The document discusses techniques for integration by parts. It provides examples of applying integration by parts to solve 25 questions involving integrals of trigonometric, exponential, and polynomial functions. For each question, it shows the steps used to derive an anti-derivative or reduction formula through repeated applications of integration by parts.
This method is limited to the numbers that have 5 in its unit’s place. With the help of these methods you will be able to find the square of any number that ends in 5 within a fraction of seconds.
This method is limited to the numbers that have 5 in its unit’s place. With the help of these methods you will be able to find the square of any number that ends in 5 within a fraction of seconds.
2AnalysisTable of ContentsQ1 – Raw Data Worksheet3Q2 – .docxrhetttrevannion
2
Analysis
Table of Contents
Q1 – Raw Data Worksheet 3
Q2 – Simulation Worksheet 12
Question 1: 15
Question 2: 15
Questions 3-7: 17
Question 8: 17
Question 10: 18
Question 9: 19
Q3 – Simulation Worksheet 19
Question 1: 22
Question 2: 22
Questions 3-7: 22
Question 8: 23
Question 10: 23
Question 9: 24
Question 11: 24
Question 12: 25
Question 13: 25
Question 14: 28
Question 15: 28
Question 16: 28
Question 17: 28
Q4 – Simulation Worksheet 29
Question 1: 32
Question 2: 32
Questions 3-7: 32
Question 8: 33
Question 10: 34
Question 9: 34
Question 11: 34
Question 12: 35
Question 13: 36
Question 14: 38
Question 15: 38
Question 16: 38
Question 17: 38
Q5 – Simulation Worksheet 39
Question 1: 41
Question 2: 41
Questions 3-7: 41
Question 8: 42
Question 10: 42
Question 9: 43
Question 11: 43
Question 12: 44
Question 13: 45
Question 14: 47
Question 15: 47
Question 16: 47
Question 17: 47
Q1 – Raw Data Worksheet
Step 1: I first put in the following formulas to get the IAT sample statistics.
To begin I ran the FIT test with the IAT values ($A$2:$A$148) and the FIT options in the image below. Choosing the Discrete type because the values are all integers. Even if selecting the various Fitting to have selected either Allow Shifted Distributions, Run Sample Independence Test, or both in addition to AIC/BIC statistics or Chi-Square statistics the results are all almost identical.
From the FIT results we see that the best fitting distribution ranked according to the Chi-Square test is Geometric, then NegativeBinomial, then IntegerUniform, and then Poisson.
Discrete
If I were to have chosen to go with the Continuous type we would have received very similar results that can be seen in the chart below. You can view that in the shape. However, notice that the best fitting distribution is no longer Geometric but Logical.
Step 2: When closing the chart, we are prompted if we want to accept the fitted distribution.
After selected yes, then we are asked to select a cell to place the fitted distribution, which I then selected cell G17. We are then also auto populated the distribution chart in red along with probability parameters. You can easily see that there is a 90% chance likely hood a drive up inter-arrival time will fall somewhere between 0 and 23 minutes.
By looking at the formula bar you can now see the formula in the call as “=PsiGeometric(0.117788461538462)” along with the uncertain variable 25.
Now that we have the uncertain variable cell being fitted to the historical data this is now converted to a cell that we can run simulations and trials from. Using the following formulas as can be seen in the below image we tie each formula to G17.
Prior to running the simulation, the cells appear as errors. Once we run the simulation the results will appear.
If we repeat the same steps for the service time, we would get IntegerUniform as the best fitting distribution.
If we decided to go with continuous, we would receive the bel.
Question 1How many parameters does a default constructor haveAn.pdfarccreation001
Question 1
How many parameters does a default constructor have?
Ans) It never has any
Explanation:
The default constructor doesn’t have any parameters.Thats why it is also called as Zero-
Argumented constructor.
Even The default constructor doesn’t return anything even void also.
___________________________________________________________________
Question 2
Which of the following statements is true about the class shown below?
class Quiz1A
{
private int[] array = new int[10];
public int[] getArray()
{
return array;
}
}
Ans) It is immutable
Explanation:
Here the getArray() method is not doing any operations inside it.It just returning an array.So it is
also called as getter method.
______________________________________________________________________
Question 3
A method that is associated with an individual object is called __________.
