11
Sequences and Series
Consider the following sum:
1 1 1
1
1
+ + +
+ i +
2 4 8 16
2
The dots at the end indicate that the sum goes on forever. Does this make sense? Can
we assign a numerical value to an infinite sum? While at first it may seem difficult
Blowout:TheDeepwaterHorizonDisaster
A Survivor Recalls His Harrowing Escape but who is ultimately to blame?
May 10, 2010 CBS News
The Deepwater Horizon drilling rig was owned and operated by the company Transocean and
was leased to client oil company BP.
Criteria For
Source Evaluation
Assessing the quality and usefulness of an article
To help you make a decision about the quality
and usefulness of any article or book, you
need to ask questions about seven factors:
1. Author
2. Content
3. Audience
4. Reada
INTRODUCTION TO
ENGINEERING DESIGN
ENGR110
The Design Process
2
Design versus other aspects of engineering
Overview of the process
Steps in the design process
ENGR 110 - Introduction to Engineering
Engineering Functions
3
Design
Manufacture - assemble com
Paraphrasing
Agenda
1. Definition
2. Purpose
3. Parts of a paraphrase
4. Recognizing a correct paraphrase
5. Creating a correct paraphrase
Definition
A paraphrase is a point by point explanation of
another persons ideas written in your own
words.
(Spatt,
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ENGR 110 Syllabus Assignment
A course syllabus is a guide for student success. It is a document that tells you where you will
end up when the semester is over, and how you will get there. Most instructors go over the
course s
Literature Review Preparation
Task Specification
This is an individual deliverable written by
each team member.
Helps the team write its background/literature
review in the proposal and the final research
report.
Each member selects two texts based on
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Assignment: Individual Literature Review Preparation Draft 1
Introduction:
Write a paragraph explaining the significance of your chosen topic in general and your particular
interest in this topic. State the main objective(s) of yo
Name:
ID:
Section:
Date:
Assignment: Individual Literature Review Preparation Draft 1
Introduction:
Write a paragraph explaining the significance of your chosen topic in general and your particular
interest in this topic. State the main objective(s) of yo
it
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Differential Equations Workshee 31 (Section 8.6)
Method of Frobenlus
Consider the equation of the form
(1) y"(I)+-P(leTI)t-q(I)y(X)=0-
Let’s assume xp(.r) and x2q(.r) are analytic functions. That is in some open interval about
x
,br
Power series expansions about x0 : 0 are easier to manipulate than expansions about nonzero
points. As the next example shows, a simple shift in variable enables us always to expand
about the origin.
leferentlal Equatlons Worksheet 30 (Soctlon 8.4
Equ
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_ . . .7 7 _ r. a . . Differential Equations Worksheet 29 (Section 8.3)
Power series solution to linear differential equation -
In this section, we study a method for obtaining a power series solution to a linear diff
THE PETROLEUM INSTITUTE
MATH 261, Differential Equations, Spring 2015
Course Syllabus
Prerequisites: Students should have exited the MATH 212 course with a grade D or higher
Instructor: Dr. Alip Mohammed
Office: 8-292
E-mail: [email protected]
Phone: ext.75
leferentlal Equations Worksheet 28 (Section 8.2)
Power serles and Analytic functlons
Power Series:
A power series about the point x0 is an expression of the form
(1) Zane—x0)" =a0 +a](x—xo)+a2(x—x0)2 + .
11:0
where x is a variable and the an ’s are const
\s- ‘ ‘ _
(“0. 0:. Ta. 43“ a 511? a»: anon S
leferentlal Equatlons Worksheet 24 (Sectlon 7.6)
Transforms of Dlseontlnuous functlons
One of the advantages of Laplace transform is that it can handle ﬁmctions that have jump
discontinuity such as step functio
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“ CONVOLU'V\ON
thhed 25 (section 7.7)
Convolutlon
Let’s consider the initial value problem
y"+y = gm, y(0) = y'(0) = 0-
Take Laplace transform by sides, we have
s2Y(s) + Y (s) = 6(3) , where Y(s) and 0(3) are Laplace transform of y(t) and g(t).
And he
Differential Equations Worksheet 27 (Section 8.1)
Introduction: Taylor polynomlal approximatlon
One of the best tools to approximate a ﬁmction f (x) near a particular point x0 is the Taylor
polynomial. The formula for the Taylor polynomial of degree n cen
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orentlal Equations Worksheet 26 (Sectlon 7.9)
Solvlng Ilnear systems with Laplace transform
We can use Laplace transform to reduce certain systems of linear differential equations with
initial conditions to a system of li
Differentlal Equatlons Worksheet 22 (Sectlon
Inverse Laplace Transform
In Section 7.2, we turned a function f (r) into F (s). In this section, we want to do the other
way round, that is, turn a function F (s) into f (I) using the inverse mapping or invers
Solving lnltial Value Problems
We now use Laplace and Inverse Laplace transforms to solve initial value proble
D.E. This can be done following two steps:
1) Take Laplace transform of both sides, to obtain J {y}
2) Take the inverse Laplace transform to obt
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Differential Equations Worksheet 18 (Section 4.
Variable-coefficient equations . ‘
In this section, we consider the non—homogeneous linear second-order equation of the form
(I) (12(t)y"+a](r)y'+an(t)y = g(r)
The following result
Q n O
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leferentlal Equatlons Worksheet 19 (Sectlon 4. '
A closer look at free mechanical vlhratlons
CASE A No friction:I no damping, free motion — no external force
rash-MAX
Recall the mass-spring system is governed by the eq
Deflnltlon of Laplace Transform
Laplace Transform
Let f (t) be a function on [0,00). The Laplace transform of f is the function F deﬁned by
the integral
(1) F(s) = j e*5‘f(t)d;.
0
The domain of F is all values of s for which the integral in (1) exists. Th
Differentlal Equations Worksheet 21 (Sectlon 7.3)
Properties of the Laplace Transform
In the previous section, we deﬁned the Laplace transform of a function f (t) as
:0
. 't
F(s)=£e"‘f(t)dt. ‘\ [and‘h‘m h S.
In this section, we discuss some properties of
Dlﬁerentlal Equatlons Worksheet 8 (Sectlon 2.6)
Solvlng by Substltutlons
Substitutions can be used to transform a DB into the forms we have learnt. This section
introduces three types of substitutions.
Type 1: Homogeneous Equations (substitute v =
If th
Differentlal Equatlons Worksheet 3 (Sectlon 1.4)
The Approxlmatlon Method of Euler (Euler’s Method or The Tangent Llne Method)
The Euler’s method is a procedure for constructing approximate solutions to an initial value
problem for a ﬁrst order D.E.
Quest
Dlﬁerentlal Equatlons Worksheet 7 (Sectlon 2.5)
Speclal Integratlng Factor
In the previous lesson, if the equation M (x, y)dx+ N (x, y)dy = 0 satisﬁes the compatibility
. . 6M 6N . . .
condition ——- (x, y) = — (x, y), then the equatlon IS exact and we kno