complex
variables
type
A
pages:37
by electrical
sele
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CHAPTER 1
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
Calculate limit of the function based on definition
Identify the differences between limits, one-sided limits and
infinite limits
Calculate limits using rules and theore
MARCH 2017
WS CH 2 Derivatives of Trigonometric Functions
1
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
Obtain formulas for the derivatives of the six basic
trigonometric functions
MARCH 2017
WS CH 2 Derivatives of Trigonomet
DERIVATIVES OF OTHER FUNCTIONS
MARCH 2017
WS CH 2 Derivatives of Logaritmic & Exponential Functions
1
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
Differentiate exponential functions
Differentiate logarithmic functions
Diffe
TECHNIQUES OF
INTEGRATION
INTEGRATING RATIONAL
FUNCTIONS BY PARTIAL
FRACTIONS
MARCH 2017
WS CH 3 Integration by Partial Fractions
1
In algebra, we combine 2 or more fractions into
single fraction by finding the common denominator
For example,
(1)
LHS
RHS
APPLICATION OF
DERIVATIVES
RELATED RATES
MARCH 2017
WS CH 2 Related Rates
1
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
To solve everyday problems using derivatives as
rates of change
MARCH 2017
WS CH 2 Related Rates
2
Exampl
APPLICATION OF
DERIVATIVES
MAXIMUM AND MINIMUM
VALUES
MARCH 2017
WS CH 2 Maximum Minimum Values & Optimization Problems
1
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
To apply differentiation in solving practical
problems usin
APPLICATIONS OF
INTEGRATION
MARCH 2017
WS CH 3 Volumes
1
VOLUMES
SOLIDS OF REVOLUTION : THE DISK METHOD
Consider the area bounded by a curve with
equation y=f(x) and the x-axis between x=a and
x=b, as shown below
y
x
MARCH 2017
WS CH 3 Volumes
2
When the
INTEGRATION
TECHNIQUES OF
INTEGRATION
MARCH 2017
WS CH 3 Integration by Parts
1
INTEGRATION BY PARTS
If we have
In this case,
, so the product
rule for differentiating
can be
expressed as
(1)
This means that
is an antiderivative
of the function on the rig
INTEGRATION
MARCH 2017
WS CH 3 Antiderivatives, Definite and Indefinite Integrals
1
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
Find a function F whose derivative is a known
function f
Integrate simple basic function
MARCH 2
INTEGRATION
TECHNIQUES OF
INTEGRATION
MARCH 2017
WS CH 3 Trigonometric Integrals
1
TRIGONOMETRIC INTEGRALS
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
Integrate trigonometry of higher orders with
odd and even powers
MARCH 201
DERIVATIVES OF OTHER FUNCTIONS
INVERSE TRIGONOMETRIC
FUNCTIONS
MARCH 2017
WS CH 2 Derivative of Inverse Trigo Functions
1
INVERSE TRIGONOMETRIC
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
Define the inverse function of sine,
Laplace Transformation
Laplace Transformation
Definition:
f (t ) Lcfw_ f (t ) = F ( s ) = f (t )e st dt
0
time domain
frequency domain
Usefulness:
differential
equations
Analogy:
log a
a b log a + log b
a
algebraic
equations
Circuit Analysis Using Lap
Arc Length
C2L2
2014 PETROLIAM NASIONAL BERHAD (PETRONAS)
Open
Presentation Title (acronym) ;
Division Name/OPU/HCU/BU (acronym) ;
Name of Presenter
All rights reserved. No part of this document may be
reproduced, stored in a retrieval system or transmi
INTEGRATION
INTEGRATION BY
SUBSTITUTION
MARCH 2017
WS CH 3 Integration by Substitution
1
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
Integrate by using substitution method
MARCH 2017
WS CH 3 Integration by Substitution
2
SUBS
CHAPTER 2
MARCH 2017
WS CH 2 Derivatives by Definition
1
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
Compute derivative by using definition
MARCH 2017
WS CH 2 Derivatives by Definition
2
The derivative of a function f with res
APPLICATIONS OF
INTEGRATION
MARCH 2017
WS CH 3 Areas Between Curves
1
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