Ans) an instance method
Explanation:
For every class we can create many objects.If we want to call a method on object, we have to call
a method by using the reference of it.That method is confined to that object only.
______________________________________________________________________
Question 4
Given the declaration Circle[] x = new Circle[10], which of the following statement is most
accurate?
Ans) x contains a reference to an array and each element in the array can hold a reference to a
Circle object.
Explanation:
This array will hold the references of Circle class objects.It will not store objects.
______________________________________________________________________
Question 5
What is the output of the program shown below?
public class Quiz1B
{
public static void main(String[] args)
{
Count count = new Count();
int times = 0;
for (int i = 0; i < 100; i++)
increment(count, times);
System.out.println(\"times = \" + times + \" count = \" + count.times);
}
public static void increment(Count count, int times)
{
count.times++;
times++;
}
}
class Count
{
int times;
public Count()
{
times = 1;
}
}
Ans) times = 100 count = 101
Explanation:
This for loop in the main method will be executed for 100 times.So the value of the variable of
main method times will increased by 1 for each iteration.As the initial value of times is 0 and
finally it will become 100.after 100 iterations
As the initial value of instance variable of Count class is 1.After 100 iterations its value will
become 101
______________________________________________________________________
Question 6
When calling a method with an object argument, ___________ is passed.
Ans) the reference of the object
Explanation:
We have to pass the reference of an object as argument while calling.not the copy or objects
contents or else.
______________________________________________________________________
Question 7
Which of the following is not a property of a constructor?
Ans) The name of a constructor can be chosen by the programmer
Explanation:
We can overload the constructor.
The constructor is called and executed when we are creating an object by using new.
New approach for wolfe’s modified simplex method to solve quadratic programmi...eSAT Journals
Abstract
In this paper, an alternative method for Wolfe’s modified simplex method is introduced. This method is easy to solve quadratic programming problem (QPP) concern with non-linear programming problem (NLPP). In linear programming models, the characteristic assumption is the linearity of the objective function and constraints. Although this assumption holds in numerous practical situations, yet we come across many situations where the objective function and some or all of the constraints are non-linear functions. The non-linearity of the functions makes the solution of the problem much more involved as compared to LPPs and there is no single algorithm like the simplex method, which can be employed to solve efficiently all NPPs.
Keywords: Quadratic programming problem, New approach, Modified simplex method, and Optimal solution.
Exercise
1
[5
points]:
Create the following classes shown in the UML diagram. Then, create PointTest.java class with main method to test all functionality of these classes.
Exercise
2
[10
points]:
The following figure shows a UML diagram in which the class Student is inherited from the class
Person
a. Implement a Person class. The person constructor takes two strings: a first name and a last name. The constructor initializes the email address to the first letter of the first name followed by first five letters of the last name followed by @tru.ca. If the last name has fewer than five letters, the e-mail address will be the first letter of the first name followed by the entire last name followed by a @tru.ca. Examples:
Name
Email Address
Jane Smith
[email protected]
Musfiq Rahman
[email protected]
John Morris
[email protected]
Mary Key
[email protected]
b. Override Object’s toString method for the Person class. The toString method should return the present state of the object.
c. Now, create a Student class that is a subclass of Person and implements Comparable interface.
d. The Student constructor will be called with two String parameters, the first name and last name of the student. When the student is constructed, the inherited fields lastName, firstName, and email will be properly initialized, the student’s gpa and number of credit will be set to 0. The variable lastIdAssigend will be properly incremented each time a Student object is constructed and the studentId will be set to the next available ID number as tracked by the class variable lastIdAssigend.
e. Override the object’s toString method for the Student class. The toString method should return the present state of the object.
Note that it should use the toString() method from its superclass.
f. The addCourse() method should update the credits completed, calculate, and update the gpa value.
Use the following values for grade:
Example GPA calculation:
GRADE CREDIT CALCULATION
(A) 4.0 x 4 = 16.00
(B) 3.0 x 4 = 12.00
(B) 3.0 x 4 = 12.00
(A) 4.0 x 1 = 4.00
(C) 2.0 x 3 = 6.00
GPA = 50.00 / 16 = 3.125; the getGPA() method should return this value.
g. Students are compared to each other by comparing GPAs. Override the compareTo() method for the student class. Note that to override the compareTo() method, the Student class must implement Comparable interface.
