International Journal for Quality Research 7(1) 127140
ISSN 1800-6450
Ripon Kumar
Chakrabortty1
Tarun Kumar Biswas
Iraj Ahmed
Article info:
Received 15 November 2012
Accepted 26 February 2013
UDC 65.018
REDUCING PROCESS VARIABILITY BY
USING DMAIC MODEL: A

Quiz
Find the critical points of the following functions
1.
2.
Lagrange
Multipliers
Locating Extreme Values
Find the extreme values of
constraint
subject to the
Find the maximum value of c such that
intersects
This happens when
have a common t

Extrema of
Functions
of Two Variables
Absolute Extrema
A function f of two variables has an absolute
minimum in a region D, if there is a point
such that
for all
in D.
in D
A function f of two variables has an absolute
maximum in a region D, if there

Tangent Planes
and Normal Lines
Example
Find the symmetric equations of the line tangent to the
curve of intersection of the surfaces
at the point
We want the line tangent to the intersection of the
two surfaces, then we need the line perpendicular

Differential Calculus
of Functions of more
than one variable
Chapter 2!
Recall
A function (of one variable) is a set of ordered pairs
such that no two distinct ordered pairs have the same rst
component. !
For each x, there corresponds a unique y. !
!
Func

Directional
Derivatives
Recall
The partial derivatives of a function f of two variables
are given by
These represent the rates of change of the function in
the positive x- and y- directions.
Directional Derivative
The rate of change of f in the dire

Quiz
Consider the function
.
1. Give the gradient of the function.
2. Find the directional derivative of f at the point
in the direction of
.
Tangent Planes
and Normal Lines
Directional Derivative
The maximum value of the directional derivative

Higher order partial
derivatives !
and Chain rule
Higher Order Partial
Derivatives
Since the partial derivatives of a function are also
functions, we can also get their partial derivatives,
called the second order partial derivatives. !
Theorem
Suppose f

Applications of
Partial Derivatives
Chapter 3
Recall
Let
The increment of f is dened as
The function f is differentiable at x if the increment
can be expressed as
where
as
Increment
Let
The increment of f is dened as
Example
Determine the

More Examples
Examples
Determine if the following are convergent or
divergent.
divergent, use test for divergence
convergent, geometric series with
Examples
one series is geometric with
the other series is geometric with
divergent, sum of a co

limits !
and !
Continuity
Theorem
If
exists, then
same limit no matter how
Suppose
If
and
approaches the !
approaches !
are curves passing through!
along
along
, and !
, then !
does not exist.!
Example
Consider two curves passing through the origin. !
Alo

Quiz
Use the denition of continuity of a function to check if the
following function is continuous at the origin. !
Partial Derivatives
Recall
If f is a function of one variable, then the derivative of f
with respect to x is given by!
Denitions
If f is a

Quiz
1. Give the denition of the Taylor series expansion of a
function f centered at a. !
2. Determine the Maclaurin series expansion of !
limits "
and "
Continuity
Recall
Let be a function dened on some open interval
containing , except possibly at a. !

INFINITE SERIES OF
CONSTANT TERMS
ILLUSTRATION
Consider the sequence
.!
Its elements are !
Consider the sums of its elements!
rst partial sum: 1!
second partial sum: 3 = 1+2!
third partial sum: 6 = 1 + 2 + 3!
We may consider the partial sums as a sequence

Multiple
Integration
Chapter 4
Multiple Integration
iterated integrals
Example
Double
Integrals
Volume of a Solid
The volume of the ij-th rectangular box is
The volume of the solid is then
Double Integral
If
for all
then the volume of the solid below the

Double
Integrals
Example
To evaluate this, we need to reverse the order of
integration first.
where
Example
Reverse the order of integration.
where
Double Integrals in
Polar Coordinates
Polar Coordinates
A point
in Cartesian coordinates can be
expressed i

