Title:
Elements of Differential Equations.
Credit:
303
Textbook:
Description