Now, test your code with the supplied client code (StudentClient.java). Note: You should not modify this client code. We will use the same client code to test your classes.
Exercise
3
[10
points]:
In this exercise, you need to implement a class that encapsulate a Grid. A grid is a useful concept in creating board-game applications. Later we will use this class to create a board game. A grid is a two-dimensional matrix (see example below) with the same number of rows and columns. You can create a grid o ...
2AnalysisTable of ContentsQ1 – Raw Data Worksheet3Q2 – .docxrhetttrevannion
2
Analysis
Table of Contents
Q1 – Raw Data Worksheet 3
Q2 – Simulation Worksheet 12
Question 1: 15
Question 2: 15
Questions 3-7: 17
Question 8: 17
Question 10: 18
Question 9: 19
Q3 – Simulation Worksheet 19
Question 1: 22
Question 2: 22
Questions 3-7: 22
Question 8: 23
Question 10: 23
Question 9: 24
Question 11: 24
Question 12: 25
Question 13: 25
Question 14: 28
Question 15: 28
Question 16: 28
Question 17: 28
Q4 – Simulation Worksheet 29
Question 1: 32
Question 2: 32
Questions 3-7: 32
Question 8: 33
Question 10: 34
Question 9: 34
Question 11: 34
Question 12: 35
Question 13: 36
Question 14: 38
Question 15: 38
Question 16: 38
Question 17: 38
Q5 – Simulation Worksheet 39
Question 1: 41
Question 2: 41
Questions 3-7: 41
Question 8: 42
Question 10: 42
Question 9: 43
Question 11: 43
Question 12: 44
Question 13: 45
Question 14: 47
Question 15: 47
Question 16: 47
Question 17: 47
Q1 – Raw Data Worksheet
Step 1: I first put in the following formulas to get the IAT sample statistics.
To begin I ran the FIT test with the IAT values ($A$2:$A$148) and the FIT options in the image below. Choosing the Discrete type because the values are all integers. Even if selecting the various Fitting to have selected either Allow Shifted Distributions, Run Sample Independence Test, or both in addition to AIC/BIC statistics or Chi-Square statistics the results are all almost identical.
From the FIT results we see that the best fitting distribution ranked according to the Chi-Square test is Geometric, then NegativeBinomial, then IntegerUniform, and then Poisson.
Discrete
If I were to have chosen to go with the Continuous type we would have received very similar results that can be seen in the chart below. You can view that in the shape. However, notice that the best fitting distribution is no longer Geometric but Logical.
Step 2: When closing the chart, we are prompted if we want to accept the fitted distribution.
After selected yes, then we are asked to select a cell to place the fitted distribution, which I then selected cell G17. We are then also auto populated the distribution chart in red along with probability parameters. You can easily see that there is a 90% chance likely hood a drive up inter-arrival time will fall somewhere between 0 and 23 minutes.
By looking at the formula bar you can now see the formula in the call as “=PsiGeometric(0.117788461538462)” along with the uncertain variable 25.
Now that we have the uncertain variable cell being fitted to the historical data this is now converted to a cell that we can run simulations and trials from. Using the following formulas as can be seen in the below image we tie each formula to G17.
Prior to running the simulation, the cells appear as errors. Once we run the simulation the results will appear.
If we repeat the same steps for the service time, we would get IntegerUniform as the best fitting distribution.
If we decided to go with continuous, we would receive the bel.
Question 1How many parameters does a default constructor haveAn.pdfarccreation001
Question 1
How many parameters does a default constructor have?
Ans) It never has any
Explanation:
The default constructor doesn’t have any parameters.Thats why it is also called as Zero-
Argumented constructor.
Even The default constructor doesn’t return anything even void also.
___________________________________________________________________
Question 2
Which of the following statements is true about the class shown below?
class Quiz1A
{
private int[] array = new int[10];
public int[] getArray()
{
return array;
}
}
Ans) It is immutable
Explanation:
Here the getArray() method is not doing any operations inside it.It just returning an array.So it is
also called as getter method.
______________________________________________________________________
Question 3
A method that is associated with an individual object is called __________.
Ans) an instance method
Explanation:
For every class we can create many objects.If we want to call a method on object, we have to call
a method by using the reference of it.That method is confined to that object only.
______________________________________________________________________
Question 4
Given the declaration Circle[] x = new Circle[10], which of the following statement is most
accurate?