Find areas of region that lie between two or
more functions
Find volume of solids
MARCH 2017
WS CH 3 Areas Bet
APPLICATION OF DERIVATIVES
INDETERMINATE FORM AND
LHOPITALS RULE
MARCH 2017
WS CH 2 Indeterminate Form & L'Hopital's Rule
1
INDETERMINATE FORM & LHOPITALS RULE
Consider
We know that is not defined when = , but we
need to know how does behaves near
In par
MARCH 2017
WS CH 2 Differentiation Techniques
1
1. Derivative of a Constant Function
Example :
MARCH 2017
WS CH 2 Differentiation Techniques
2
2. Power Rule
If n is any integer, then
Example :
MARCH 2017
WS CH 2 Differentiation Techniques
3
3. Constant Mu
THE CHAIN RULE
MARCH 2017
WS CH 2 The Chain Rule
1
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
Determine the derivatives of composition of functions,
using the known derivatives of simpler function, by using
the Chain Rule
MA
Portfolio Assignment Reflection
In my Portfolio, I have redesigned certain parts of my paper. I have added 2 more pictures which are
the ripple on water and the superposition principle explained. I added the ripple on water picture
because I wanted to hig
Flow Area of Pipe week 14
The code is as follows
Option Explicit
Public Function FlowAreaPipe(pdiam As Double, depth As Double)
Dim r As Double
r = pdiam / 2
If (depth <= 0) Then
FlowAreaPipe = 0
Exit Function
End If
If (depth >= pdiam) Then
FlowAreaPipe
PLANE,
NORMAL
LANE AND
ULATING P
OS C
G PLANE
RECTIFYIN
C2L5
NORMAL PLANE
At any point t, the plane through the curve R(t ) and normal to the tangent vector T (t ) is called the normal plane
to the curve at t. The unit normal vector N (t ) and the binorma
Cylindrical and Spherical
Coordinates
C1L6
1
Learning Outcomes
At the end of the lesson, the
students should be able to:
1. Convert the cylindrical coordinates
to rectangular coordinates or vice
versa.
2. Express the spherical coordinates to
rectangular c
Distance Involving Planes and Lines
C1L5
1
Learning Outcomes
At the end of the lesson you should be able
to:
1. Find the angle between two intersecting
planes.
2. Find the distance involving planes and lines.
2
Distance Involving Planes and Lines
1.
2.
3.
Distance Involving Planes and Lines
C1L5
1
Learning Outcomes
At the end of the lesson you should be able
to:
1. Find the distance involving planes and lines.
2
Distance Involving Planes and Lines
1.
2.
3.
Cases:
Distance between a point and a plane
Distan
Planes in Three Space
C1L4
1
Learning Outcomes
At the end of the lesson you should be able
to:
1. Define a plane.
2. Find the equation of the plane.
2
Planes
In mathematics, a plane is a flat surface.
Although a line in space can be determined by
a poin
APPLICATIONS OF
INTEGRATION
MARCH 2017
WS CH 3 Arc Length
1
ARC LENGTH
Consider a function = () such that () and
() are continuous on the interval [, ].
The length of the curve = () can be
approximated by connecting a finite number
of points on the curve
TECHNIQUES OF
INTEGRATION
INTEGRATION INVOLVING
INVERSE TRIGONOMETRIC
FUNCTIONS
MARCH 2017
WS CH 3 Integration involving Inverse Trigo Functions
1
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
Derive integrals of inverse trigon
INTEGRATION
MORE ON INTEGRATION BY
SUBSTITUTION
MARCH 2017
WS CH 3 Integration by Substitution
1
SUBSTITUTION IN
DEFINITE INTEGRALS
There are 2 ways of doing this
MARCH 2017
WS CH 3 Integration by Substitution
2
METHOD 1
First evaluate the indefinite inte
IMPLICIT DIFFERENTIATION
MARCH 2017
WS CH 2 Implicit Differentiation
1
LEARNING OBJECTIVES :
At the end of the module , STUDENTS should be able to :
compute the derivatives of implicit functions
MARCH 2017
WS CH 2 Implicit Differentiation
2
IMPLICIT FUNC
LOGARITHMIC DIFFERENTIATION
MARCH 2017
WS CH 2 Logarithmic Differentiation
1
LEARNING OBJECTIVES :
At the end of the module , you should be able to :
Decide when to use logarithmic differentiation
Compute derivative of certain functions using logarithmi