Quiz
1. Express the double integral
iterated integral where
as an
D is the triangular region with vertices (0,0), (1,2),
(0,3)
2. Set-up the iterated integral of the volume of the solid
below the surface
above the region bounded
by
.
Sketch the two region

AN INTERACTIVE PROGRAM
TO BALANCE ASSEMBLY LINES
John J. Bartholdi, III
1992; April 3, 2003
Abstract
We describe the development of a program to balance 1- or 2-sided
assembly lines for a manufacturer of utility vehicles. The program is
highly interactive

Weights and Measures Program
Requirements
A Handbook for the
Weights and Measures Administrator
Authors:
Carol Hockert, Chief
Henry V. Oppermann
National Institute of Standards and Technology
Weights and Measures Division
Gaithersburg, MD 20899-2600
U. S.

International Conference on Product Lifecycle Management
1
Line Balancing in the Real World
Emanuel Falkenauer
Optimal Design
Av. Jeanne 19A bote 2, B-1050 Brussels, Belgium
+32 (0)2 646 10 74
E.Falkenauer@optimaldesign.com
Abstract: Line Balancing (LB) i

ProBalance Automotive Tutorial
ProBalance 1.6
Mixed Model Automotive Line Balance Tutorial
Copyright 2006, Proplanner
1/19
ProBalance Automotive Tutorial
2/19
Tutorial Objective:
This tutorial will show you how to create a Mixed Model Two-sided balance fo

CASE STUDY
THE KEARNEY COMPANIES, INC.
GULF COAST LOGISTICS COMPANY
SHIFTS INTO HIGH GEAR WITH
DIGITAL COMMUNICATIONS
MOTOTRBO DRIVES SAFETY AND EFFICIENCY FROM DOCK TO DOOR
The Kearney Companies, Inc. (KCO) is one of the largest third party logistics (3P

Principles of Checkweighing
A Guide to the Application and Selection of
Checkweighers
Third Edition
Copyright 1997
by Hi-Speed Checkweigher Company, Inc.
HI-SPEED
A Mettler Toledo Company
We Wrote The Book on Checkweighing
1
Introduction
Welcome to the Pr

Global Perspectives on Engineering Management
May 2013, Vol. 2 Iss. 2, PP. 70-81
Selection of Balancing Method for Manual
Assembly Line of Two Stages Gearbox
Riyadh Mohammed Ali Hamza*, Jassim Yousif Al-Manaa
Mechanical Engineering Department, Gulf Univer

A Practical Approach to solving
Multi-objective Line Balancing
Problem
Dave Sly PE, PhD
Prem Gopinath MS
Proplanner
Agenda
Introduction & Examples
Objectives & constraints
Academic versus Industry focus
Solution approaches
Mixed Model Balancing
Operator C

University of the Philippines Rural High School
College of Arts and Sciences
College, Laguna 4031
Tel. /Fax No. 049-573-0093/049-501-0389
OFFICE OF ADMISSIONS AND REGISTRATION
GENERAL INFORMATION ON VALIDATION EXAMINATION FOR TRANSFER APPLICANTS
for the A

University of the Philippines Rural High School
College of Arts and Sciences
College, Laguna 4031
Tel. /Fax No. 049-573-0093/049-501-0389
OFFICE OF ADMISSIONS AND REGISTRATION
GENERAL INFORMATION ON VALIDATION EXAMINATION FOR TRANSFER APPLICANTS
(GRADE 8)

Manual Work Design
BULAON, Micaela
CALUBAYAN, Faye
RONQUILLO, Alyssa
SANTOS, Marc Neil
CAYAON, Harmon Ric
DELA CRUZ, Jayvee
GARACHICO, Nhicko Jerome
TOLENTINO, Andrea Nicole
Abstract
Manual work design was introduced by the
GIlbreths and further developed

Triple Integrals
Triple Integrals
Other Solids
Other Solids
Example
Rewrite the integral using the order
Bounds:
Example
Set-up the integral that will give the volume of the
solid bounded by
and
Surface to the right
Surface to the left
Projection on the x