Ans) x contains a reference to an array and each element in the array can hold a reference to a
Circle object.
Explanation:
This array will hold the references of Circle class objects.It will not store objects.
______________________________________________________________________
Question 5
What is the output of the program shown below?
public class Quiz1B
{
public static void main(String[] args)
{
Count count = new Count();
int times = 0;
for (int i = 0; i < 100; i++)
increment(count, times);
System.out.println(\"times = \" + times + \" count = \" + count.times);
}
public static void increment(Count count, int times)
{
count.times++;
times++;
}
}
class Count
{
int times;
public Count()
{
times = 1;
}
}
Ans) times = 100 count = 101
Explanation:
This for loop in the main method will be executed for 100 times.So the value of the variable of
main method times will increased by 1 for each iteration.As the initial value of times is 0 and
finally it will become 100.after 100 iterations
As the initial value of instance variable of Count class is 1.After 100 iterations its value will
become 101
______________________________________________________________________
Question 6
When calling a method with an object argument, ___________ is passed.
Ans) the reference of the object
Explanation:
We have to pass the reference of an object as argument while calling.not the copy or objects
contents or else.
______________________________________________________________________
Question 7
Which of the following is not a property of a constructor?
Ans) The name of a constructor can be chosen by the programmer
Explanation:
We can overload the constructor.
The constructor is called and executed when we are creating an object by using new.
New approach for wolfe’s modified simplex method to solve quadratic programmi...eSAT Journals
Abstract
In this paper, an alternative method for Wolfe’s modified simplex method is introduced. This method is easy to solve quadratic programming problem (QPP) concern with non-linear programming problem (NLPP). In linear programming models, the characteristic assumption is the linearity of the objective function and constraints. Although this assumption holds in numerous practical situations, yet we come across many situations where the objective function and some or all of the constraints are non-linear functions. The non-linearity of the functions makes the solution of the problem much more involved as compared to LPPs and there is no single algorithm like the simplex method, which can be employed to solve efficiently all NPPs.
Keywords: Quadratic programming problem, New approach, Modified simplex method, and Optimal solution.
Exercise
1
[5
points]:
Create the following classes shown in the UML diagram. Then, create PointTest.java class with main method to test all functionality of these classes.
Exercise
2
[10
points]:
The following figure shows a UML diagram in which the class Student is inherited from the class
Person
a. Implement a Person class. The person constructor takes two strings: a first name and a last name. The constructor initializes the email address to the first letter of the first name followed by first five letters of the last name followed by @tru.ca. If the last name has fewer than five letters, the e-mail address will be the first letter of the first name followed by the entire last name followed by a @tru.ca. Examples:
Name
Email Address
Jane Smith
[email protected]
Musfiq Rahman
[email protected]
John Morris
[email protected]
Mary Key
[email protected]
b. Override Object’s toString method for the Person class. The toString method should return the present state of the object.
c. Now, create a Student class that is a subclass of Person and implements Comparable interface.
d. The Student constructor will be called with two String parameters, the first name and last name of the student. When the student is constructed, the inherited fields lastName, firstName, and email will be properly initialized, the student’s gpa and number of credit will be set to 0. The variable lastIdAssigend will be properly incremented each time a Student object is constructed and the studentId will be set to the next available ID number as tracked by the class variable lastIdAssigend.
e. Override the object’s toString method for the Student class. The toString method should return the present state of the object.
Note that it should use the toString() method from its superclass.
f. The addCourse() method should update the credits completed, calculate, and update the gpa value.
Use the following values for grade:
Example GPA calculation:
GRADE CREDIT CALCULATION
(A) 4.0 x 4 = 16.00
(B) 3.0 x 4 = 12.00
(B) 3.0 x 4 = 12.00
(A) 4.0 x 1 = 4.00
(C) 2.0 x 3 = 6.00
GPA = 50.00 / 16 = 3.125; the getGPA() method should return this value.
g. Students are compared to each other by comparing GPAs. Override the compareTo() method for the student class. Note that to override the compareTo() method, the Student class must implement Comparable interface.
Now, test your code with the supplied client code (StudentClient.java). Note: You should not modify this client code. We will use the same client code to test your classes.
