Two-Fo
orce
1. Member BD exerts a f
D
force (with a 960-N ver
rtical component and u
unknown ho
orizontal
component on memb ABC, de
t)
ber
etermine (a the magn
a)
nitude of th force exe
he
erted by
member BD and its ho
D
orizontal com
mponent , (b force Q

POWER SERIES
POWER SERIES
A power series is a series of the form !
where
.!
The series may converge for some values of x, and may
diverge for other values of x.!
POWER SERIES
The sum of the series is a function !
whose domain is the set of all x for which

TAYLOR
AND !
MACLAURIN SERIES
TAYLOR SERIES
Suppose a function f can be expressed as a power series.!
Note that
.!
If we dierentiate f, we get !
Again, note that
.!
Dierentiating again, we get !
Again, note that
.!
Repeating the process, we get !
TAYLOR S

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

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. !

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

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

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

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

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

DIFFERENTIATION AND
INTEGRATION !
OF POWER SERIES
THEOREM
If the power series
convergence R, !
then
has radius of
is dierentiable when!
and!
EXAMPLE
Find the power series representation of !
Recall:!
Dierentiating both sides, we get!
The radius of converg

QUIZ
Determine whether the following series is convergent or
divergent. !
1. !
2.!
ABSOLUTE CONVERGENCE
ABSOLUTE CONVERGENCE
The series
is absolutely convergent if !
is convergent. !
The series
is conditionally convergent !
if it is convergent, but not ab

Department of Engineering Science
College of Engineering and Agro-Industrial Technology
University of Philippines Los Baos
ES
Laboratory Examination (SAMPLE)
Engineering Science 26. Computer Application in Engineering
Name: _ Lab Section: _-_ Batch: _
Stu

SHEAR AND BENDING MOMENT DIAGRAMS
Exercise Problems
Determine the maximum Shear and Moment of the
following beams. Use both Method of Sections and Area
Method to counter check your answers.
1.
2.
3.
4.
5.
6.
7.
P = 5N
8.
9.
10.
Answers for Nos. 1 to 5
Ite

Engineering Science 11
nd Exam Review
2
ENGR. PAOLO ROMMEL P. SANCHEZ
Asst. Prof. 1, Engineering Science Department
CEAT, UPLB
Problem 1
Given the following simple truss supported by a pin at A and a roller at G,
determine the forces in members IJ and IE

MATH 38
Section F
Mathematical Analysis III
Lecture: WF 3 4 PM
2nd Semester 2012-2013
Recitation: T 3 4 PM
COURSE DESCRIPTION: Infinite series, techniques and applications of partial
differentiation and multiple integration.
COURSE OBJECTIVES: Upon comple

Exercise on Triple Integrals
20 March 2013
Write your answers to the following on clean sheets of bond paper.
1. Evaluate the following iterated integrals.
a.
b.
1
z
x+z
0
0
0
6xz dy dx dz
/2
2
9r 2
0
0
0
r dz dr d
2. Give an iterated integral that wil

ALTERNATING SERIES
ALTERNATING SERIES
If
for all natural number n, !
then alternating series are of the form!
The terms of an alternating series alternate in signs. !
ALTERNATING SERIES TEST
If an alternating series satises the following:!
i) !
ii) !
then

INFINITE SERIES
Unit I!
RECALL
Sequence!
- list of numbers in a denite order!
- function whose domain is the set of natural numbers!
A sequence
has the limit L, written as !
if for every
there exists
if
, then !
such that !
If the limit of a sequence exis

INFINITE SERIES OF
POSITIVE TERMS
COMPARISON TEST
Let
If
then
and
be series of positive terms. !
is convergent and
is also convergent. !
for all n, !
COMPARISON TEST
If
is divergent and
then
is also divergent. !
for all n, !
EXAMPLE
Let us compare this

QUIZ
1. Consider
and
.!
How are these two dierent and when are each of them
convergent?!
2. Determine if
is convergent or divergent. !
Write a! necessary solutions and explanations."
INFINITE SERIES OF
CONSTANT TERMS
THEOREM
If
is convergent, then !
Proof

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

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

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

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

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

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

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

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
[email protected]
Abstract: Line Balancing (LB) i

ES 13: Mechanics of Deformable Bodies I
Lecture No. 19
Beam Deflections using
Area-Moment Method (AMM)
Euler-Bernoulli Equation
Moment-Area Theorems
Area-Moment Method (AMM)
Moment Diagram by-Parts
Maximum Beam Deflection
Jessica M. Junio
Institute of Civ

ES 13: Mechanics of Deformable Bodies I
Lecture No. 18
Beam Deflections using Double
Integration Method (DIM)
Governing Equation for Euler
Bernoulli
Macaulay Functions
Double Integration Method (DIM)
Jessica M. Junio
Institute of Civil Engineering
Univ

Analytic
Geometry
MATH 36 A
MATHEMATICAL ANALYSIS I
Lecture 15
18 March 2015
Limits &
Continuity
Derivatives
Applications of
Derivatives
Integration
1
CHAPTER 3
THE DERIVATIVE AND DIFFERENTIATION
Outline
1. Definition of the Derivative and its
Geometric R

Unit 3
Applications of the
Definite Integral
Unit 3.1
Volume of Solids of
Revolution
Unit 3.1.1
Volume of Solids of
Revolution
(Using Disks)
Solid of Revolution
A solid of revolution is a solid obtained by
revolving a region in a plane about a line in
the

Print this file using A4 and write your solutions on it.
EXERCISE 10
Center of Mass
Week 13 MATH 37 X
Due on November 3, 2016 in the lecture class
NAME:_ RECIT SECTION:_ SCORE:_
SET-UP only. All or nothing per set-up.
1. A 3-ft long rod has a density that