Exercise
3
[10
points]:
In this exercise, you need to implement a class that encapsulate a Grid. A grid is a useful concept in creating board-game applications. Later we will use this class to create a board game. A grid is a two-dimensional matrix (see example below) with the same number of rows and columns. You can create a grid o ...
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It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
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The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
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Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
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Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
Palestine last event orientationfvgnh .pptxRaedMohamed3
An EFL lesson about the current events in Palestine. It is intended to be for intermediate students who wish to increase their listening skills through a short lesson in power point.
Embracing GenAI - A Strategic ImperativePeter Windle
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This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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1. mathcity.org http://www.mathcity.org
1 Prepared by: Aqeel Nawaz aqeel_755@yahoo.com 03349587500
Techniques of Integration
EXERCISE 4.3
Integration By Parts means 4
brackets in such a way that:
Or
Q No. 1
Applying By Parts
Q No. 2
Q No. 3
Q No. 4
In we use long division and get
Q No.5
Here we have,
2. mathcity.org http://www.mathcity.org
2 Prepared by: Aqeel Nawaz aqeel_755@yahoo.com 03349587500
Q No. 6
_______________________________________
Q No. 7
In trigonometry we write:
and
So,
------(1)
Hence,
Alternative method (after step 4):
Applying by parts on first
______________________________________________
Q No. 8
___________________________________________
Q No. 9
3. mathcity.org http://www.mathcity.org
3 Prepared by: Aqeel Nawaz aqeel_755@yahoo.com 03349587500
Alternative method of QNo. 9 is at the end of the exercise (page9)
Q No. 10
Applying by parts on first angle,
Q No. 11
Hence our integral becomes as follows,
We will apply By Parts technique upon 1st and 3rd
integral:
------- (1)
And
------- (2)
Putting values in we get:
__________________________________________
Q No. 12
Multiply Dr
and Nr
by
Diff. w.r.t
4. mathcity.org http://www.mathcity.org
4 Prepared by: Aqeel Nawaz aqeel_755@yahoo.com 03349587500
Applying integration by parts
Re-back substitution, using eqs. (1) & (2)
Q No. 13
Hence our integral become,
Re-back substitution, using eqs. (1) and (2)
_____________________________________________
Q No. 14
Using by parts formula,
Using by parts formula,
5. mathcity.org http://www.mathcity.org
5 Prepared by: Aqeel Nawaz aqeel_755@yahoo.com 03349587500
__________________________________________________
Q No. 15
Q No. 16
Integrating first integral by parts,
______________________________________________
Q No. 17
6. mathcity.org http://www.mathcity.org
6 Prepared by: Aqeel Nawaz aqeel_755@yahoo.com 03349587500
Integrating again by parts,
____________________________________________
Q No. 18
Integrating by parts
Integrating by parts again
Hence
Q No. 19
Integrating by parts,
_____________________________________________
Q No. 20
Hence our integral becomes:
Apllying by parts,
7. mathcity.org http://www.mathcity.org
7 Prepared by: Aqeel Nawaz aqeel_755@yahoo.com 03349587500
Q No. 21 Show that
Hence evaluate,
Let,
Integrating by parts,
As required.
Now, put n=3
By long division we get,
So becomes,
Q No. 22 Find a reduction formula for
and apply it to evaluate
.
Applying by parts formula,
Which is the required reduction formula.
Now, put
Again put
8. mathcity.org http://www.mathcity.org
8 Prepared by: Aqeel Nawaz aqeel_755@yahoo.com 03349587500
Q No. 23 Find a reduction formula for
and where n is a
positive integer.
We separate a single powerof sinx. As follows:
Applying by parts formula
And,
We separate a single powerof sinx. As follows:
Applying by parts formula
______________________________________________
Q No. 24 Find a reduction formula
for is an
integer. Hence evaluate .
Integrating by parts,
Again by parts,
9. mathcity.org http://www.mathcity.org
9 Prepared by: Aqeel Nawaz aqeel_755@yahoo.com 03349587500
Which is the required reduction formula,
Put n=4 & a=4
Now put n=2 and a=4,
_______________________________________
Q No. 25 Find a reduction formula for
And n is an integer greater than 1. Hence
evaluate,
Which is the required reduction formula.
Now put m=3 and n=2
Integrating by parts,
______________________________________________
Alternative method Q No. 9
Put
So
__________________________________________
Alternative method Q No. 11
Applying by parts on first integral
